Page 340 - Engineering Electromagnetics, 8th Edition

322 ENGINEERING ELECTROMAGNETICS

The reflected power is then found by substituting the reflected wave voltage into
(76a), where the latter is obtained by multiplying the incident voltage by :
2
1 ( V 0 )( V ) 1 | | |V 0 | 2 −2αL
∗
∗
0
e
P r = Re e −2αL jθ = e cos θ (76b)
2 |Z 0 | 2 |Z 0 |
The reflected power fraction at the load is now determined by the ratio of (76b)to
(76a):
P r 2
∗
= =| | (77a)
P i
The fraction of the incident power that is transmitted into the load (or dissipated by it)
is therefore
P t 2
= 1 −| | (77b)
P i

2
The reader should be aware that the transmitted power fraction is not |τ| ,asone
might be tempted to conclude.
In situations involving the connection of two semi-infinite transmission lines
having different characteristic impedances, reflections will occur at the junction, with
the second line being treated as the load. For a wave incident from line 1 (Z 01 )to
line 2 (Z 02 ), we find

Z 02 − Z 01
= (78)
Z 02 + Z 01
2
The fraction of the power that propagates into the second line is then 1 −| | .

EXAMPLE 10.5
A 50- lossless transmission line is terminated by a load impedance, Z L = 50 −
j75 .If the incident power is 100 mW, find the power dissipated by the load.

Solution. The reflection coefficient is
50 − j75 − 50
Z L − Z 0 − j.93
= = = 0.36 − j0.48 = 0.60e
Z L + Z 0 50 − j75 + 50
Then
2
2
P t = (1 −| | ) P i = [1 − (0.60) ](100) = 64 mW

EXAMPLE 10.6
Two lossy lines are to be joined end to end. The first line is 10 m long and has a loss
rating of 0.20 dB/m. The second line is 15 m long and has a loss rating of 0.10 dB/m.
The reflection coefficient at the junction (line 1 to line 2) is = 0.30. The input