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I find some text about the winding of a toroid core as below:

enter image description here

But what's the math and physics behind these?

BTW: Can someone suggest some good books about the design of an inductor/transformer not only in theory but also about "how to make them by hand".

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  • \$\begingroup\$ Some context behind the picture is important or it becomes guesswork. \$\endgroup\$ – Andy aka Dec 20 '16 at 14:27
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    \$\begingroup\$ The quality of this advice varies with the permeability of the core, and what you're trying to achieve. C is better than A for mains on high u iron, B is never good, but it's not too bad on high u iron, to understand A it would have to be on quite low permeability at high frequency on a non-conductive core. \$\endgroup\$ – Neil_UK Dec 20 '16 at 15:10
  • \$\begingroup\$ That gap comes to mind for (mains) spacing too. If that's the case and you need to fit more turns, there are coil formers for the job. Sometimes you can get away with a zip tie too. \$\endgroup\$ – winny Dec 20 '16 at 15:49
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These winding instructions give you a single-layer inductor that minimizes parasitic effects of capacitance. The (B) version suffers from capacitance from one turn to its neighbours. The (C) version minimizes neighbour-capacitance but suffers from capacitance from end-to-end. The (A) version compromises between (B) & (C). Minimizing capacitance increases self-resonant frequency (important where the inductor is used in wide-bandwidth circuits).
For toroids used in high-Q tuned circuits, core relative permeability is often quite low (perhaps a factor of ten higher than air). So not every turn is linked to all other turns as it would for high-permeability cores. Although the rule where inductance is proportional to turns^2 is often applied, turns couple more tightly when bunched together, and couple less tightly when spread. In circuit simulators, the coupling factor (k) is certainly less than one, and is often less than 0.5 for low-permeability cores.
The math is not worth the effort. Besides, tolerance on toroid core permeability is notoriously poor.

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