IDEAL TRANSMISSION-LINE FUNDAMENTALS - Advanced Signal Integrity for High-Speed Digital Design - page 80

telegrapher's equations (3-12) and (3-18):

∂v(z,t)
∂z

=− L ∂i(z,t)
∂t

(3-12)

∂i(z,t)
∂z

=− C ∂v(z,t)
∂t

(3-18)

Assuming that the digital signals can be decomposed into sinusoidal harmon-
ics using the Fourier transform, the telegrapher's equations can be expressed in
time-harmonic form where the voltage and current have the forms v(t) = V 0 e jωt
and i(t) = I 0 e jωt . Similar representations for the time-harmonic fields were dis-
cussed in Section 2.3.3. Consequently, the time-harmonic forms of the telegra-
pher's equations are

dv(z)
dz

=− jωLi(z) (3-25)

di(z)
dz

=− jωCv(z) (3-26)

Taking the derivative of (3-25) with respect to z produces

d 2 v(z)
dz 2

di(z)
dz

=− jωL (3-27)

and substituting (3-26) into (3-27) allows us to write an equation only in terms
of voltage,

d 2 v(z)
dz 2

+ ω 2 LCv(z) =

0 (3-28)

which is the loss-free transmission-line wave equation for voltage . Equa-
tion (3-28) is a second-order differential equation with the general solution
given by

v(z) = v(z) + e − jzω √ LC + v(z) − e jzω √ LC

(3-29)

The term v(z) + e − jzω √ LC describes the voltage propagating down the transmis-

+ z -direction and v(z) − e jzω √ LC describes the voltage propagating sion line in the

in the − z -direction. Note the similarity of (3-29) and (2-41), which is the solu-

tion equivalent to the wave equation for the electric field. In Section 2.3.4, a
propagation constant for a wave traveling in an infinite medium was defined that
completely describes the medium where the electromagnetic wave is propagating:

= α + jβ

γ (2-42)