E = -jwA - Grad(V)
E = electric field
A = Vector potential = Integral[ G(w) J dv ]
V = Scalar Potential = Integral[ G(w) p dv ]
G(w) = free space Green's function = exp(-jwr)/r
w = angular frequency, j = sqrt(-1)
J = current density
p = charge density
r = distance from observer to volume element
dv = volume element
First discretize the source current densities:
J dv ~= Sum[n, In]
For a perfect electric conductor:
E(tangent) = 0 -> Grad(V) = -jwA (tangent to the conductor surface)
Substituting the sum for J and interchanging the sum and the integral:
Grad(V) = -jw Sum[n, An]
Integrate over a line segment representing the tangent on the destination
conductor (m) :
Integral[Grad(V) dXm] = -jw Sum[n, Integral[An, dXm]]
= -jw Sum[n, In * Lmn]
where Lmn is the (mutual) partial inductance of the n,m conductor pair.
Of course the integral on the left is the potential difference across the
the mth conductor segment. So the partial inductance is just the potential
difference (voltage) across the mth conductor due to a 1 Amp/second
current ramp on the nth conductor. As has been pointed out, measuring
these voltages can be tricky due to the interaction of the probe with the
fields, but the concept itself is straightforward. Note that no loop
is required. The charges piling up at the ends can be ignored for the
calculation. Even for the case of a loop, it is a charge pileup that is
needed to create the scalar potential gradient necessary to cancel the
EMF due to the jwA.
Partial inductance is a useful bookeeping device for keeping track of all
the of the magnetic interactions in complex multi-conductor configurations.
The fact that a given value is difficult to measure in isolation leads to
confusion, but does not invalidate the technique.
Viewlogic Advanced Development Group
(Quad Design Technology)
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>From Edlund@EXCHANGE.eng.pko.dec.com Tue Jun 17 14:09:46 1997
From: "Edlund, Greg" <Edlund@EXCHANGE.eng.pko.dec.com>
To: "'J. Eric Bracken'" <firstname.lastname@example.org>
Cc: "'email@example.com'" <si-list@silab.Eng.Sun.COM>
Subject: RE: Partial Inductance
Date: Tue, 17 Jun 1997 16:46:45 -0400
X-Mailer: Microsoft Exchange Server Internet Mail Connector Version 4.0.994.63
Content-Type: text/plain; charset="us-ascii"
I myself don't have a problem with currents that don't form a loop.
After all, current is just charge moving past a point (or through a
surface, or through a volume). And if you're moving along with the
charge, it isn't a current at all, right? Witness the quarter-wave
dipole antenna. When a time-dependent magnetic field cuts the wire, a
current is induced, right? There doesn't need to be a loop for the
charge in the copper to move around.
Now about partial inductances: seems like this is kind of a
mathematical construct to let us think about inductive voltage drops in
much the same way we think about Ohmic drops, i.e. that they add up to
zero around a loop. (Most of us would rather think in terms of i and v
rather than E and B.) OK - I'll buy that. But you need to DEFINE the
rest of the current loop in order for the partial inductance to have any
meaning, don't you? After all, inductance is defined by current loops.
Otherwise you have people saying things like, "The inductance
per-unit-length of a 1 mil diameter wire in free space is blah, blah,
blah..." Antennas might not need a loop to have current, but SPICE
surely does. And I can't talk about - or accurately simulate - the
voltage on the chip-side of a 1 mil diameter bond wire unless I know
where the current is coming from and where it's going.
p.s. Wouldn't it be cool to take a short course called "Maxwell's
Equations Applied to Modeling Packages, Connectors, and Transmission
Lines?" Then those of us with a dim memory of the material could speak
with more authority!
Greg Edlund , Principal Engineer
Alpha Server Signal Integrity
Digital Equipment Corp.
129 Parker St. PKO3-1/20C
Maynard, MA 01754
(508) 493-4157 voice
(508) 493-0941 FAX
>From: J. Eric Bracken[SMTP:firstname.lastname@example.org]
>Sent: Monday, June 16, 1997 2:58 PM
>Subject: Re: Partial Inductance
>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
>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 email@example.com
>Pittsburgh, PA USA 15219-1119 http://www.ansoft.com
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