From: Howard Johnson ([email protected])
Date: Fri Mar 16 2001 - 12:15:30 PST
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
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
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
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.
Dr. Howard Johnson
At 08:31 AM 3/15/01 -0800, you wrote:
>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.
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