CROSSTALK - Advanced Signal Integrity for High-Speed Digital Design - page 177

are then used to drive the isolated lines in a simulation or hand analysis such as
a lattice diagram. The simulated or calculated response is then converted back
from modal voltages and currents to line voltages and currents. The material that
follows describes the modal decomposition and analysis technique.

4.4.1 Modal Decomposition

If the matrices LC and CL can be diagonalized, the diagonal components are
orthogonal to each other. This allows us to represent the line voltages as a linear
combination of modal voltages. To do this it is first necessary to determine the
relationship between line voltages and modal voltages. We can use the coupled
voltage wave equation (4-33) to derive a voltage transform matrix T v ;

v = T v v m (4-65)

where T v is a matrix that contains the eigenvectors of LC , which translate normal
line voltages in v to modal voltages in v m . Once we have done so, we can simulate
the system as n isolated lines using v m as input rather than as n coupled lines
with v as input. Similarly, equation (4-34) for the coupled current wave is used
to derive T i :

i = T i i m (4-66)

where T i is a matrix that contains the eigenvectors of CL , which translate normal
line currents in i to modal currents in i m . From (4-28) and (4-31), the matrix
equations for an n × n system become

v

i

v

i

d
dz

− jω L
− jω C0

0

= (4-67)

To derive the modal inductances, we start with equation (4-28) and substitute the
modal voltages, currents, and transform matrices for the line voltage and current:

d
dz ( T v v m ) =− jω LT i i m

(4-68)

Multiplying both sides by T − v gives

d

dz ( T − v T v v m ) =− jω T − v LT i i m
d v m

(4-69)

dz =− jω T − v LT i i m

The modal inductance matrix is defined as

T − v LT i L m = (4-70)