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Background

The commonly known skin effect formulas are derived and only apply to solid conductors. The commonly used "skin depth" only applies in these cases. It is for this reason that in some applications tubes are used, as these are much more weight-efficient than the same diameter wire at a high-enough frequency.

At 1MHz the skin depth of copper wire is 65µm which means that only 40% of the volume of a 1mm diameter wire is carrying 95% of the current, with >35% of it in the outside 20%.

From the skin-depth formulas it is known that a lower conductivity material (e.g., aluminum) has a skin depth that is considerably larger than a higher conductivity one (e.g., copper). As the formula predicts, skin depth is inversely proportional to the square root of the conductivity. If we carry this to its logical consequences, it should be the case that for a conducting tube (which has an insulation core) skin depth should be larger than for an equivalent solid conductor.

As an alternative intuition a thin-walled insulated-core conductor would have nearly twice the surface area of a solid conductor. So it should asymptotically approach nearly half the resistance at a high enough frequency.

In effect, as can be seen in this paper from HB Dwight in 1922 (possible paywall), the increase in resistance w.r.t. frequency for a tube whose wall thickness is 20% of its diameter is more than a factor of two lower than for a solid wire.

Skin effect in tubes and wires

From the above curves it can be seen that a tube with t=200µm and d=1mm, due to the increased actual skin depth, should have less than 50% of the impedance increase than a solid d=1mm wire (do note that the curves are normalized w.r.t \$ F / R_{dc} \$, so interpretation is a bit tricky).

Similar effects (although not as dramatic) can be observed with individually-insulated stranded wire.

Application

In medium-frequency applications, as for example switching power supplies, it is common to use Litz Wire a multi-stranded insulated wire which reduces the losses due to skin effect but becomes less and less effective at higher frequencies (~1MHz) because of the proximity effect and the capacitive coupling of the individual strands.

Probably more gains (particularly with respect to proximity effects) could be obtained if there were multiple individual strands embedded around the periphery of a non-conducting core.

Question

Have I missed something in the theory?

If not, why isn't insulated core wire (either tubes or strands around a core) being commercially exploited for high-frequency inductor applications?

Addendum

As John Birckhead answer points out, flat wire has basically the same advantages with none of the disadvantages (e.g., fill factor). But this leads me to ask:

Why isn't insulated-core flat wire being used for these applications? It should have the same advantage of flat wire with nearly half the resistance at high enough frequencies. Are the possible gains inconsequential?

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    \$\begingroup\$ Comments are not for extended discussion; this conversation has been moved to chat. Any conclusions reached should be edited back into the question and/or any answer(s). \$\endgroup\$ – Dave Tweed Feb 28 at 19:12
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    \$\begingroup\$ I'll need to read it further, but I found this set of pages on Litz wire. Just a note. \$\endgroup\$ – jonk Feb 28 at 19:37
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No, you are correct in the theory, but your approach leads to an unnecessary increase in volume when compared to using flat wire, which is both easier to manufacture and provides a similar advantage for skin effect and the advantage of volumetric efficiency.

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    \$\begingroup\$ I have never seen flat wire being used in RF applications, transformers, or inductors while Litz wire is quite common. Could you expand your answer to point to those and how it compares? \$\endgroup\$ – Edgar Brown Feb 28 at 19:23
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    \$\begingroup\$ See mouser.com/pdfdocs/bourns_ic046_flatwire_inductor_appnote.pdf \$\endgroup\$ – John Birckhead Feb 28 at 19:27
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    \$\begingroup\$ To your question, Litz wire also has poor volumetric efficiency for high-current applications because of the insulation and the way the wires cross each other in the coil. It is also difficult to terminate at high currents to get an even current distribution. It is useful at low currents when you are not space constrained, because flat wire is tricky to wind. \$\endgroup\$ – John Birckhead Feb 28 at 19:58
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    \$\begingroup\$ If my intuition is correct, a flat wire with a non-conductive core would have less impedance at higher frequencies than a flat wire (and it should be relatively easy to construct by flattening a thin insulation-filled tube). So, although this points in the right direction and does answer the main aspect of the question, it does not fully address it. Are the gains insignificant, or the application space non-existent? \$\endgroup\$ – Edgar Brown Feb 28 at 20:26
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    \$\begingroup\$ First of all, thanks for a really interesting and well-stated question, (near to my heart being a magnetics guy). It's just easier flattening the wire down to the skin depth - there would only be a marginal difference because there is insulation between two layers just as there would be insulation in your scenario in the center of the proposed conductor, and you can get the same cross section with a wider flat wire. It would be an interesting study to determine how much if any advantage is gained - it does "feel" like inter-winding capacitance might be less. \$\endgroup\$ – John Birckhead Feb 28 at 20:42
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Wikipedia entry for Litz wire contains a direct answer your question of "Why aren't hollow tubes used instead?":

One technique to reduce the resistance is to place more of the conductive material near the surface where the current is by replacing the wire with a hollow copper tube. The larger surface area of the tube conducts the current with much less resistance than a solid wire with the same cross-sectional area would. The tank coils of high power radio transmitters are often made of copper tubing, silver plated on the outside, to reduce resistance. However tubing is not flexible and requires special tools to bend and shape.

The article goes on to describe why Litz wire provides an alternate solution.

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Induction Heating (Industrial) commonly uses hollow copper tubing for the inductor.

When you are running 1000 kW or higher, you better believe the copper loss needs to be minimized.

Additionally, the hollow core is used for water cooling.

Sometimes the copper is called "hollow bar". It comes in rectangular or round. It is not uncommon to order a "mill run" to get the hollow bar and thickness desired.

Image from luvata dot com

enter image description here

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    \$\begingroup\$ At what frequency is this normally operating? \$\endgroup\$ – Edgar Brown Feb 28 at 22:55
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    \$\begingroup\$ @EdgarBrown . .Induction heaters are used from 50 Hz to several megahertz. Mostly below 50 kHz though. \$\endgroup\$ – Marla Feb 28 at 23:30
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There is indeed a variant on insulated core wire for very high frequency applications. It's called a waveguide. It's a hollow tube used for conducting RF. I am of the understanding that the signal travels in the inside of the conducting shell rather than the outside, but the idea of only needing the conducting shell as thick as the skin effect dictates is there.

Not much use for inductors though.

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    \$\begingroup\$ A waveguide is a rather different thing. The RF is travelling in the air itself, and being "reflected" off the inside (to simplify), rather than actually travelling in the metal of the waveguide itself, \$\endgroup\$ – mbrig Feb 28 at 23:01
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    \$\begingroup\$ @mbrig: the difference is not as big as you think. Also using a solid wire at high frequencies most of the power is flowing in the field around the conductor, not inside the conductor. Cf. Poynting vector. \$\endgroup\$ – Curd Mar 1 at 9:22

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