Re: Equivalent lumped inductance and toroidal coils

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

Original poster: "Gerry Reynolds" <gerryreynolds-at-earthlink-dot-net> 

Hi Paul,

One hairbrained idea that might work is to represent the coil mathmatically
in a suitable coordinate system (for a helical coil maybe this would be a
cylindrical coordiates).  One would target a turn, integrate the
differential current element over the total wiring path to evaluate the H
field at a sampling of points within the cross section of that turn.  Once
the H field is sufficiently sampled, the total flux threading that turn can
be evaluated by integrating B.dA

The H field at the sample point would be:

H(r,phi,z) = integral [wire beginning to wire ending] {(Ids X R}/4piR^2}

Ids X R is a vector cross product of the vectors ds and R (R vector is from
ds to the point being evaluated) and the R in the denominator is the scalor
of the vector R.  With a helical symetry, the integral may be reducable to a
closed form.  If symmetry is lacking then the H field integral could be
evaluated numerically.

The total flux thru a turn from all the current elements would then include
self and mutual coupling (no distinction would then be made).    A current
profile could be assumed or perhaps starting with the base current (or
current at an antinode), compute the current loss do to the displacement
current.


This probably wouldn't be done for helical coils but I use it to explain the
concept.  A lot of number crunching and maybe this is just brut force, but
what the heck.

Gerry R



 > Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk>
 >
 > Gerry wrote:
 >
 >  > Is this another way of saying differential current elements
 >  > or is there something different here?
 >
 > It's a way of representing mutual inductances between
 > short segments, rather than loops.  I don't really understand
 > it or know how to implement it.  There's something about it
 > in some tuturial notes I found in
 >
 >   http://videos.dac-dot-com/videos/38th/22/22_1/22_1.pdf
 >
 > (it also mentiones the quasi-static approximation that we use
 > when modelling TCs, which decouples the E and H fields allowing
 > us to use inductance and capacitance as a substitute for field
 > theory).
 >
 > Antonio wrote:
 >
 >  > I am thinking about doing "exact" calculations using as
 >  > core the mutual inductance between two straight segments.
 >
 > I wonder if that's the same thing as 'partial inductances'?
 > Maybe it's like equ 9 in the pdf above.  From Rosa, 1908.
 >
 > The idea seems to be that you break up the network into
 > segments, work out the partial inductance between each
 > possible pair, and then the required loop mutual inductance
 > are produced by simply summing the relevant partial terms
 > as per equ 8.  Doesn't sound any different, really, to what
 > we already do with circular filaments.  I expect Antonio will
 > say it is the same thing that he is doing now with the
 > Neumann integration.
 >
 > Often, with these inductance things, I don't really understand
 > the formulas involved or try to derive them - I just close my
 > eyes and plug them in to the computer.
 > --
 > Paul Nicholson
 > --
 >
 >