1. Analyze an incremental section of the proposed transmission line mode shown in Figure 1 and obtain the coupled differential equations for the voltage and the current and compared with eqs. (1-
2) and (1-4) in the lecture notes. Comment on the comparison of the two different models.
Figure 1
2. Prove that the time-average power flow on a transmission line is expressible by
 
2
  and the time-average stored electric and magnetic energy densities are respectively
 

1
4
2
W m
 

1
4
2
3. A transmission line of length

connects a load to a sinusoidal voltage source with an oscillation frequency f
. Assuming that the velocity of wave propagation on the line is c , for which of the following situations is it reasonable to ignore the presence of the transmission line in the solution of the circuit:
(a)


(b)
 
(c)


(d)
 
20 ,
50
20
1 ,



10 kHz
60
300
Hz
MHz

100 GHz
4. Using a slotted line, the following results were obtained: distance of first minimum from the load is
4 cm; distance of second minimum from the load is 14 cm; voltage standing-wave ration is 2.5. If the line is lossless and
Z
0

50

, find the load impedance.

5. Show that on a lossless transmission line, the impedance at voltage minimum is and at a voltage maximum is
Z in

R max

Z S
0
, where
Z
0
and
S
 
Z in

R min
 characteristic impedance and VSWR.
6. A transmission line is short-circuited ( a) Find the expression for
 
and b) Sketch and
I
  c) Find VSWR on the line.
Z
L

0
).
as a function of
Z
0
,
 z and
V
 .
Z S
0


30
 j 60


. The 7. A 50Ω lossless transmission line is terminated in a load with impedance wavelength is 5 cm. Find
Z
L
(a) the reflection coefficient at the load
(b) the standing-wave ratio on the line
(c) the position of the voltage maximum nearest the load
(d) the position of the current maximum nearest the load.
8. The peak instantaneous voltage on a certain lossless transmission line must be kept below
V
0 volts in order to avoid voltage breakdown. The characteristic impedance is
Z
0
. If the VSWR is S , how much power can transmitted by the line without risking breakdown?
9. The complex power transmitted through a lossless transmission line at a particular point
P
P



3 j j

 z
 z
0 is
(a) Find the complex power carried by the backward propagating wave;
(b) Find the VSWR on the line;
(c) Find the possible value(s) of the reflection coefficient

 
0
.
10. Use the Smith chart to find the reflection coefficient corresponding to a load impedance of (a)
Z
L

3 Z o
; (b)
Z
L


2

2
 j Z o ; (c)
Z
L
 
2 jZ o
; (d)
Z
L

0
.
11. Use the Smith chart to find the normalized load impedance corresponding to a reflection coefficient of (a)
 
.
 
; (b)
 
.
 
30

; (c)
 
0 ; (d)
12. On a lossless transmission line terminated in a load
   j
Z
L
.

100

, the standing-wave ratio was measured to be 2.5. Use Smith chart to find the corresponding values of
13. A lossless 50-Ω transmission line is terminated in a load with
Z
L

Z o
.

50

25
 j

. Use the Smith chart to find
(a) the reflection coefficient at load

L
;
(b) the voltage standing-wave ratio;
(c) the input impedance at 0.35λ from the load;
(d) the input admittance at 0.35λ from the load;

(e) the shortest line length for which the input impedance is purely reactive;
(f) the position of the first voltage maximum from the load.
14. A lossless 100-Ω transmission line of length the input impedance is
Z in
 
. j

.
3 8

is terminated in an unknown load impedance. If
(a) Use the Smith chart to find
Z
L
;
(b) What length of open-circuit line could be used to replace
Z
L
?
15. Using a slotted line on a 50-Ω lossless transmission line, the VSWR was measured to be
VSWR

.
and
V max occurred only at 10cm and 24cm from the load. Use the Smith chart to find
Z
L
.

from the load, a resistor with resistance R

is terminated in a load with

30

Z
L



 j

. At
is connected as shown in Figure 1. Use the
Smith chart to find
Z in
.
Z in
0.3λ
Figure 1
R
0.3λ
Z
L
17. Use the Smith chart to find
Z

50
 o
.
Z in of the feed line shown in Figure 2. All lines are lossless with
0.3λ
0.3λ
Z in
0.7λ
Figure 2
Z
1


50
 j50
Z
2




50
 j50


18. If the load impedance of a transmission line is purely reactive, i.e.,
Z
L
 j x
L
, where number, can it can matched by adding a shunt susceptance somewhere in the line? Why? x
L
is a real