true doughnut vs rectangular loop for signal level torroidal inductor?

[coppercone2]

I have never actually seen a true doughnut core inductor, the closest I have seen is a R-core transformer that had pretty rounded edges.
I understand that with a transformer, due to the non circular bend, you typically have fringing occur, so flux carrying capability of corner is not used well or something like that. But that is kind of like piecewise.

But if a torroid is uniformly wrapped, does it matter at all? Obviously a die is easier to  form, but do they actually have truly 'donut' shaped inductors?

http://www.solomoncorp.com/circular-vs-rectangular-windings-part-2/

This makes me think that magnetic forces may act as some kind of modulation of the ESR of the inductor depending on the winding geometry, and that there may be less microscopic corner forces.
"During a fault, an incredible amount of radial and axial forces are generated, which causes stress on the transformer coil. The current in the primary and secondary windings flow in opposite directions, this creates the repulsive forces between the two windings. The result is the coil tends to expand outwards and assume a circular shape under the influences of short-circuit stresses. As such, the coil that is originally circular in shape is fundamentally the most durable and least likely to distort during a fault. A rectangular coil construction will try to “round itself out” under these same repulsive forces between the two windings. This can cause the rectangular shaped coil to dramatically change shape, resulting in damage to the coil’s insulation.

The axial force in transformer coils occurs when the electrical centers of the primary and secondary windings are unbalanced. This unbalance causes a force to be exerted from primary to secondary windings. The force tends to make the primary and secondary windings slide past one another in what is called “telescoping” of the coil. This causes substantial damage to a transformer coil which typically results in transformer failure."


Since no transformer is perfectly balanced there is always some kind of strain going on I guess... all I can think of is if you consider the inductor a transmission line, the farther it is from round the more there is going to be formed a distributed Rhigh RLow network, where the resistance should be higher in the parts near the corner then the other parts.. wheras in a round core you get a much more uniform stress. This seems like a bunch of tiny insignificant impedance mismatch networks. But from a RF prospective you only want bends of infinite radius. Not sure what happens to electrically small bends. Is there even a single electron flowing in the transmission mode that can be effected? Maybe you can see it better on nano-coils.

This makes me think of something going on in a common mode choke. But how about single windings? What is the distortion PPM?? (since the mechanical will lag the electrical this is a result that results in distortion of small signals, I think).

Are there other factors at play? And of course there is some kind of non-ideal thermal gradient that forms, even so it seems like you would need some kind of minor cooling system in the center of the torroid because of themal profile vs the exterior which does not radiate on itself.

A ring will reheat the other side more then a doughnut, which aims its interior energy radially, I think?

How about a pill shaped rod? Will the tip get hotter because it has no heat sink?

[coppercone2]

https://superlab.stanford.edu/poster/3Dpassives_Rivas.pdf

any idea what 'ideal shape' is based on that document? the FEA is not very illumninating imo, but q appears to be significantly better despite the higher inductance, but its not expected to be accurate.

the magic search word is 'optimal cross section'

http://www.butlerwinding.com/toroidal-inductors-theory-2/



Toroidal inductors with a round core cross section are better performers than toroidal inductors with a rectangular cross section. The cancellation is more complete for the round cross section. The round cross section also gives a shorter turn length per unit of cross sectional area, hence lower winding resistances. Good turn-to-turn coupling is dependent on the winding being wound a full 360 degrees around the core. As winding turns are positioned further away from the core less complete turn-to-turn coupling will occur. Turns on the outer layers see a core cross sectional area that includes some non-magnetic area (air, insulation, copper). This added area generates some leakage inductance that adds to the inductance expected from the core.

throw some d's on that ?
https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/27492/1/Thesis-1987-Belahrache.pdf

any thoughts on distortion rather then just maximum inductance? would you need a mu-gradient or something, so it saturates linearly rather then having increased flux on the inner rim? how does it look like with change in frequency?

[T3sl4co1l]

Distortion:

Sure, why not.  Or, more generally, if not distortion per se, then non-LTI behavior.  For example, thermal expansion of the core/winding will not cause distortion as such (i.e., on a per-cycle basis), but the change in value will eventually change the signal amplitude/phase, and therefore it is not a time-invariant system (i.e., the inductance parameter varies over time, in this case due to temperature).

This will be a ~ppm scale effect, at best, for the most part.  The core, if present, will distort significantly more than this.

Mechanical effects are significant in some types of ferrite, which exhibit relatively high magnetostriction (strain (change in size) resulting from magnetic field magnitude).  In this case, per-cycle effects are perceptible, and electro-mechanical coupling is seen.  Much like a speaker or piezo, mechanical resonances manifest as peaks and dips in the electrical impedance.

Regarding cross section, rectangular is probably more popular because of higher power density.  Also, increased core permeability makes the wire conductivity much less important.  And for almost all materials, core loss dominates*, anyway.

There aren't any EMI/radiation or near field problems with an odd cross section, as long as it is still a solid of revolution.  The toroid core depends on its rotational symmetry, not on the shape of the cross section (but yes, winding resistance will be least for the circular case).

*In a real power inductor application, ripple might be a modest fraction of DC bias, so that total copper losses can be comparable.  The difference is due to DC heating.  The AC heating is still dominated by the core.

For an uncored inductor, a square winding path is of course worse.  Curious about that paper, that they went to the trouble of fabricating these things, but they made the one mistake that would've won them another factor of 2: rounded conductor cross section with even spacing of turns.

This is easy to demonstrate with a helical winding calculator: set pitch = 2 * wire diameter and observe the Q.  Then squash it together, say pitch = 1.1 * wire diameter, and observe again.  (The inductance is higher in this case, but the AC resistance is MUCH higher, and the Q lower.)

They also claim to cancel out the solenoidal field from a single layer progressive toroid winding, by stacking two oppositely.  Uh, right.  That makes a quadrupole, which drops off faster at a distance but is most definitely not a cancellation!

The other links aren't very substantial.  Transformer winders follow drawings; they don't need much design expertise in house, let alone theoretical knowledge.  No proof is given, only claims made.

Tim

[coppercone2]

well the last one is just a masters thesis so it may have been cobbled together in 3 months before it was due

I actually got the idea of a D-shape inductor without any math just looking at the flux density between the round and square ones in the superlab paper, because I kinda imagined the 'high field point' on the inner torroid causing a spot heat problem and noticed it did not seem to matter on the other side, i am sure you notice the 'integral operation' of smushing the increased density into a wider area in your head, but I was unsure if they should be mitered or not. I assume the circular cross section has a higher flux density band on the most interior orbit then the square one even if its not easily apparent from the tiny FEA, since proximity increases and available area decreases.

could you make a material that has varying Mu across a dimension (like fiber-laser optical glass with variable refraction index) to prevent the concentration of field in the end, so you have equal flux density? Not sure what that would do though, I assume since you need to layer different materials you would have some kinda weird area where flux density is uniform at some particular frequency at a very specific amplitude level?

Like nesting dolls I guess, so you would maybe paint different material layers on then sinter it. Or use a few materials throughout the whole thing and vary their proportions towards the inside. But since they all have a different saturation flux vs mu it would be a very noticeably amplitude specific thing like a MOV i think.

yadda.icm.edu.pl/yadda/element/bwmeta1.element.baztech-ec374557-78a3-434a-ab90-14bb2ce05cf0/c/Kasikowski_impact_MAM_4_2017.pdf

This paper says there is like a 70% flux gradient towards the inner shell and gives some spot temperature measurements (gradient only). I assume it might be even worse with a round CSA one. Interestingly the situation is completely different with the rod, where I imagine max flux is in the center in a ring around the axis?

[coppercone2]

pretty

[T3sl4co1l]

Of course a nonuniform winding gives a nonuniform field; and a gapped core, even more so.

Tim