ELECTRICAL PROPERTIES OF DIELECTRICS - Advanced Signal Integrity for High-Speed Digital Design - page 285

Similarly, from equations (2-76), (2-59) and (3-1), we can write the capacitance
in the form

ε S

E ·

d s

Q
v =

C = (6-43b)

− a

E · l

d

Dividing (6-43a) by (6-43b) yields the following ratio:

G
C =

σ dielectric
ε

(6-44)

From (6-15) and (6-16b),

| δ | ε ω σ dielectric = tan

which is substituted into (6-44) to yield

G(ω) = tan | δ | ωC (6-45)

where G ( ω ) is the frequency-dependent conductance in units of siemens per unit
length, ω = 2 πf , and C is the capacitance per unit length for the transmission

line.
The characteristic impedance, which was defined in equation (3-33), can be
calculated by dividing the series impedance as defined by (5-68) by the parallel
admittance defined by (6-42) for a short section of transmission line of length z :

Z series
Y shunt

R ac

+ jωL total
G + jωC

Z 0 = = ohms (6-46)

( ohms ) 2

Note that the units in (6-46) are √ ohms /( 1 / ohms ) = ohms.

The propagation constant can be derived by inserting the complex values of
the series impedance and shunt admittance into the loss-free formula derived in
Section 3.2.4 in (3-30), which takes the form

=

+ jω √ LC =

(j ωL)(j ωC) = Z lossless Y lossless

γ = α + jβ = 0

Substitution of Z series and Y shunt in place of the loss-free values of the series
impedance and the parallel admittance yields the propagation constant for a trans-
mission line with a lossy dielectric and a conductor with a finite conductivity:

= α + jβ = Z series Y shunt = (R + j ωL)(G + jωC)

γ (6-47)