Page 352 - Engineering Electromagnetics, 8th Edition

334 ENGINEERING ELECTROMAGNETICS

and the reflected wave is equal in amplitude to the incident wave. Hence, it should
not surprise us to see that the VSWR is

1 +|− j1|
s = =∞
1 −|− j1|
and the input impedance is a pure reactance,
− j300 cos 288 + j300 sin 288 ◦
◦
Z in = 300 = j589
300 cos 288 + j(− j300) sin 288 ◦
◦
Thus, no average power can be delivered to the input impedance by the source, and
therefore no average power can be delivered to the load.

Although we could continue to find numerous other facts and figures for these
examples, much of the work may be done more easily for problems of this type by
using graphical techniques. We encounter these in Section 10.13.

D10.4. A50W lossless line has a length of 0.4λ. The operating frequency is
300 MHz. A load Z L = 40 + j30 is connected at z = 0, and the Thevenin-
equivalent source at z =−l is 12 0 Vin series with Z Th = 50 + j0 . Find:
◦
(a) ;(b) s;(c) Z in .
◦
Ans. 0.333 90 ; 2.00; 25.5 + j5.90

D10.5. For the transmission line of Problem D10.4, also find: (a) the phasor
voltage at z =−l;(b) the phasor voltage at z = 0; (c) the average power
delivered to Z L .

Ans. 4.14 8.58 V; 6.32 −125.6 V; 0.320 W
◦
◦
10.13 GRAPHICAL METHODS: THE SMITH
CHART

Transmission line problems often involve manipulations with complex numbers, mak-
ing the time and effort required for a solution several times greater than are needed
for a similar sequence of operations on real numbers. One means of reducing the
labor without seriously affecting the accuracy is by using transmission-line charts.
Probably the most widely used one is the Smith chart. 3
Basically, this diagram shows curves of constant resistance and constant reac-
tance; these may represent either an input impedance or a load impedance. The latter,
of course, is the input impedance of a zero-length line. An indication of location along
the line is also provided, usually in terms of the fraction of a wavelength from a voltage
maximum or minimum. Although they are not specifically shown on the chart, the
standing-wave ratio and the magnitude and angle of the reflection coefficient are very

3 P. H. Smith, “Transmission Line Calculator,” Electronics, vol. 12, pp. 29–31, January 1939.