Re: Equivalent lumped inductance and toroidal coils

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

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

Gerry wrote:
 > Yes, R=0 is a problem if you assume all the current is all
 > concentrated at R=0.

Worse, it is a problem if *any* current is at R=0, so splitting the
single filament up into several spread over the current-carrying
area of the wire doesn't help - you'll still come up against the
need for self-inductance for each filament.  (I suppose if you just
leave out the self inductances you'll have some error, but only
slight if you have many filaments per wire.)

Not only that, but as soon as you represent the wire with more than
one filament, you have to decide how the total current is to be
shared across the bundle of paralleled filaments.   You can't
in general assume say uniform distribution over the surface.

This problem extends to networks of separate conductors which offer
alternative conduction paths through the field.  For example an
OLTC primary might consist of several circular loops all in
parallel.

Therefore, when you come to calculate the total inductance with say
a Neumann sum over the source wires, you can't just assign a nominal
1 amp of current to each wire.

Instead, you have to work algebraically, by using symbols I1, I2, ...
and so on to represent unknown currents in each parallel branch.

Then you work out the mutual inductance coefficient between each
pair of branches, and the self inductance of each branch,  and when
you have all these, a matrix calculation must then be carried out to
compute the ratios between I1, I2,... and so on.  In other words
you're computing how the current divides up when faced with the
pattern of mutual and self inductances of the various alternative
pathways.  In this way you would determine how the current spread
itself over the surface of your conductor represented as a tube
of filaments, or in the other example, how the current shared
between the parallel turns.

I think this is involved in the method of 'partial inductances'.

It gets worse still if you want to calculate the high frequency
behaviour, because now you can't just assume a uniform current
in each branch.  I1, I2, etc become functions of position instead
of constants!
--
Paul Nicholson
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