Re: Formula for self-inductance?


Wimpie wrote:

When you convert the Biot-Savart formula to a current sheet version,
the B-field will not blow up to infinity, hence the inductance. Most
EM-field simulators for planar structures use the current sheet
approximation. Of course, this is also an approximation, because in
real world, current doesn't flow in infinity thin sheets. One example
is a hair pin inductor made of strip wire. When the distance between
the strips approaches zero, internal flux determines the inductance
and you should use a full 3D current distribution.

I believe that if you are not willing to limit the application of your
formulas to be developed, the problem will turn out into a mathematics
problem rather then a physics problem. Also you have to incorporate
dynamic fields (almost complete Maxwell equations) because current
distribution depends heavily on dI/dt and material properties.

Here's my points.

First off, I only intend to limit solutions (for now) to low
frequencies as well as ignoring current distributions due to qVxB as
occurs in heavy current bus bars.

Next, I have severe reservations about using flux to calculate self-
inductance. Sometimes this works. Sometimes not. This "flux linkages"
is a rather vague concept and is clearly not a fundamental one.

Next. I find that in the case of straight wire segments (yeah, I know
this is sort of a mathematical concept because a loop is required to
complete the circuit, but still...) things are particularly bad. The
idea is that self-inductance is merely mutual induction of the current
in a wire BACK upon that same wire. But hey, Biot-Savart says that
along a straight wire, B along that wire is pretty much zero. There is
no B field at the wire to induce anything into it.
Flux methods give infinite values for inductance. But clearly straight
wires DO have real and finite amounts of self-inductance per unit
length!

Next. So ignoring that problem and noting that the Neumann equation
does provide for induction between a source current and target wire,
things start to look good, but then we have the "blow up" problem as
one moves the target toward the source current location. Still, the
basic idea seems correct.

Next, If one uses current density as a source instead of current the
idea is that as one moves toward the source things won't blow up. This
does not seem to be the case.

So finally, as you can see, what I'm looking for is the self-
inductance equivalent to the Neumann formula. The idea is that this
formula should take into account the geometry of the conductor, and
allow it to be broken down into small elements which are then
integrated in some formula dependent only on geometry (or perhaps a
few material properties) and barf out the self-inductance of that
particular conductor shape. It it takes a computer program to do
this, that's OK. If you have to do finite differences, that would
work. But what I"m missing here is the THEORY or more explicitly the
kind of equation that you'd plug into the program to make the whole
thing work.

Let me point out, Mutual inductance between two single loops is a
snap. You can usually get away with the thin filament thing at the
center of the wires to simplify integrations. The Neuman formula works
like a charm to find the mutual inductances between each pair of loops
if there are many loops. However, if you look at the mutual inductance
between two coils of wire, you find that you not only need the mutual
inductance between all pairs of loops, but you need the self-
inductance of each loop of wire as well. The thin filament thing won't
work because the inductance of a very thin filament is infinite! You
can, of course approximate this self-inductance by using , nearest M,
or the Jackson or P-C ideas I mentioned above, but it would be better
to have a REAL calculation for this value that comes from the wire
geometry, cross-section and all the rest.

You see the problems here?

I'm wondering what might happen if one actually did take Maxwell's
equations and went after a time-dependent solution of self-inductance
for a given geometry? Could some kind of formula of the type I'm
talking about come from that? And then if that were true the next
question would be can it be reduced to some low-frequency version?

Benj

.

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