**From:** Josh Nickel (*[email protected]*)

**Date:** Wed Nov 17 1999 - 18:42:55 PST

**Next message:**Bob Davis: "RE: [SI-LIST] : Cables with driven shields, was "FCAL DB9 cable shield""**Previous message:**Chris Cheng: "RE: [SI-LIST] : FCAL DB9 cable shield"**In reply to:**Mike Jenkins: "Re: [SI-LIST] : Differential Pair Theory"**Next in thread:**Mike Degerstrom: "Re: [SI-LIST] : Differential Pair Theory"**Maybe reply:**Mike Degerstrom: "Re: [SI-LIST] : Differential Pair Theory"

On Mon, 15 Nov 1999, Mike Jenkins wrote:

*> Chris,
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*>
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*> I'm sure you will get multiple responses to your question. Here's
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*> a theoretical view. (Apologies to the math-phobics out there.)
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*>
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*> Looking onto a single-ended line is a one-port network. Differential
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*> turns this into a two-port. The dif'l voltage is defined typically
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*> as V(port1) - V(port2) and the common mode as [V(port1) + V(port2)]/2.
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*> Some simplifying assumptions (which may or may not be true and should
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*> be checked for each application) are:
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*> 1) The dif'l pair is symmetric (i.e., port1 and port2 can be
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*> interchanged and the 2-port looks the same).
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*> 2) The signal is dif'l only (i.e., V(port1) = - V(port2)). Your
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*> example would probably fail this assumption badly, as ground
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*> loops would induce common mode signals.
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*>
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*> Back to the 2-port thing....The input impedance is now a 2x2 matrix
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*> rather than a single number:
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*>
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*> | V(port1) | | Z11 Z12 | | I(port1) |
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*> | | = | | * | |
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*> | V(port2) | | Z21 Z22 | | I(port2) |
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*>
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*> Z12 = Z21 for passive networks like this dif'l line. The dif'l
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*> impedance is Z11+Z22-2*Z12. If the pair is symmetric, Z11 = Z22,
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*> so the dif'l impedance is 2(Z11-Z12). On your board, if the two
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*> traces are separate, Z12 is neglible, so Z11=50 ohms (100 ohms dif'l).
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*> On your shield-less cable, Z11 and Z12 are large, but Z11-Z12=50 ohms.
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*> These two structures (PCB and twisted pair) cannot be matched both
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*> for dif'l and common mode signals.
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*>
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*> Hope that gives you a framework to start with.
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*>
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*> Regards,
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*> Mike
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*>
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(*** switch to fixed font mode ***)

Mike:

I wanted to clarify your last comment; I was not sure whether you

meant "both modes cannot be matched" or the "two structures cannot both be

matched".

In the case of two-coupled microstrip lines over a ground plane, such a

termination may be found which matches both modes. If we consider the

above equation, the Z matrix may be diagonalized (under certain conditions

- I'm thinking of microstrip in particular) to yield a modal

characteristic impedance matrix. Then the Ohm's law state equation

above may be transformed to the modal Ohm's law equation :

| V(mode1) | = | Z_char_mode1 0 | | I(mode1) |

| | = | | * | |

| V(mode2) | = | 0 Z_char_mode2 | | I(mode2) |

If both lines are terminated to ground with Z_char_mode1, the "even" mode,

then we will match mode 1, meaning that any modal voltage content of an

arbitrary signal incident upon the termination will experience no

reflection. Same holds for mode 2, the "odd" mode.

However, both modes may be matched if we can find a termination that

equals the characteristic impedance matrix (Z_char) from Ohm's state

equation. Such a termination would necessarily be a 3-element network,

realized by the following procedure:

Y_termination = Inverse(Z_char) = | Z22 -Z12 |

| | / (Z11*Z22-Z12*Z21)

| -Z21 Z11 |

Thus, the matching network may be realized by:

terminating line 1 to ground with (Z11*Z22-Z12*Z21)/Z22 ohms

terminating line 2 to ground with (Z11*Z22-Z12*Z21)/Z11 ohms

interconnecting lines 1 and 2 with (Z11*Z22-Z12*Z21)/Z21 ohms

Note that the off-diagonal elements of the admittance matrix are negative,

by virtue of simple network theory. They are also equal by reciprocity.

This matching theory may be generalized to an arbitrary number of lines,

n, which are characterized by (n x n) distributed impedance and admittance

matrices or (n x n) characteristic impedance/admittance matrices. For n >

2, however, exact matching obviously becomes somewhat impractical, but if

termination interconnections are restricted to nearest neighbors the

approximation is usually pretty good (especially in microstrip). We've

done some research on this topic, so I felt it necessary to point out the

the possibilities of "multimode matching".

Josh G. Nickel

Graduate Reseach & Teaching Assistant

University of Illinois at Urbana-Champaign

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**Next message:**Bob Davis: "RE: [SI-LIST] : Cables with driven shields, was "FCAL DB9 cable shield""**Previous message:**Chris Cheng: "RE: [SI-LIST] : FCAL DB9 cable shield"**In reply to:**Mike Jenkins: "Re: [SI-LIST] : Differential Pair Theory"**Next in thread:**Mike Degerstrom: "Re: [SI-LIST] : Differential Pair Theory"**Maybe reply:**Mike Degerstrom: "Re: [SI-LIST] : Differential Pair Theory"

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