**From:** Howard Johnson (*[email protected]*)

**Date:** Fri Mar 16 2001 - 12:15:30 PST

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To Sainath Nimmagadda,

Wow! An actual Maxwell's equation question... I may

not be able to give you a complete answer, but hopefully

I can start you down the right path. Those of you

not interested in philosophical questions about

Maxwell's equations may want to skip this message.

The principle in question is the "minimum energy"

principle. My recollection of Maxwell's equations

(specifically I *think* it's the ones that say

the Laplacian of both electric and magnetic

fields are zero within source-free regions)

is that the distributions of charge and current

in a statics problem fall into a pattern

that satisfies all the boundary conditions around

the edges of the region of interest,

satisfies the Laplacian conditions in the middle,

AND ALSO just happens to store the *minimum*

amount of energy in the interior fields.

In other words, you aren't going to get huge,

unexplained, spurrious magnetic fields in

the middle of an otherwise quiet region (unless

you believe in vaccuum fluctuations, which is

a different subject entirely...).

For a parallel-plate capacitor, the minimum-stored-

energy principle means that charge

pumped from one plate to the other will distribute

itself fairly evenly across the two plates, producing an

even distribution of electric field intensity everywhere.

That's the minimum-stored-energy configuration.

The energy stored in a capacitor is E=(1/2)*C*(V^^2),

where C is the capacitance and V^^2 is the voltage

squared.

If you divide the plate in two, making two capacitors each

with half the capacitance and half the total charge, the

total energy stored remains the same:

E = (2 capacitors)*(1/2)*(half the capacitance)*((same voltage)^^2)

If you bunch the same charge all together on just

one of these half-capacitors, you'd have twice the electric field

intensity (twice the voltage) on that capacitor,

but no voltage on the other.

The energy on the charged half would be

E' = (1/2)*(half the capacitance)*((double voltage)^^2),

which works out to twice as big as E. If you try all combinations

of charge distribution, you'll find that the

way to minimize the total stored energy in this system

is to distribute the charge equally on both capacitors.

I use the capacitor analogy because I find most electrical

engineers have an easier time visualizing electric-field

problems than magnetic-field problems.

Let's shift gears now to look at currents. Imagine I

divide a conducting path (like a printed-circuit trace)

into a multitude of skinny, parallel pathways.

Now look to see in what pattern the current distributes

itself. At frequencies high enough that the inductance of

the traces is more significant that the resistance, but not

so high that we have to worry about excessive radiative

losses or non-TEM propagation modes, the answer is this:

the current distributes itself in that pattern

that minimizes the total energy stored in the magnetic field.

The stored energy for inductive problems is: E = (1/2)*L*(I^^2),

where where L is the system inductance and I^^2 is the

total current squared. As you can see, stored magnetic

energy E and inductance L are directly proportional to each

other. Therefore, the minimum-stored-energy distribution of

current and the minimum-inductance distribution of current are

the same.

Notice that I have assumed in this treatment that the

resistance is insignificant and there are no significant

capacitances to worry about. Both assumptions apply

pretty well to the problem of figuring the inductance

of a bypass-capacitor via, the issue that started this

whole discussion.

In answer to what might logically be your next

question, "Why do electromagnetic fields tend towards

the minimum-stored-energy distribution?", I can only say

that I'm not sure anyone really knows -- we just observe

that this is the way nature seems to operate. Perhaps

someone more well-versed in electromagnetic theory

can provide an answer.

It's possible that by assuming the current is *NOT* in

the minimum-energy distribution you could prove some

impossibility, like a perpetual-motion machine or

something, that would convince you of the absurdity of

the situation, but that won't really increase your

understanding unless you also intuitively believe that

nature is not absurd. Further discussion of *that* issue

is probably best left to physicist-philosophers.

I hope this brief answer is helpful to you, and doesn't

just stir up a lot of other doubts.

Best regards,

Dr. Howard Johnson

At 08:31 AM 3/15/01 -0800, you wrote:

*>Dear Howard,
*

*>
*

*>Please see below:
*

*>
*

*>Howard Johnson wrote:
*

*>
*

*>> Dear Itzhak Hirshtal and Brian Young,
*

*>>
*

*>> The difficulties with approximating the inductance
*

*>> of a via are even worse than you
*

*>> may have suspected. Both approximations are flawed whether
*

*>> you use +1 or -3/4, (or, as I have also seen, -1).
*

*>>
*

*>> The issue of the exact constant (1, -3/4, or something
*

*>> else) depends critically on your assumption about
*

*>> the path of returning signal current. (Current always
*

*>> makes a loop; when signal current traverses the via,
*

*>> a returning signal current flows SOMEWHERE in
*

*>> the opposite direction.). It is a principle
*

*>> of Maxwell's equations that high-speed returning signal
*

*>> current will flow in whatever path produces the
*

*>> least overall inductance.
*

*>
*

*>My question is on this last statement. I like to understand which
*

*>Maxwell's equation suggests this and how? Thanks.
*

*>
*

*>Sainath
*

*>
*

*>
*

*>
*

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**Next message:**Michael Nudelman: "RE: [SI-LIST] : Re: approximations for partial self inductance - WHY"**Previous message:**Howard Johnson: "[SI-LIST] : Re: approximations for partial self inductance"**Next in thread:**Doug McKean: "Re: [SI-LIST] : Re: approximations for partial self inductance - WHY"**Reply:**Doug McKean: "Re: [SI-LIST] : Re: approximations for partial self inductance - WHY"**Maybe reply:**[email protected]: "Re: [SI-LIST] : Re: approximations for partial self inductance - WHY"

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