RE: [SI-LIST] : Differential Pair Theory- an answer

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From: Eric Bogatin ([email protected])
Date: Mon Nov 15 1999 - 19:33:39 PST


I'll take a stab at answering some of your diff pair questions below. They
are actually related to a paper that Mike Resso and I have just drafted on
TDR analysis of diff pairs for DesignCon 2000, and also topics I cover in
the classes I teach. You can find out about the public courses on my web
site at There are two in early Dec that
go into much more detail on diff impedance than I have included below.

I have tried to develop a simple intuitive explanation of diff pair and diff
impedance, that is understandable, but doesn't compromise the underlying
engineering principles. I have added some specific comments below after your
questions. Of course, this is only the quick version- coincidently, I have a
more complete paper I am currently drafting that will be posted on my web
site at the end of this month.


Eric Bogatin
Training for Signal Integrity and Interconnect Design
26235 W. 110th Terr.
Olathe, KS 66061
v: 913-393-1305
f: 913-393-1306
pager: 888-775-1138
e: [email protected]
web: <>

> -----Original Message-----
> From: [email protected]
> [mailto:[email protected]]On Behalf Of Chris Bobek
> Sent: Monday, November 15, 1999 3:28 PM
> To: Si-list
> Subject: [SI-LIST] : Differential Pair Theory
> Hi,
> I am trying to understand how differential pairs behave and how you
> terminate them. I've read app notes and books that tell you how to
> terminate them, but I don't know how they derived the values and
> methodology. I would just like a general understanding of what's going
> on. I understand how and why termination works on normal, single-ended
> lines.
> As an example, suppose you had a 100ohm, unshielded, twisted pair cable
> that ran for 5000 feet between two boards. Once the pair entered the
> board, it was routed over several inches to a differential receiver.
> The two boards do not share a ground (suppose they are battery powered).
> 1) How did I know to route the traces on the board as two 50 ohm
> traces?
> 2) How did I know to terminate the pair by placing a 100ohm resistor
> between the two traces at the receiver?
> 3) Could I terminate each line separately as if they were two,
> single-ended lines (say, using a 50ohm resistor to ground on each line
> at the receiver?)
> 4) Where does the return current flow as the signal is launched down
> the line from the driver, across the twisted pair, across the diff pair
> on the board, and to the receiver?

The short answer is, that to first order, the two traces on the PCB are
really a single ended transmission line. The single ended impedance between
the two signal lines, as a coplanar transmission line, is what we also call
its differential impedance. It is strongly affected by the proximity of the
floating metal plane below. Its single ended impedance will be about 100
Ohms, if the plane is in proximity so as to create a characteristic
impedance of one trace to the plane of 50 Ohms. This may not be a very
comfortable answer to some, so here's a bit more detail:

You seem to have a good feel for the impedance environment of the twisted
pair. You recognize that the twisted pair lines represent a single ended
transmission line, with characteristic impedance of 100 Ohms. To connect
them to the two traces on the second board, you would merely solder one
twisted line wire to the signal trace of one of the board traces, and the
other wire to the second signal line trace pair. The transmission line
composed of the two traces on the board is also a single ended transmission
line. However, the presence of the "floating metal plane" below the traces
strongly influences the characteristic impedance of the transmission line
composed of the coplanar printed traces above the plane.

Suppose the plane were not there, and you had the two printed traces on the
FR4 substrate. They would make up a coplanar transmission line. If they were
5 mil line width and separated by 10 mils, let's say, they would have a
single ended characteristic impedance of about 150 Ohms. Now, slowly bring
the floating metal plane up from below. The presence of the plane will
decrease the characteristic impedance of the two printed traces. The
capacitance per length between the traces will increase, since there is now
an additional capacitive coupling path between the two traces through the
floating metal plane, and the loop inductance per length between them will
decrease because of induced eddy currents in the floating metal plane.

First think about the capacitive coupling through the floating plane. When
the plane is far away, the capacitance between the traces is dominated by
the field coupling from one trace to another, call it C21. The series
capacitance between one trace and the floating plane, C11, and then the
other trace and the floating plane, C22, is small. As the plane is brought
closer, C21 will decrease very slightly, but C11 will increase pretty
rapidly. When the separation to the bottom trace is about on the order of
the trace to trace separation, the total capacitance between the traces
begins to be dominated by the trace to plane separation. So, when the plane
is in proximity to the traces, the coupling between traces is not so
critical, it is the trace to plane separation that determines the
capacitance between the traces, and hence the characteristic impedance
between the two traces.

To first order, the characteristic impedance of the two coplanar traces is
going to be related to 1/Ctotal. And Ctotal is 1/2 C11 + C12. When the plane
is close and the traces are far apart from each other, Ctotal is mostly 1/2
C11. If C11 corresponds to a 50 Ohm line, you can see why the single ended
impedance of the coplanar is 100 Ohms, since there are two of these in
series. When the traces are spaced close together, Ctotal gets larger by the
trace to trace coupling capacitance, and the single ended impedance of the
coplanar lines (differential impedance) decreases. That's why diff impedance
is affected by the trace to trace coupling, but is dominated by the trace to
plane impedance. In most situations, C11 is much larger than C12.

In the case of a standard board stack up, with the floating plane positioned
so as to give the two signal lines a line to plane single ended impedance of
50 Ohms, even at a separation = line width, C11 is about 10x C12- so the
single ended impedance of the coplanar lines is really dominated by the
coupling capacitance through the floating plane. The details of this
analysis and the connection to modes, even and odd impedance and
differential impedance is covered in more detail in the classes I teach.

If you had a coplanar transmission line, with a characteristic impedance of
100 Ohms, and you had to terminate it, you would not blink before you said,
just connect a 100 Ohm resistor between the signal and return path. That's
just like the case above, its just that the characteristic impedance of the
coplanar line is reduced by the proximity of the metal plane. A resistor
across the two coplanar lines is all it takes. In this perspective, you can
also see that connecting two, 50 Ohm resistors from the signal paths to the
floating metal plane is the same as connecting the two 50 Ohm resistors
between the signal lines directly. Its still 100 Ohms total between the two
signal lines.

Now let's look at the current distribution. It's all related to the induced
eddy currents in the floating plane. When the plane is in close proximity,
the propagating signal front, which is the voltage between the two coplanar
signal traces (with its associated current based on the impedance the signal
sees), will induce a mirror image propagating current loop in the plane. The
closer the traces are to the floating metal plane, the larger the induced
current is in the plane. This also contributes to decreasing the
characteristic impedance of the two coplanar traces.

But look at the induced currents in the plane. If there is a current, I, in
one trace, there is -I in the other trace. However, there is an induced
current of -I in the floating metal plane directly below the first trace,
and I in the floating plane below the second trace. Even though there is no
DC connection to the floating plane, there will be induced eddy currents in
it, and these will affect the characteristic impedance of the coplanar
traces. Mike Resso and I will be showing a really neat, simple demo of this
at the DesignCon 2000 Conference.

If you were to do something to prevent these eddy currents from flowing,
such as cut gaps transverse to the traces in the floating plane (a common
technique to minimize eddy currents), you will be increasing the
characteristic impedance of the coplanar single ended transmission line. It
will no longer be 100 Ohms, it may go back up to 140 Ohms, for example.

Think about the current distribution in the two traces and the plane. In
cross section, it sure looks like the induced eddy current is a "return
current" for the signal trace above it. But, I don't like using this term
when there are multiple current paths. All the currents contribute to
affecting the characteristic impedance the signal sees. If they all aren't
carefully controlled, the impedance the signal sees will change, and signal
integrity will be affected. Keeping track of the various currents involves a
more complicated analysis that introduces the concept of modes.

Now think about what is going on at the source board. A signal is launched
into the coplanar pair. But, since they have a plane underneath, their
single ended characteristic impedance is only 100 Ohms. While the coplanar
traces have the uniform plane beneath them, they will have this constant
impedance, and the induced eddy current distributions will be distinct under
each trace. If the plane should be suddenly ended, and the 100 Ohm twisted
pair cable now attached to the end of the coplanar lines, the signal will go
from one 100 Ohm environment to another 100 Ohm environment. Of course,
there is also fringe fields and other sources of discontinuity at the
transition. However, it can be kept small. What happened to the induced eddy
currents in the plane below? The loop was closed and the currents blended
together and cancelled out at the edge.

The same thing happens in reverse on the receiver board- the induced eddy
current loop starts when the signal reaches the edge of the board,
propagates down with the signal and comes together under the SMT 100 Ohm
terminating resistor connecting the two signal traces.

I apologize for taking so long in this explanation. If anyone is interested
in the details, I invite you to attend the short courses I have coming up in
San Jose in Dec. Alternatively, I will be posting an article in my December
downloads, "What really is Differential Impedance?" This is along with the
one on "What really is characteristic impedance?" If you sign in on my web
site, you will be automatically notified when they are posted.

as always, I welcome comments.


> Thank you very much,
> Chris
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