Re: Equivalent lumped inductance and toroidal coils

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Antonio Carlos M. de Queiroz" <acmdq-at-uol-dot-com.br> 

Tesla list wrote:

 > Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk>

 > Agreed.  This case is attractive due to the symmetry and
 > uniformity of the coils.   I'm hard pushed to suggest any
 > reason why either the voltage or current amplitude
 > distributions should depart from a sinusoidal function
 > of position.

A toroidal coil looks similar to a transmission line formed into
a loop. With magnetic excitation from a primary coil, apparently
the first resonance would result in two opposite maxima at 90
degrees with the position of the driving coil, and notches
where the driving coil is and the diametrally opposite position.
A weird form of a resonant transformer with balanced output.
Yes, it looks as the voltage profile will be something as
a full sinusoidal cycle along the loop, starting from the
driving point.
I don't see how to drive a toroidal coil so it forms a single
maximum, or any odd number of maxima.

 > Well I'll code something to do the low frequency inductance.

I was looking at your formula, and comparing to a formula in the
NBS Circular 544, that looks similar, maybe with a different
definition of the angles. I can't access your picture with this
address:
http://www.abelian.demon.co.uk/tmp/pn110704.gif

The formula for loops whose axis intersect doesn't cover all
the possibilities. A formula for two parallel
loops, noncoaxial, is necessary too. Imagine, for example, how
to calculate the mutual inductance between radially opposite turns
in a toroidal coil. The Circular has one, quite
horrible with Gamma functions and hypergeometric series...

 > And if the amplitude distributions really are sinusoidal, then
 > we also get the HF inductance quite easily.

I imagine that at the first resonance the current will follow a
cosinusoidal profile along the toroid, a full cycle.

 > Yes, we cannot use our usual solution in terms of charge
 > rings.  But I wonder perhaps if the symmetry and lack of
 > end effects might just lead to a closed formula for the
 > elements of the capacitance matrix, with respect to some
 > natural surface tiling of the toroid, rather than ring
 > by ring.

Maybe, but it's already complicated to calculate the charge
distribution in a toroid with constant surface potential. In
this case the potential would vary as a full sinusoidal cycle
along the toroid.

Antonio Carlos M. de Queiroz