Re: differential impedance

Eric Wheatley (alterra@adnc.com)
Wed, 09 Apr 1997 18:26:58 -0700

Hello all,

At 11:26 AM 4/9/97 CDT, you wrote:
>
>We've seen specifications on differential impedance, but no one has
>been able to define what that number represents. We use the rule of
>thumb that the differential impedance is 2 times the odd mode
>impedance. Does that make sense? Can someone give me a real
>definition of differential impedance and it's relation to odd mode
>impedance?
>
>Thanks,
>
>--
>Weston Beal Signal Integrity Engineer
>beal@twisto.compaq.com beal@bangate.compaq.com
>"A little bit of knowledge is a dangerous thing"
>

It is *not* true that the differential impedance is twice the odd mode
impedance for a general two conductor transmission line, but if the line is
symmetrical then this rule is true.

I hope the following explanation is helpful:

An arbitrary transmission line with N conductor plus a ground has N
different TEM modes that can propagate in either direction (call it forward
and reverse) on the line. The line is characterized by an N x N impedance
matrix and an N x N admittance matrix. At any plane of interest there are N
total voltages (Vforward + Vreverse) and N total currents (Iforward + I
reverse) which are functions of time.

In a three conductor line of interest here (two signal lines and a ground)
there is a 2 x 2 impedance matrix and two TEM modes. Lets say the impedance
matrix is

Z11 Z12
[Z} =
Z21 Z22

----- Differential Mode Impedance -----

The differential mode is defined as the case where only a forward wave
exists between conductor 1 and ground and only a reverse wave of equal
amplitude exists between conductor 2 and ground. The currents in the ground
conductor are equal and opposite so there is no net current in the ground
conductor in this case. The current in conductor one i1 is equal and
opposite the current in conductor two i2. The voltage on conductor one v1
is equal and opposite the voltage on conductor two v2.

Since [V] = [Z] * [I} we get

v1 = Z11 * i1 + Z12 * i2
v2 = Z21 * i1 + Z22 * i2

Ssubtracting these two equations and using the fact that Z12 = Z21 gives

(v1-v2) = (Z11 - Z21) * i1 + (Z12 - Z22) * i2

Eliminating i2 and defining the differential voltage Vdiff gives :

Vdiff = (v1 - v2) = 2 * v1 = (Z11 + Z22 - 2 * Z12) * i1

The differential impedance is given by

Zdiff = Vdiff / i1 = Z11 + Z22 - 2 * Z12

If the line is symmetrical then Z11 = Z22 so Zdiff = 2 * (Z11 - Z12)

----- Common Mode Impedance -----

The common mode is defined as the case where only a forward wave exists
between conductor 1 and ground and only a forward wave of equal amplitude
exists between conductor 2 and ground. The currents in the ground conductor
are equal and additive so there is a net current in the ground conductor of
2 * i1 in this case. The current in conductor one i1 is equal to the
current in conductor two i2. The voltage on conductor one v1 is equal to
the voltage on conductor two v2.

Again since [V] = [Z] * [I} we have

v1 = Z11 * i1 + Z12 * i2
v2 = Z21 * i1 + Z22 * i2

Adding these two equations and using the fact that Z12 = Z21 gives

(v1 + v2) = 2 * v1 = (Z11 + 2 * Z21 + Z22) * i1

Defining the common mode voltage Vcomm gives :

Vcomm = v1 = (Z11 + Z22 + 2 * Z12) * i1 / 2

The common mode impedance is the ratio of the common mode voltage divided by
the total current in both conductors

Zcomm = v1 / (2 * i1) = (Z11 + Z22)/4 + Z12/2

If the line is symmetrical then Z11 = Z22 so Zcomm = (Z11 + Z12) / 2

----- Odd Mode Impedance -----

This concept applies to the same line conditions as for the differential
mode above, that is i1 = -i2. The odd-mode impedance is defined as the
ratio of the differntial mode voltage on one conductor divided by the
current in that conductor. Thus

Zodd1 = v1 / i1 = Z11 - Z12
Zodd2 = v2 / i2 = Z22 - Z21

These are not the same for the two conductors unless the line is symmetrical
in which case Zodd1 = Zodd2 = Z11 - Z12

----- Conclusion -----

Your rule of thumb that differntial impedance is 2 times the odd mode
impedance is not true in general but is true for a symmetrical line.