DIFFERENTIAL SIGNALING - Advanced Signal Integrity for High-Speed Digital Design - page 300

ill effects of nonideal reference planes. Additionally, when the fields are confined
between the conductors (i.e., strongly coupled to the virtual reference plane), they
are less apt to fringe out to other signals, which helps reduce crosstalk.

7.4 PROPAGATION OF MODAL VOLTAGES

For a multiconductor system, in Chapter 4 we discussed how all digital signal
states are composed of linear combinations of the modal voltages. For a single
differential pair, the two modes are the even and odd modes. Ideally, the differ-
ential pair is driven with lines 1 and 2 180 ◦ out of phase so that all of the energy
is contained solely in the odd mode. However, if common-mode noise is cou-
pled onto the differential pair, some energy will exist simultaneously in the even
mode. This is easy to show mathematically using modal analysis, as discussed
in Section 4.4. For example, consider a differential pair with line voltages v D +
and v D − , which are composed of linear combinations of the odd- and even-mode
voltages:

v D +
v D −

[ T V ] v odd

v even

= (7-2)

where [ T V ] is a matrix containing the eigenvectors of the product LC as devel-
oped in Section 4.4.1.
If the differential pair is driven exactly 180 ◦ out of phase and there is no noise
present, the odd- and even-mode voltages are calculated, where

0 . 707

0 . 707 0 . 707

[ T V ] =

− 0 . 707

from Example 4-4:

1 0 . 707 v odd v even

1 0 . 707 0 . 707

= (7-3)

− − 0 . 707

resulting in

v odd = 1 . 41443 V

v even = 0V

which proves that all the energy is contained on the odd mode.
However, if common-mode noise is present, energy is introduced into the even
mode. If v noise is the voltage noise introduced to each leg of the pair, the even
and odd modes are calculated:

+ v noise 0 . 707 v odd v even

1 + v noise 0 . 707 0 . 707

= (7-4a)

− 1 − 0 . 707

v odd = 1 . 41443 V (7-4b)

v even = 1 . 41443 v noise V (7-4c)