Page 339 - Engineering Electromagnetics, 8th Edition

CHAPTER 10 Transmission Lines 321

The phasor voltage at the load is now the sum of the incident and reflected voltage
phasors, evaluated at z = 0:

V L = V 0i + V 0r (71)
Additionally, the current through the load is the sum of the incident and reflected
currents, also at z = 0:
1 V L 1
I L = I 0i + I 0r = [V 0i − V 0r ] = = [V 0i + V 0r ] (72)
Z 0 Z L Z L
We can now solve for the ratio of the reflected voltage amplitude to the incident
voltage amplitude, defined as the reflection coefficient, :

V 0r Z L − Z 0
≡ = =| |e jφ r (73)
V 0i Z L + Z 0
where we emphasize the complex nature of —meaning that, in general, a reflected
wave will experience a reduction in amplitude and a phase shift, relative to the inci-
dent wave.
Now, using (71) with (73), we may write

V L = V 0i + V 0i (74)
from which we find the transmission coefficient, defined as the ratio of the load voltage
amplitude to the incident voltage amplitude:

V L 2Z L
τ ≡ = 1 + = =|τ|e jφ t (75)
V 0i Z 0 + Z L
A point that may at first cause some alarm is that if is a positive real number,
then τ> 1; the voltage amplitude at the load is thus greater than the incident voltage.
Although this would seem counterintuitive, it is not a problem because the load current
will be lower than that in the incident wave. We will find that this always results in
an average power at the load that is less than or equal to that in the incident wave.
An additional point concerns the possibility of loss in the line. The incident wave
amplitude that is used in (73) and (75) is always the amplitude that occurs at the
load—after loss has occurred in propagating from the input.
Usually, the main objective in transmitting power to a load is to configure the
line/load combination such that there is no reflection. The load therefore receives all
the transmitted power. The condition for this is = 0, which means that the load
impedance must be equal to the line impedance. In such cases the load is said to be
matched to the line (or vice versa). Various impedance-matching methods exist, many
of which will be explored later in this chapter.
Finally, the fractions of the incident wave power that are reflected and dissipated
by the load need to be determined. The incident power is found from (64), where this
time we position the load at z = L, with the line input at z = 0.
1 V 0 V 0 ∗ 1 |V 0 | 2 −2αL
e
P i = Re e −2αL jθ = e cos θ (76a)
2 |Z 0 | 2 |Z 0 |