High Freq XFMR winding optimization for DC resistance ?

[CataM]

Yes, I figured out FvM was referring to a single wire afterwards.

I was initially posting about "transformer windings" where it is assumed windings are wound in layers and hence only 1D H field is possible, hence Dowell expression is somehow accurate.

The strange thing is Easy Peasy's claim of 15% rise of their Rac with respect to Rdc with the claimed wire size. Obviously he is hiding some details.

 

[The Electrician]

Here's the diagram from my book (K. Simonyi, Theoretische Elektrotechnik)
View attachment 146698

The x-axis unit r0 is r/δ, wire radius divided by skin depth. R/R0 is Rac/Rdc ratio. Inner inductance is also plotted.
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I get the same result:

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Yes, I figured out FvM was referring to a single wire afterwards.

I was initially posting about "transformer windings" where it is assumed windings are wound in layers and hence only 1D H field is possible, hence Dowell expression is somehow accurate.

The strange thing is Easy Peasy's claim of 15% rise of their Rac with respect to Rdc with the claimed wire size. Obviously he is hiding some details.

He says " the sec wires are paralleled at the transformer terminations to give up to 50A in some apps."

He must be using wire larger than .7mm for the secondary, or a lot of strands of .7mm bundled. Look at my measurements showing the rise in Rac/Rdc just by winding the .7mm wire into a single close wound layer. I'll do some more measurements in a few days.

 

[Easy peasy]

Also, just for completeness, if you have a resonant converter with basically sinusoidal current, then harmonics don't come into play, for square wave currents the harmonics add very little, as the first is I/3, 2nd is I/5 and so on, the I^2 R product of these and vector summation amounts to <5% extra losses ...

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per the above, #22, what are the y axes for the 2 other plots? all the information was in my posts, just have to read more carefully ...

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really only a good FEA look at the current distribution will tell you how a Tx is best wound for lowest skin depth losses...

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The skin depth at 70kHz is about 0.25mm (skin depth being that depth where the current density falls to 1/e, ~36.8%), so in an interleaved Tx, we are using at least .25mm on each side of the wire, leaving a "band" of 0.7-0.5 = 0.2mm that looks like it may not carry much current...?

But, in a single layer winding, where does the proximity effect try to push the current density? if you can map out the magnitude of each effect and super-pose them you will see why there is one ( and generally only one) optimal winding geometry that reduces AC frequency effect losses in Tx windings ( different for chokes ).

Also - now consider the square wave (current) converter, after the switching edge the current is near constant ( i.e. DC ) until the next switching edge. The current in the wire does not "know" when that next edge will be - so how are the losses due to AC effected? For a sine wave current the current is always changing at the cos(freq) rate - so the wire "knows" or "can see" the instantaneous rate of change of current and the losses due to self induced mag fields and current density displacement are right there, forcing the current to the outside and increasing wire resistance.

This is not the case for square wave current. Really what is transpiring is that the Rac is very high during the transient switching edge, when the di/dt is very high, and then this effect decays in the time afterwards, until and to when the current is static (no changing mag field, or very little), thus higher frequencies of square wave current give higher losses. To say that there is a fundamental sine wave + harmonics is not strictly correct, but it is the accepted engineering way of calculating losses (or trying to) in transformers (and other magnetics) to this day. As the results give answers that are close to real, it is accepted, and: it is very hard to measure these losses accurately - deepening the problem.

To illustrate; consider a Tx running at 100kHz, square wave current, with a large core (much larger than needed) at the end of a half cycle the fets are commanded to stay on for 50uS say, instead of switching after 5uS, as they normally would, how do you calculate the Rac in the 5uS after the last switch? do you calculate it at the 100kHz fundamental rate? or at the DC rate? or at a 10kHz rate (assuming we move to 10kHz switching, 50uS + 50uS) - you see my point?

Polite answers welcome ...

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See also:

https://docplayer.net/47452601-Qual...y-various-winding-configurations-part-ii.html

 

[FvM]

per the above, #22, what are the y axes for the 2 other plots?

See post #18, it's wire inner inductance and inductance increase due to skin effect, not relevant for the present discussion in the first order, although playing a role for current sharing of paralleled windings. It's simply printed in the book along with Rac.

I don't yet fully understand your point with square waves and harmonic currents. Presumed the current waveform is mainly determined by the external circuit, you calculate Rac for each harmonic component separately (due to orthogonality) and sum the losses.

Similarly, the possibly unequal current sharing in parallel windings can be calculated for each harmonic component separately.

Also, just for completeness, if you have a resonant converter with basically sinusoidal current, then harmonics don't come into play, for square wave currents the harmonics add very little, as the first is I/3, 2nd is I/5 and so on, the I^2 R product of these and vector summation amounts to <5% extra losses ...

Accounting for 3th, 5th and 7th harmonic with the discussed 70 kHz 0.7 mm wire scenario and the Rac for each frequency, I get 35% increased losses compared to DC, or 25% compared to AC losses with 70 kHz fundamental only (resonant converter case).

Additionally, there will be a surcharge for the proximity effect in the top and bottom winding layers, pulling the current towards the adjacent winding. Also current asymmetry at the winding borders.

Finally current sharing between paralleled windings must be analyzed in detail. I don't see an obvious reasoning why it should be exactly equal.

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Yes, I figured out FvM was referring to a single wire afterwards.

I was initially posting about "transformer windings" where it is assumed windings are wound in layers and hence only 1D H field is possible, hence Dowell expression is somehow accurate.

The assumption is that you get almost similar current distribution in a single wire and interleaved single layers because proximity effect cancels out. There are some deviations from the ideal picture, top and bottom layer, also borders. 15% versus 8 % of the no proximity case seems realistic.

 

[Easy peasy]

Dear FvM, as you say, you presume. Is your calc for the harmonics based on the Rac you get in an interleaved Tx ... ? I think not.

 

[FvM]

Do you see a possibility that Rac in the interleaved configuration can be smaller than the value achieved in a single wire (only proximity effect)? If not, the Rac curve in post #18, #20 and #22 must be assumed.

 

[Easy peasy]

It is a certainty the Rac is affected by the mag fields of conductors nearby, I have already alluded to same, all the answers are in the above for the careful reader...

 

[FvM]

I didn't yet hear a reasoning (in any post of this thread) how Rac can be decreased below the value determined by basic skin effect.

 

[CataM]

The assumption is that you get almost similar current distribution in a single wire and interleaved single layers because proximity effect cancels out. There are some deviations from the ideal picture, top and bottom layer, also borders. 15% versus 8 % of the no proximity case seems realistic.

That assumption is incorrect (almost is not correct, it is very far in fact for the example we are discussing) for the case of layers in transformers. Dowell expression must be used and is widely accepted in industry (also by the document you provided in one of your earlier posts).

Skin effect in single wire < skin effect in layer made of many turns

 

[FvM]

Dowell doesn't consider the case of interleaved single layers, as far I'm aware of. But whether Rac in this configuration can come near to single wire or not, you surely agree that it can't be smaller, which was the question of the recent debate.

 

[CataM]

But whether Rac in this configuration can come near to single wire or not, you surely agree that it can't be smaller, which was the question of the recent debate.

Yes, I do agree.

 

[Easy peasy]

I think it would be instructional to read the link I posted, Rac can be mitigated / reduced to a very great extent by careful choice of geometry.

 

[FvM]

Thanks for hinting to the Stan Zurek paper again. A brief reading shows that it's all about how to eliminate proximity effect as far as possible, leaving basic skin effect. See my considerations in post #24, #26 and #28.

I'll surely study the paper in detail because I didn't yet succeed in proving the effect in FEM analysis. The full text of part I and II can be read at researchgate.

 

[Easy peasy]

given that the current is not uniform in the skin - in the paper - then the skin effect is being altered - more careful "looking" required...

 

[The Electrician]

In part 2 of Stan Zurek's paper (available at: )

on page 4 in paragraph 4 he says: "The resulting DC resistance is an order of magnitude lower, and therefore the AC resistance and the high-frequency loss are also reduced by the same factor.", referring to Figure 15 on page 5.

It doesn't appear to me that the AC resistance and high frequency loss are reduced by the same factor (an order of magnitude).

Here is that Figure 15. I've place red arrows at the 1 kHz loss points on the green oval and blue oval curves. At 1 kHz they are indeed a factor of 10 different. But, at 1 MHz they appear to differ by a factor of about 1.6 (I'm estimating this), definitely not a factor of 10. Am I not seeing this correctly?

Dr. Zurek has another interesting paper here: https://hrcak.srce.hr/file/254588

 

[Easy peasy]

Note that wire 0.42mm coherent, has constant low losses to 300kHz ...

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I'm not sure that paper (Zurek) reveals anything useful for real world compact switch-mode Tx's in the 100 - 400kHz bracket...