Page 328 - Engineering Electromagnetics, 8th Edition

310 ENGINEERING ELECTROMAGNETICS

lines. In such studies, the effect of the transmission line on any signal can be deter-
mined by noting the effects on the frequency components. This means that one can
effectively propagate the spectrum of a given signal, using frequency-dependent line
parameters, and then reassemble the frequency components into the resultant signal in
time domain. Our objective in this section is to obtain an understanding of sinusoidal
propagation and the implications on signal behavior for the lossless line case.
We begin by assigning sinusoidal functions to the voltage functions in Eq. (14).
Specifically, we consider a specific frequency, f = ω/2π, and write f 1 = f 2 =
V 0 cos(ωt + φ). By convention, the cosine function is chosen; the sine is obtainable,
as we know, by setting φ =−π/2. We next replace t with (t ± z/ν p ), obtaining
V(z, t) =|V 0 | cos[ω(t ± z/ν p ) + φ] =|V 0 | cos[ωt ± βz + φ] (26)

where we have assigned a new notation to the velocity, which is now called the phase
velocity, ν p . This is applicable to a pure sinusoid (having a single frequency) and will
be found to depend on frequency in some cases. Choosing, for the moment, φ = 0,
we obtain the two possibilities of forward or backward z travel by choosing the minus
or plus sign in (26). The two cases are:
V f (z, t) =|V 0 | cos(ωt − βz) (forward z propagation) (27a)

and
V b (z, t) =|V 0 | cos(ωt + βz) (backward z propagation) (27b)
where the magnitude factor, |V 0 |,is the value of V at z = 0, t = 0. We define the
phase constant β, obtained from (26), as

ω
β ≡ (28)
ν p
We refer to the solutions expressed in (27a) and (27b)as the real instantaneous
forms of the transmission-line voltage. They are the mathematical representations of
what one would experimentally measure. The terms ωt and βz, appearing in these
equations, have units of angle and are usually expressed in radians. We know that ω
is the radian time frequency, measuring phase shift per unit time, and it has units of
rad/s. In a similar way, we see that β will be interpreted as a spatial frequency, which
in the present case measures the phase shift per unit distance along the z direction.
Its units are rad/m. If we were to fix the time at t = 0, Eqs. (27a) and (27b)would
become
V f (z, 0) = V b (z, 0) =|V 0 | cos(βz) (29)

which we identify as a simple periodic function that repeats every incremental dis-
tance λ, known as the wavelength. The requirement is that βλ = 2π, and so

2π ν p
λ = = (30)
β f