[hsdd] High-Speed Digital Design Newsletter
- From: "Dr. Howard Johnson" <howiej@xxxxxxxxxx>
- To: <hsdd@xxxxxxxxxxxxx>
- Date: Thu, 7 Aug 2003 15:44:17 -0700
MINIMUM-INDUCTANCE DISTRIBUTION OF CURRENT
HIGH-SPEED DIGITAL DESIGN ? online newsletter ?
Vol. 6 Issue 7
In preparation for writing my new text, "High-Speed
Signal Propagation", I took the time to read some
classic works in electrical engineering. One particular
work stood out as a magnificent tribute to man's
ability to solve complex problems by intuition and
analogy. "A Treatise on Electricity and Magnetism", by
James Clerk Maxwell, is available from www.amazon.com
in a modern, reprinted edition. This is a two-volume
set, vol. I?ISBN 0486606368 and vol II?ISBN 0486606376.
This newsletter tries to impart a sense of the
intuitive power of Maxwell's approach to field
calculations. If you want to experience the full
effect, GET HIS BOOK. Although Maxwell's electrical
vocabulary (from the 1880's) can at times be difficult
for people in the modern world to interpret, the
clarity of his vision is stunning. What particularly
appeals to me is the fact that he developed his entire
theory without access to any computing resources more
powerful than a pencil. He did not have Mathematica, or
field-calculation software, or a three-dimensional
visualization package. Everything happened in his head.
I like that kind of engineering. That's the kind of
intuitive understanding I teach in class.
The following exchange took place during a
protracted discussion on the si-list about the
distribution of current on a reference plane
underneath a pcb trace. Sainath is trying to grasp
the relation between the inductance of a microstrip
trace distribution of current on the reference
plane. He wonders if the phrase "least inductance
distribution of current" implies that perhaps there
exist other distributions of current with other
values of inductance.
Those of you who, like I, have struggled a lot with
magnetic field theory may find Sainath's questions very
familiar.
______________________________________________________
MINIMUM-INDUCTANCE DISTRIBUTION OF CURRENT
Sainath Nimmagadda writes to the si-list:
?I get the themes that inductance is a one number
affair and current returns through the least
inductance path. [But] Is there a contradiction in
these themes?
Let me borrow the following from your [referring to
Andrew Byers] previous mail.
"If you were to put your integrating surface on
the other side of the trace, extending up from
the top of the trace, you theoretically would
have to make the area of the surface extend to
infinity to "catch" all the field lines."
For this case, is the inductance of the microstrip
going to be infinity (because of the infinitely-
large surface)? Or any other value? Or does the
inductance remain the same as it was when
integrating [the total flux on the] surface below
the trace?
______________________________________________________
Dr. Johnson replies:
The mysteries of magnetic-field integration are indeed
sometimes difficult to comprehend. In answer to your
question about the surface of integration, the best
mental image for this appears in the famous work by
James Clerk Maxwell,
"A Treatise on Electricity and Magnetism". The first
volume of this work (Electricity) is available on
www.amazon.com in the form of a modern reprint of an
old Dover version, circa 1954.
>From the preface of that book, here is the key idea
that renders sensible this whole business of
integration of magnetic field intensity over a surface:
"Faraday, in his mind's eye, saw lines of force
traversing all space".
The "lines of force" concept underlies all of
electromagnetism. What you need to know about Faraday's
"lines of force" idea, in the context of your problem
having to do with evaluating the inductance of your
trace, is that magnetic lines of force form continuous
loops having no beginning and no end. The total number
of lines extant is a measure of the total magnetic flux
produced by a magnetized structure.
Of course you can re-normalize any magnetic field
picture to produce a different number of lines by
declaring each line to represent a different quantity
of flux, for example 1/10th the original amount would
produce 10x the number of lines, etc. Presumably you
have scaled the flux represented in your (mental)
magnetic field picture in such a way as to produce a
manageable number of lines that is at once enough to
represent accurately the pattern of field intensity and
also not too many to clutter the image. Keep in mind,
however, that regardless of the number of lines, there
are a finite number of them and each is a continuous
entity forming a complete, unbroken loop.
In Maxwell's view, integrating the magnetic flux
passing through a surface is simply a matter of simply
COUNTING how many lines pass through it.
For example, consider a closed surface (a sphere) in
space. Any particular line that enters the ball must,
since it cannot end within the sphere, exit at some
other point. Therefore, when counting the number of
lines penetrating the surface, since each line must
both enter (a positive count) and exit (a negative
count), the sum of entrances and exits penetrating the
sphere must be zero. From this simple idea, Maxwell
derives the idea that the integral of flux over any
closed surface (of any shape) must be zero.
[Mathematical aside--you may be familiar with certain
complications having to do with the integration of
field vectors penetrating a surface whereby you have to
dot product the field intensity direction vector with a
vector normal to the surface--these difficulties
disappear when you simply "count lines", which is the
beauty of Faraday's brilliant intuitive approach. When
the surface is tilted so that the lines intersect the
surface at an oblique angle, the number of lines
penetrating each square area of surface is naturally
reduced. This reduction is precisely accounted for, in
multidimensional vector calculus, by the dot product.]
Now let's apply the line-counting analogy to your trace-
inductance problem. Imagine a certain finite number of
magnetic lines of force wrapped around your trace.
[I'll assume the reference plane is infinite in the x-y
directions. The plane is located at z=0, and the trace
is at z=1. Since the plane is infinite, no lines of
force exist below z=0.]
Assume I hook up my inductance meter to one end of the
trace. Connect the other end of the trace to the
reference plane. Now stretch an imaginary "soap bubble"
in the region between the trace and the reference
plane. Beginning at my end of the trace, the edges of
the bubble touch the trace all along its length,
following along at the end down to the reference plane,
returning along the plane to the source. For
completeness, let's also consider how at the source the
edges of the bubble also must track along the ground
lead of my inductance meter up to the instrument and
then back down the signal lead of the instrument to the
beginning of the trace. We'll assume the meter is tiny
compared to the size of the trace so we don't have to
worry too much about the shape of the source end of the
bubble (this is a serious real-life complication in the
measurement of tiny inductances).
Next step: apply 1-amp of current to the trace, and
count the number of field lines penetrating the soap
bubble. Since the bubble is an "open" shape (i.e., it
is bounded at the edges in such a way that it does not
enclose any space), you will record some non-zero
amount of flux penetrating the bubble. NOW comes the
cute part of this mental experiment. I want you to blow
on the bubble, stretching it. It's still anchored at
the edges, but no longer a flat sheet. The remarkable
thing that happens is that the number of magnetic field
lines penetrating the bubble does not change. It
doesn't matter how you stretch or modify the shape of
the bubble, or how far you blow it out of position, as
long as you don't change where the bubble is anchored
around the edges, you haven't changed the number of
lines penetrating it. That property (of the total flux
not changing regardless of the exact shape of the
surface of integration used) is essential to
understanding how to calculate inductance.
To prove that distorting the bubble doesn't change the
total flux, Maxwell imagines two surfaces, A and B,
both anchored to the trace and plane just like your
soap bubble. When connected together, these two
surfaces A and B form a single closed surface.
Therefore, using our earlier reasoning about the
sphere, the total number of lines penetrating the
combined object A+B (that is, coming into A and leaving
through B) must equal zero--from which you may
correctly deduce that when measured separately the
total flux passing through A must precisely equal the
total flux passing through B.
In a minute, I'm going to directly address your
question about making "the area of the surface extend
to infinity to catch all the field lines", but first I
need to go over one more detail. That detail has to do
with how a 2-dimensional surface with infinite extent
acts kinds of like a closed surface, in that it
partitions space into two regions. Instead of the
regions being "inside" and "outside" as they are for an
ordinary closed surface, the regions are "this side"
and the "other side", but the partition exists just the
same. I bring this up because the partition idea helps
you see why the total flux penetrating any infinite
plane must equal zero. Just like with the sphere, any
line of flux that passes through the infinite sheet to
the other side (a positive count) must eventually make
its way back (a negative count), making the total
number of crossings equal zero. I'm now going to apply
this idea (finally) to your problem.
I want you to turn your mental picture so you are
looking at the side of the trace (a broadside view of
your soap bubble). Color the bubble pink. Now, pick
some particular line of magnetic flux that penetrates
the pink region. If it passes through the pink region
then there are two possibilities for how it returns to
its source (completing the loop): either it comes back
through the pink region, in which case it cancels
itself out contributing nothing to the total count of
flux penetrating the pink region, or it comes back SOME
OTHER WAY. The only other way back is through the
"white space" that you see above, below, and to the
sides of the apparatus. Therefore, if you erect a white
curtain above, below, and to the sides of the
apparatus, covering all the space you see that isn't
already pink (looking from your perspective like a
photographic negative of the pink region), and if the
sheet is anchored at its edges along the trace and
plane precisely coincident with the edges of the pink
soap bubble, then you may rightly conclude that any
flux that contributes to the total flux count in the
pink region must also penetrate the white sheet. In
other words, you can count the flux passing through the
pink region, or count the flux passing through the
white sheet, either way you get the same answer. This
property directly relates to the discussion above about
the infinite plane partitioning space. As long as the
pink and white surfaces, when combined, form an
infinite partition of space, the total flux through
that partition must be zero, ergo; the flux through the
pink and white surfaces must be the same. This is what
I think Andy was talking about when he said that if you
extended the area of integration to infinity you could
catch all the flux.
The total flux passing through the pink region in
reaction to a current on the trace of 1 amp is defined
as the inductance of the circuit formed by the trace
and its associated reference plane.
I hope my rather lengthy discussion helps you sort out
some of the paradoxes associated with magnetic-field
integration.
Buried in the definition of inductance is the
assumption that current always assumes minimum-
inductance distribution. We say, "Current always
follows the path of least inductance", or more
precisely, "Current at high frequencies, if not altered
by significant amounts of resistance, always assumes a
distribution that minimizes the inductance of the loop
formed by the signal and return paths". If you put
something in the way of your current that alters the
distribution of current on the return path (like a hole
in the reference plane), then the current assumes some
alternate distribution that must necessarily raise the
inductance of the configuration (moving to any
distribution other than the minimum-inductance
distribution must necessarily raise the inductance).
Regarding your interest in visualizing the distribution
of current in a "least-inductance" configuration, let
me propose an analogy that I find quite helpful in
working through that problem.
First, replace your dielectric medium (the space
between the trace and reference plane) with a slightly
resistive material. I like to imagine salt water
occupying that space. Leave the trace open-circuited at
both ends, and apply 1-V DC to the trace. A certain
pattern of current will flow through the salt water to
the reference plane. I'll bet you could draw a picture
showing the pattern of current flow in this situation.
Start with a cross-sectional view of the trace. Suppose
you use 100 lines for the picture, each line
representing a certain fraction of the total current.
Each line emanates from the trace and terminates on the
plane (unlike magnetic lines of force these current
density lines have beginnings and endings). A great
density of lines will flow directly between the trace
and plane, with the lines feathering out to lower and
lower densities as you work your way further from the
trace. The lines always leave the surface of the trace
in a direction perpendicular to the surface of the
trace, and land perpendicular to the reference plane.
Here's why I like this exercise: Your picture of the DC
current flow exactly mimics the picture of lines of
electric flux in a dielectric medium operated at high
frequency. I find many people have no difficulty
imagining how DC currents would behave in salt water,
which turns out to be precisely the same problem as
figuring out how AC currents behave in a dielectric
medium.
Now we get to the part of this discussion about the
density of current in the reference plane. Your
electric-field picture shows a great density of current
flowing from trace to plane at a position directly
underneath the trace, and less and less density of
current flowing to positions on the plane remote from
the trace. This picture shows precisely how the current
gets from trace to plane (i.e., it flows through the
parasitic capacitance between trace and plane). If you
assume that once the current arrive on the plane it
flows parallel to the trace (making the cross-sectional
picture the same at each position along the trace, as
required by symmetry), then you can see that the
picture also shows the density of current flowing on
the plane as a function of position. Most of the
current flows on the reference plane right under the
trace, with less and less as you move away from the
trace (it happens to fall off approximately
quadratically for microstrips, even faster for
striplines).
Of course, you are going to want to know "why" current
should behave in such a manner. The principle in
question here 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, spurious magnetic fields in the middle of
an otherwise quiet region (unless you believe in vacuum
fluctuations, which is a different subject
entirely...).
The stored energy for inductive problems is:
E = (1/2)*L*(I^^2),
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 vary in direct proportion to one
another. Therefore, the distribution of current on the
reference plane that minimizes the total stored
magnetic energy and the distribution of current that
minimizes the inductance are one and the same.
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.
By assuming the current is *NOT* in the minimum-energy
distribution you can demonstrate the existence of a
mode of current that, if it got started, would rapidly
leads to a lower-energy state. Presumably such a mode
would start up immediately, so that any state but the
lowest-energy state is unstable. This demonstration
would convince you of the absurdity of the non-minimum
energy situation only if you also intuitively believe
that nature is not absurd. Further discussion of *that*
issue is probably best left to physicist-philosophers.
I hope this discussion is helpful to you, and doesn't
just stir up other doubts.
For further reading, try the following articles: "High-
Speed Return Signals", "Return Current in Plane",
"Proximity Effect", "Proximity Effect II", "Proximity
Effect III", and "Rainy-Day Fun", (see www.sigcon.com ,
under "archives", look for the alphabetical index).
Best Regards,
Dr. Howard Johnson
______________________________________________________
High-Speed Signal Propagation: Advanced Black Magic
is here! This is an all-new sequel to (not just an
update of) the classic book High-Speed Digital
Design: A Handbook of Black Magic. See preface and
table of contents at www.sigcon.com , under "books".
Check for it at www.barnesandnoble.com , or
www.amazon.com , ISBN 013084408X.
Dr. Johnson will present an all-new two-day seminar
Advanced High-Speed Signal Propagation in San Jose,
CA, October 27-28, 2003. Make a reservation for a
private on-site version of this course by calling
the Signal Consulting business office at
509.997.0750. A full schedule of cities and dates
appears at: www.sigcon.com .
Questions & Comments: all students who attend High-
Speed Digital Design or Advanced High-Speed Signal
Propagation seminars can talk directly with Dr.
Johnson about any signal integrity issues that may
interest them.
If you have an idea that would make a good topic for
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