# [SI-LIST] Re: si-list Digest V3 #194

• From: "Dr. Howard Johnson" <howiej@xxxxxxxxxx>
• To: "Si-List@xxxxxxxxxxxxx" <si-list@xxxxxxxxxxxxx>
• Date: Tue, 22 Jul 2003 10:58:27 -0700

```Those of you interested in magnetic field theory may find
Sainath's questions about the integration of magnetic flux a
fascinating subject; others may find this a good time to
step out for a cup of tea...

Dear Sainath,

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 as
a modern reprint of an old Dover version, circa 1954. I read
a copy of the work in preparation for writing my latest
book, "High-Speed Signal Propagation", and found it most
enlightening.

>From the preface of Maxwell's 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".

It's the "lines of force" concept that makes everything
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 also 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 dissappear 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 really 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 really 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.

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 an 2-dimensional
surface with infinite extent acts kinds of like a closed
surface, in that it partitiions 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
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 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 errect 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 anchored at its edges along the
trace and plane precisely coincident with the edges of the
pink soap bubble, 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 this 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 which 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 the exact distribution of current
in the "least-inductance" configuration, let me propose an
analogy that I find quite helpful in working through that
problem. This analogy I've developed in the course of making
up laboratory demonstrations for my new class on Advanced
High-Speed Signal Propagation.

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
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 magetic 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 perpindicular to the
surface of the trace, and land perpindicular 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--and it's the same
problem 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, spurrious magnetic fields in the middle
of an otherwise quiet region (unless you believe in vaccuum
fluctuations, which is a different subject entirely...).

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 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.

"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

By assuming the current is *NOT* in the minimum-energy
distribution you can demonstrate the existance of a mode of
current that leads to a lower-energy state, but that
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 a lot of 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 http:\\sigcon.com, under "archives",
look for the alphabetical index).

Best regards,
Dr. Howard Johnson, Signal Consulting Inc.,
tel +1 509-997-0505,  howiej@xxxxxxxxxx
http:\\sigcon.com  -- High-Speed Digital Design articles,
books, tools, and seminars

-----Original Message-----
From: si-list-bounce@xxxxxxxxxxxxx
[mailto:si-list-bounce@xxxxxxxxxxxxx]On Behalf Of Sainath
Sent: Thursday, July 17, 2003 11:44 PM
To: andrew.c.byers@xxxxxxxxxxxxxx
Cc: si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] Re: si-list Digest V3 #194

Hi Andy,

Thanks again. I get the themes that inductance is a one
number affair
and current returns through the least inductance path. Is
there a

Let me borrow the following from your 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 infinite surface)? or any other value?
remains same
as what it was for the integrating surface below the trace?

Sainath

---------Included Message----------
>Date: Thu, 17 Jul 2003 17:37:12 -0700
>From: <andrew.c.byers@xxxxxxxxxxxxxx>
>To: <gigabit@xxxxxxxxxx>
>Cc: <si-list@xxxxxxxxxxxxx>
>Subject: RE: [SI-LIST] Re: si-list Digest V3 #194
>
>Hello Sainath,
>
>Clearing up some terminology here.
>
>"Least inductance" refers to the path that the current will
travel
because
>it has the least inductance of all possible paths in the
system.
Current
>will never choose an alternate path of "most inductance".
BUT you can
have a
longer.
For
>example a two wire line with no ground plane where the
wires are
extremely
>far apart. Huge loop, huge inductance. But still the
smallest loop for
that
>system. For a microstrip, a path of More Inductance would
be if there
were a
>gap in the ground plane under the microstrip line. The
current would
be
>forced to diverge around the gap. This path would be more
inductive
than a
>solid ground plane, but the current would still be
following the path
of
>least inductance for that particular case.
>
>The main challenge in most systems I've dealt with is
making sure that
>return current paths have the least inductance possible.
The simplest
way to
>do this is go differential. Then you carry your virtual
ground with
you
>everywhere. If single ended, then be very conscious about
where the
return
>currents flow and try to provide a short path. Plenty of
this
>
>Not sure if this clears up your last question, hope it
helps though.
>
>- Andy
>
>
>
>-----Original Message-----
>Sent: Thursday, July 17, 2003 4:01 PM
>To: Byers, Andrew C
>Cc: si-list@xxxxxxxxxxxxx
>Subject: RE: [SI-LIST] Re: si-list Digest V3 #194
>
>
>Andy,
>
>Thanks. I appreciate the extra effort to explain detail of
integration.
>In short, you've explained the current loop formed by a
signal path on

>trace and signal return path beneath the trace and on the
ground plane.

>Such a return path, with its minimum loop area, is widely
known to
>provide the path of "least" inductance for high-frequency
currents(for

>example, Black Magic book). If inductance is thought of as
one number,

>what does "least inductance" refer to? Which is the path of
"most"
>inductance for the microstrip? No doubt, I'm missing
somethig.
>
>Sainath
>
>---------Included Message----------
>>Date: Thu, 17 Jul 2003 10:02:49 -0700
>>From: <andrew.c.byers@xxxxxxxxxxxxxx>
>>To: <gigabit@xxxxxxxxxx>, <beneken@xxxxxxxxxxxx>
>>Cc: <si-list@xxxxxxxxxxxxx>
>>Subject: RE: [SI-LIST] Re: si-list Digest V3 #194
>>
>>Sainath,
>>
>>As Thomas pointed out, inductance is the ratio of magnetic
flux to
>current
>>in the conductor. Magnetic flux is the integral of B dot
dA, or the
>magnetic
>>field [dot product] the surface you are integrating over.
The "dot
>product"
>>is the same as multiplying the B-field by the area by the
cosine of
>the
>>angle between the B-vector and the normal to the area. So
if the
>B-vector is
>>perpendicular to the area surface, then the B-vector is
parallel to
the
>unit
>>normal vector of the area surface, cosine of this zero
degree angle is

>1,
>>and you simply multiply B*area. Here's an example to
illustrate.
>>
>>You have a rectangular metal trace over a ground plane,
length in the
>>z-direction, height in the y, width in the x. Stretch a
rectangle in
>the yz
>>plane between the trace and the ground plane. Make it any
length
>(smaller if
>>you are simulating with EM tool). If we assume perfect
conductors (ie

>no
>>internal-conductor magnetic fields), then all of the
magnetic field
>>associated with that signal trace will pass through this
rectangle. It

>is
>>kind of like a net. Magnetic field lines always have to
end up in the

>same
>>place they started, completing the circle. Also, in this
configuration,
>all
>>your field lines are perpendicular to the integrating
rectangle. So
>>inductance is flux/I = B*A/I. In this case, you will
actually have
z-length.
>>
>>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.
>>By placing it between the signal line and the return path,
you capture

>all
>>the field lines. So you have one number for inductance if
you account

>for
>>all the B field lines. An inductance "distribution" would
indicate
that
>you
>>are not catching all the magnetic field lines with your
integrating
>surface.
>>
>>
>>This might open up a talk about internal inductance, when
you have
>magnetic
>>field lines (ie current) INSIDE the conductors. As
frequency
increases,
>the
>>current crowds to the surface, and the internal inductance
diminishes.

>But
>>at lower or intermediate frequencies, this internal
inductance can be

>a
>>contributing factor. For PCB's, this is typically in the
low MHz
range.
>But
>>for square conductors on silicon, measuring a few microns
wide and a
>few
>>microns high, the internal inductance might have to be
considered up
>to
>>several GHz. Does this affect you? Do you electrical
models consider
>this
>>effect? How about internal inductance of the ground plane?
Interesting

>stuff
>>here.
>>
>>Salud,
>>
>>Andy Byers
>>
>>-----Original Message-----
>>Sent: Thursday, July 17, 2003 9:25 AM
>>To: beneken@xxxxxxxxxxxx
>>Cc: si-list@xxxxxxxxxxxxx; gigabit@xxxxxxxxxx
>>Subject: [SI-LIST] Re: si-list Digest V3 #194
>>
>>
>>Thomas,
>>
>>Thank you. I agree, you get one value of inductance for
one
>integration.
>>If you repeat this for a number of 'concentric spheres',
you will get
a
>
>>number of inductances- ranging from minimum to maximum.
Does that make

>
>>sense?
>>
>>Sainath
>>
>>---------Included Message----------
>>>Date: Thu, 17 Jul 2003 12:04:57 +0200
>>>From: "Thomas Beneken" <beneken@xxxxxxxxxxxx>
>>>To: <si-list@xxxxxxxxxxxxx>
>>>Subject: [SI-LIST] Re: si-list Digest V3 #194
>>>
>>>Hello Sainath,
>>>
>>>inductance is the proportional factor between the current
and the
>>magnetic
>>>flux. So far Your idea is ok. But calculating magnetic
flux from
>>magnetic
>>>field requires an integration across a closed surface
surrounding
the
>>>conductor carrying the current. So - as You see - You
will not get a
>>>inductance distribution over conductor length but only an
integral
>>value for
>>>the conductor enclosed in the chosen sphere.
>>>
>>>Sorry,
>>>Thomas
>>>
>>>> Msg: #12 in digest
>>>> Date: Wed, 16 Jul 2003 11:55:35 -0800
>>>> Subject: [SI-LIST] Microstrip Inductance
>>>>
>>>> Hello experts:
>>>>
>>>> For a microstrip, we know the magnetic field
distribution(for
>>>> example,
>>>> Fig. 2.3 Stephen Hall's book) and current density
>>>> distribution(Fig. 4.5
>>>> same book). Given these, how would you obtain the
inductance
>>>> distribution?
>>>>
>>>> Sainath
>>>
>>>
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