CROSSTALK - Advanced Signal Integrity for High-Speed Digital Design - page 172

voltage difference across the differential segment ( dz ) which travels back toward
the source:

= L M dz di 1
dt

v b − v f (4-54)

In the same way that we defined a capacitive coupling coefficient, we define the
inductive coupling coefficient as the ratio of the mutual inductance between lines
to the self-inductance of the line:

L M
L 0

K L ≡ (4-55)

Application of Ohm's law at the driven end of line 1 ( i 1 = v 1 /Z 0 ) yields

L 2 M (C g + C M )
L 0

dz L M
Z 0

dv 1
dt =

L 0
L 0

dv 1
dt

v b − v f = dz

= dz L 0 (C g + C M ) L 2 M
L 0

dv 1
dt

which reduces to another expression relating the forward and backward coupled
noise to a coupling coefficient, in this case the inductive coupling coefficient:

dz K L
ν p

dv 1
dt

v b − v f = (4-56)

Since (4-53) and (4-56) give us two expressions with two unknowns, we can
solve them for v b and v f and take dz → 0 in the limit, to get

dv f
dz =

K C − K L
2 ν p

dv 1
dt

(4-57)

dv b
dz =

K C + K L
2 ν p

dv 1
dt

(4-58)

Integrating (4-57) from z = 0to z = l gives an expression

Forward Crosstalk

for forward crosstalk:

1
2 (K C − K L )

l
ν p

dv 1
dt

v f = (4-59)

By approximating dv 1 / dt as the ratio of the voltage swing v and a 10 to 90%
rise time t r , we have our final expression for the forward crosstalk:

1
2 (K C − K L )

l
ν p

v
t r

v f = (4-60)