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Would there be any advantages to having magnet wire with a square or hexagonal cross section? Could this allow stronger or more tightly coupled magnetic fields in electrical motors or transformers?

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Yes. Square wire is available in the sizes for which it has sufficient advantage to be worthwhile. I believe that flat rectangular strap is also used in some situations. Search wire manufacturer's web sites to se what is available.

Here is a picture of square wire in a motor or generator rotor:

enter image description here

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  • \$\begingroup\$ So why isn't this kind of wire always used? Is it simply because it is more difficult to manufacture? It seems like this kind of wire might have less heat dissipation because there would be no air gap but maybe not since copper wire conducts heat better than air. \$\endgroup\$ Jan 16, 2017 at 16:33
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    \$\begingroup\$ @DanielCaoili, more expensive. And the cross section of square or rectangular magnet wire is not exactly square.. it dimples in the middle section so I'm not sure the thermal conductivity is that much different. The big advantage is filling more space with copper. \$\endgroup\$ Jan 16, 2017 at 16:46
  • \$\begingroup\$ square wire is used in microwave impedance control circuits but magnet wire in this shape would suffer from abrasion and loss of insulation. although the fill factor may be better. \$\endgroup\$ Jan 16, 2017 at 16:50
  • \$\begingroup\$ Square and rectangular wire has lower resistance and better fill in a given winding window and much better thermal capacity. \$\endgroup\$ Jan 16, 2017 at 17:16
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    \$\begingroup\$ However, winding coils with square or hexagonal wire is more expensive. The wire cannot be allowed to twist, and especially square wire must be carefully laid down, since it will tend to snag on its corners if alignment is not perfect, and will not self-pack easily. Round wire can be wound under tension with slight bias against the advancing layer and will self-organize very nicely. Square, not so much, although hexagonal will do better. Plus, the sharp corners provide ample opportunity for insulation failure during handling. \$\endgroup\$ Jan 16, 2017 at 19:02
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At least one loudspeaker manufacturer, Morel, already does, to decrease their voice coil resistance without increasing the gap between magnetic pole pieces. (Their web server appears broken at the moment)

Morel See also Why are copper cables round?

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  • \$\begingroup\$ Perhaps all larger loudspeakers do this, certainly all the 12" and up that I've seen. \$\endgroup\$
    – tomnexus
    Jan 16, 2017 at 19:56
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It is usually used on the big stuff. Our motor rewinders always offer to rewind with square wire, saying we will get more power if we do. We have to constantly tell them not to, as it will imbalance the system.

Damage to the wires' insulating varnish is less a concern because these commutators are typically epoxy dipped.

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