Totally unsuitable wire used for SMPS transformer (skin effect)?

[KeepItSimpleStupid]

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.

That would probably keep "The Electrican" happy. It appears "The Electrician" isn;t your average sparky. He's reminding me of someone else on this boad.

But,, I do agree, leave the "sacrid terms" or "by definition"," alone,

But things such as "boiling point" depend on pressure, so why can't "effective resistivity" depend on frequency. The term "effective" makes it less sacrid.

 

[MrAl]

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.

For a better example, say we have a one foot copper wire cut down the middle, length wise, that forms two half circle cross sections. We throw one half section away. We do the same with a brass wire same length, same diameter, throw the second section away. We are left with one half copper and one half brass, and we force them together with insulating clamps so that they form a completely new wire, one half lengthwise is copper and one half brass. What is the DC resistivity? We cant say it is that of copper and we cant say it is that of brass, but it does in fact have some bulk resistivity and that number we can use in everyday design life in order to use that kind of wire in our designs, and the resistivity would be a single number just as with any other wire.

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".


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.

 

[Flyback]

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.

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.

It really makes you wonder why people go in for eg litz wire etc for smps's with switching frequency less than 100KHz. Surely all that's needed is two or three strands of enamelled copper wire.?...using 30, or even 60 strands of 0.1mm wire, as in the example in the top post, seems completely ridiculous.

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

 

[KeepItSimpleStupid]

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.

Your calculator specifies exactly that, the wire AWG where Rac=Rdc whch, I guess would be an ideal wire size unless your using silver plated copper tubes likes some equipment I worked on (13.56 MHz)

 

[MrAl]

Hi,

I am still looking at the accuracy issue here. I find numbers published by the (old) National Bureau of Standards that shows that the quick calculation g=1361/(6641*pi*sqrt(f)*R) might be closer to the real value. This looks strange though, but it is hard to question given the source. So the lower numbers seem to be more accurate overall.
I'd have to find out how they came up with their numbers to find out if it is really true that it can be that low, even given the fact that skin effect is defined by the reduction in current density to 1/e. Very strange.
Given the numbers posted by Electrician however it's too tempting to agree with the lower numbers.
But then we are left with the question why any design would show litz wire as an ingredient. Perhaps they need more overall conduction because of higher current. We cant use #26 wire for 10 amps, so we'd have to use litz wire with at least #26 wire, and if we use that, why not go down as far as #30 or better. Would 2 percent more resistance affect the design that much? That can only be answered after looking at the actual application and the goals of that design.

I am now looking at a simple curve fit for the NBS data to a witch to see if we can come up with a really simple formula which is accurate enough for any application. I think it will work out well.

 

[Flyback]

so "g" means RAC/RDC I presume?
I cant find that formula on any web page (g = 1361/(6641*sqrt(f)*R))
Isnt it amazing that this hasn't been well worked out and a common formula available everywhere.
The RAC/RDC figures from The Electrician just sound too low....nobody would ever use multi strand wire for sub 200khz smps's
-but The electrician backs them up with measurement......but still the rac/rdc just seems far too low....I mean, who would need litz for a 100khz smps with the electricians figures.?

 

[MrAl]

Hi again,

Well that is a valid question, and after a good night sleep (or sort of) i think i found the answer.
First, the NBS numbers are fairly low too, so we cant ignore that.
So the answer must be what i was saying before about the higher harmonics of the square wave. At 100kHz the effect might not be as much as i original found in a #26 wire, but at 300kHz there is much more effect, even using the NBS numbers. The effect would take the resistance up 10 to 12 percent higher at 300kHz (third harmonic). At 500kHz (fifth harmonic), about 27 percent higher.

So as before, the higher harmonics get hit harder. With a square wave it could be significant, and with a triangle wave a bit less because the amplitudes of the harmonics are lower to begin with.

Of course there is also the proximity effect which we havent calculated yet. When wires are bundled together the effect on the resistance is higher. Unfortunately i only have data for two wires at a time.

I believe that we can find a single witch that will satisfy the requirements over a significant range, but until then a combination of a single witch and linear curve works out pretty well, so here is a better formula...

First calculate x from:
x=2*pi*r*sqrt(2*f/172400)

where f is frequency in Hertz and r is wire radius in millimeters,

then for 0<=x<=4 use:
K=(3.2*x^4+576)/(x^4+576)

and for 4<x<=10 use:
K=(204*x+151)/576

and then we have:
Rac=Rdc*K

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.

If you dont care about the accuracy too much even this works for 0<=x<=4:
K=(17*x^4)/6400+1

or even the simple:
K=5.2e-4*(x+2)^4+1

 

[The Electrician]

Let me address the question Flyback asked in post #1. 30 x 0.1mm wire has a cross sectional area equivalent to a single strand of 23.3 gauge solid wire. Flyback is asking if the AC resistance of the 30 x 0.1 mm wire is enough better than a solid wire to justify its greater cost.

Plotting the Rac/Rdc of an isolated 23.3 gauge wire, we get this result:

At 85 kHz, Rac/Rdc for this wire is about 1.04. If that's all there were to it, the cost of the 30 x 0.1 mm wire would only be justified in a case where extremely high efficiency was required.

But, there's more to it. I don't have any 23.3 gauge wire, but I do have some 22 gauge wire. I used a 3 foot piece of 22 gauge wire in previous postings, and I'll show some more measurements with that same piece of wire. First, here's a sweep of the AC resistance of the isolated (not wound into a coil or transformer) piece of wire. The scale is different than it was in my previous plots, 100 mΩ at the top of the plot:

Now, I have wound the same piece of wire into a single layer solenoid on a 1/2 inch diameter plastic rod. This plot shows the AC resistance of the isolated wire and the single layer solenoid superimposed on the same plot:

Next, I wound the same piece of wire into a coil with 3 layers on a 1/4 inch plastic rod. This plot shows the AC resistance of all 3 geometries: an isolated wire, a single layer coil, and a 3 layer coil. The coils have much higher AC resistance because in addition to skin effect, proximity effect comes into play when the wire is wound into a coil which could be an inductor or one winding of a transformer. Notice that the scale is changed again; the top of the plot is now 200 mΩ. Also notice that the curves are not perfectly coincident at the left side of the plot. This is because I didn't insert the ends of the wire exactly the same distance into the fixture each time. Just a millimeter difference makes a detectable change in the measured resistance:

The Rac/Rdc of the 3 layer winding is about 5.5 at 100 kHz, but is only about 1.1 for an isolated wire. With multiple layers, the so-called "layer effect" comes into play which enhances proximity effect even more.

So the answer to Flyback's original question is that the reason for the expensive 30 x 0.1 mm wire is that the proximity effect causes the Rac/Rdc of the winding in a transformer to be much greater than if the wire were not wound.

 

[The Electrician]

Imagine a semi-infinite flat copper surface; for example, if half the universe were filled with copper and presenting a flat surface to the half that wasn't full of copper. When an electromagnetic wave would impinge on that flat surface, the current density in the copper would decrease with distance penetrated into the copper. The mathematical function describing the decay of the current density would be the exponential function. At some depth below the surface, the current density would have decayed to a value 1/e = .368 times the current density at the surface; this is called the skin depth. As you go deeper into the copper, the current density continues to decrease until it vanishes into the atomic noise level, because there is an infinite depth of copper for it to penetrate.

If the copper isn't infinitely deep, for example, a sheet of copper .001" thick, at low frequencies (high audio perhaps) the current density hasn't attenuated to near zero by the time we get to the other side of the copper sheet. A thin copper sheet can't provide much shielding against low frequencies.

The situation with a copper wire is not like the infinitely thick copper half universe, or a finite but very thick copper sheet.

At frequencies where there is noticeable attenuation (but not too much) of the surface current density, what happens is that the current density decreases as we move from the surface of the wire toward the center, and if we continue to proceed past the center and back toward the surface on the other side from our starting point, the current density begins to increase again until we reach the surface. The current density at the center of the wire doesn't reach the atomic noise level.

If the frequency is low enough (but we still see some attenuation), the current density at the center of the wire will not ever decrease to as little as 1/e=.368 times the surface value of the current density. In this case, what is the skin depth? When we were considering the very deep half universe of copper, the current density always reached 1/e times the surface density.

So it happens that the very concept of skin depth is defined in terms of an AC current decaying in a thick piece of copper with a large plane surface of very large extent (infinite in the ideal case). This is not the situation where a wire with circular cross section is concerned. The surface of the wire which the AC current is penetrating is not flat--not a plane. This leads to behavior which is very different from the case of penetration into a plane surface when the skin depth is not much different from the wire radius. In that case, the rate of decay of the current density is not well approximated by an exponential function; it happens that Bessel functions must be used.

But when the skin depth is very small compared to the wire radius, then the exponential approximation is good. See this explanation from Ramo, et al:

Figure 5.16 (b) shows the case where the exponential approximation fails. The solid line shows the actual current density with depth, and the dashed line shows the exponential approximation. When the attenuation is not very great, the difference in the two curves is fairly great. When the attenuation is great, the difference in the two curves is not very large.

What all this means, is that calculating Rac/Rdc when the skin depth is nearly the same as the wire radius by means of an exponential decay in the current density will have a large error. If the skin depth is small with respect to the radius, then the exponential functionality will give reasonably accurate results.

As it happens, finding analytical expressions for various geometries of conductors is extremely difficult, but besides the case of a flat plane surface the other simple geometry where an exact analytical expression is known, is the case of a wire of circular cross section. Ramo, et al. give the result, which involves Bessel functions:

The tables in the old NBS circular No. 74 were produced like this using Bessel functions; here's a plot rather than tables. This is the same graph found in Terman's "Radio Engineer's Handbook":

For cases where the analytical solution hasn't been found, numerical FEA analysis comes to the rescue. This is especially useful for transformers and inductors where proximity effect is dominant.

 

[The Electrician]

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.

As I posted above, there is an exact analytical formula. This formula uses σ for the conductivity of copper, which is taken to be a constant; it does not vary over the cross section of the wire. For the case of 26 gauge wire (radius taken to be .0202 cm) at 100 kHz, your K is given by:

We can also plot Rac/Rdc over a wide frequency range. Here is the plot showing the variation of Rac/Rdc from very low frequency to 3 MHz. The shape of the curve is unexpected. Rac/Rdc has a concave up shape at first, then becoming concave down at higher frequencies:

Lest anyone be skeptical that this mathematical result is not the truth, here is a sweep of a piece of 11 gauge solid copper wire:

 

[The Electrician]

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.

I'm well aware that you don't mind using conductance and conductivity interchangeably. In technical writing about electricity this is an error. They are two different things.

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.

Copper doesn't have a different resistivity "property" when it is exposed to AC until the frequency of the AC reaches the optical range. For the frequencies we're concerned with in this thread the resistivity and conductivity are constant. Even if it did change with frequency, at a given frequency such as 100 kHz as you've been considering, it wouldn't be different at different depths in the copper as your formula assumed it was.

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

Why would I hate it? I have nothing against material properties that vary with position or direction. But when it is incorrectly assumed that they do, I point out the error.

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".

Other people can misuse the word. His second use of the word is apparently in connection with the fact that at RF frequencies the "sheet conductance" is proportional to the square root of the FREQUENCY, not the DC value of something. If he can make that mistake, then he can also make a mistake in his usage of the word conductivity.

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

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.

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.

These things don't sound unfamiliar to me; they sound wrong and I don't see any need to open my mind to any more error than already gets in. I will always argue against using well defined technical terms to mean something completely different from their standard and well accepted meanings.

The well known solution to the skin effect behavior in wires of circular cross section takes resistivity (and conductivity) to be constant, not varying with frequency or with depth. The fact that your solution assuming variation in resistivity gets a wrong answer is evidence against its validity.

 

[The Electrician]

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

Of course, Rdc is always less than Rac. But for a low enough frequency, it's not much less.

Using the analytical formula, when the skin depth is equal to the radius, Rac/Rdc = 1.00519

If the skin depth is twice the radius, then Rac/Rdc = 1.00033

 

[MrAl]

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)

 

[Flyback]

Thanks Mr Al, excellent analysis, I have copied your above posts into my Stock course folder, reference bits and pieces that I use in projects etc

 

[The Electrician]

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!

 

[The Electrician]

Flyback, did post #28 explain to your satisfaction why litz wire is used in the subject transformer?

 

[Flyback]

Yes Thankyou The Electrician, that was very gratefully received and absorbed.....the proximity effect being depicted excellently there too.
I also found your graphing and measurement of great interest throughout.
Its interesting that Power.com (formally powerint.com) rarely use litz wire in their PI Expert software designs for the transformers......if I do it for a 60W offline flyback, there certainly is nowhere near 30x0.1mm litz wire getting used.

#28 did explain well, though I am expecting to see multi strand wires used for an offline flyback smps, just not as many as 30*0.1mm.....maybe say four to 8 strands of ECW.......30 strands sounds really way over the top..........it also sounds very expensive, and tends to need to be custom made, litz wire is rarely available off the shelf.

 

[The Electrician]

Flyback, here's some more material on the topic of layer effect.

https://www.scribd.com/doc/142353901/Skin-Proximity-Effect

**broken link removed**

Prof Sullivan at Dartmouth has many papers on skin and proximity effect.

and here's the figure from Dowell's paper showing how more layers dramatically increase Rac/Rdc:

 

[Flyback]

yes I see what you mean, on that subject , it seems to me that 30 strands of 0.1mm all couped up together is going to present a big layer effect problem.

 

[MrAl]

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!

Hello again Electrician,

Well thank you, and wow, you've been busy today
Thanks for doing those plots as they reveal a lot about the approximation. It was intended to be used for x=1 to x=3 so it seems ok for that i guess.

Also, thanks for posting your interesting data on the skin effect. My EE book doesnt cover it quite as good as i would have liked because it concentrates mainly on power line frequencies like 60 Hertz and wire diameters like 4 centimeters.