Re: Partial Inductance

J. Eric Bracken (bracken@ansoft.com)
Mon, 16 Jun 1997 14:58:54 -0400

The point about "piling up" of charges is not strictly necessary to
the discussion of partial inductance (and I now regret starting this
digression), but since it's provoked so much discussion I thought I
would clarify what I was saying. Stop reading this *NOW* if you don't
want to hear electromagnetics lingo.

If you try to solve Maxwell's equations with a current distribution
that just terminates in the middle of nowhere, then Maxwell's
equations will actually pile up charges on the ends of that
distribution for you.

This is because Maxwell's equations have "charge continuity" built
into them. Remember this one?

curl H = J + epsilon * dE/dt

If you take the divergence of this equation, you get

div curl H = 0 = div J + d[ div( epsilon * E ) ] / dt

But div( epsilon * E ) = Q (this is another of Maxwell's equations, known
as Gauss' law), and so we get

div J + dQ / dt = 0

There's a big divergence to J where the current distribution abruptly
terminates, and so there's a dQ/dt there to balance this.

Fortunately, even time-varying charges don't contribute to the magnetic
vector potential A. It's computed by integrating over all the currents

A = Integral[ (Green's function) * J ]

The vector potential is used in the energy definition of inductance:

Wm = (1/2)*Integral[ A * J ] = 1/2 * L * I^2.

So, the piling-up charges DO NOT affect the inductance calculation in
any way. Therefore, they may safely be neglected, and indeed they are.

Sorry to drag anybody through this muck, who, really, didn't give
a damn.


--
J. Eric Bracken, Ph.D.                          Tel: 1.412.261.3200 x135
Group Leader, Signal Integrity R&D              Fax: 1.412.471.9427
Ansoft Corp., Four Station Square, Suite 660    bracken@ansoft.com
Pittsburgh, PA USA 15219-1119                   http://www.ansoft.com