CROSSTALK - Advanced Signal Integrity for High-Speed Digital Design - page 154

The same approach will give the wave equation for the current:

d 2 i
dz 2 + ω 2 L 0 C 0 i =

0 (4-25)

Equations (4-24) and (4-25) should be familiar as the wave equations for a
uniform lossless transmission line.

4.2.2 Coupled Wave Equations

Our goal now is to generalize equations (4-24) and (4-25) to the n -coupled-line
case. Figure 4-8 describes the circuit for a subsection of a pair of coupled lines.
Our derivation follows the same method that we used above for the isolated
transmission line. We start by applying KVL to develop equation (4-26). Note
that the voltage drop on line 1 depends on the mutual inductance L M and the
current on the adjacent line i 2 (z) , in addition to the line self-inductance L and
driving current i 1 (z) . We can also create a corresponding expression (4-27) for
the voltage drop on line 2:

v 1 (z) − v 1 (z + dz) =− jωL 0 i 1 (z) dz − jωL M i 2 (z) dz
=− jω [ L 0 i 1 (z) + L M i 2 (z) ] dz

(4-26)

v 2 (z) − v 2 (z + dz) =− jω [ L 0 i 2 (z) + L M i 1 (z) ] dz (4-27)

We can write the equations for the voltage drop across the coupled subcircuit in
compact matrix form, where the boldface symbols represent the compact matrix:

d v
dz =− jω Li

(4-28)

where

v 1 (z) − v 1 (z + dz)
v 2 (z) − v 2 (z + dz)

d v
dz =

1
dz

taken in the limit as dz → 0

L 0

L 0

L M

L =

L M

i 1 (z, t)
i 2 (z, t)

i =

Applying the isolated line method to the current change caused by the capaci-
tances of the coupled line yields

i 1 (z) − i 1 (z + dz) =− jω(C g + C M )v 1 (z) dz + jωC M v 2 (z) dz
=− jω [ (C g + C M )v 1 (z) − C M v 2 (z) ] dz

(4-29)