ECE 451
Homework 1
Solutions
IR
+
Zo = 50 
VR
ZR = 75 
-
l = 3/4
1. Given the lossless transmission line shown above,
(a)
Find the reflection coefficient R and the standing wave ratio (SWR) for this
loaded transmission line.
(b)
Sketch the standing wave patterns for the magnitude of the voltage along this
transmission line in terms of VR.
(c)
Determine the impedance at the input of this loaded transmission line.
(d)
If a sinusoidal generator 10 o V, which has a source impedance of 100  is
connected to this loaded transmission line, what is the time average power
delivered to the 75- load.
(a)  R 
SWR 
ZR  Zo 75  50

 0.2
ZR  Zo 75  50
1  0.2
 1.5
1  0.2
(b)
2
Zo2  50 
2500
(c) Zin 


 33.33 
ZR
75
75
100  33.33
Vs Zin
1
(d) P  Re  Vin I*in  

2
2
2
2  Zs  Zin 
2 100  33.33
2
P  0.0937 watts
2. A transmission line of length 0.625 and characteristic impedance 50  is connected to an
unknown load impedance ZR. The input admittance is measured to be Yin= 0.004 - j0.008 -1.
Using the Smith chart find:
(a)
the normalized and the actual load admittance
(b)
the normalized and the actual load impedance
(c)
the load reflection coefficient
(d)
the voltage standing wave ratio (VSWR)
(e)
the distance, in wavelengths, of the first voltage maximum and minimum, with
reference to the load.
Yin = 0.004 –j 0.008 mhos
Yo = 0.02 mhos
yin = 0.2 –j 0.4
(a)
yR =1 – j2
YR=0.02 – j0.04 mhos
(c)  R  0.7  135
(d) VSWR = 5.66
(b) zR = 0.2 + j0.4
ZR=10 +j20 
(e) first maximum is 0.188 from load
first minimum is 0.438from load
3. Consider the lossless transmission line network shown below. The operating frequency is
2000 MHz and the propagation velocity on the transmission line is 0.3 m/ns.
(a) Using the Smith chart, determine the SWR on the section of line of length l.

vo
0.3

 15cm
fo 2  10 9
15  j 40   z R 
15  j 40
 0.3  j 0.8
50
From Smith chart, VSWR=5.7
SWR = 5.7
(b) Using the Smith chart find two values for the length l such that Z(l) is equal to
Zo ± jX.
l1=0.186-0.112=0.074=0.07415=1.11 cm
l2=0.313-0.112=0.201=0.20115=3.015 cm
Z(l1) = Zo + jX, l1 =1.11cm
Z(l2) = Zo - jX, l2 =3.015cm
(c)
Determine the value of series inductance L and the proper length of the
transmission line section (l1 or l2) that insures Zin = Zo
Impedance is capacitive at l2 with z=1-j1.9 or Z=50+j95 . Can be compensated with inductor
such that
L
95
 7.55 nH
2  2  10 9
L = 7.55_ (nanoHenries)
4. Answer the following questions using the attached Smith chart. Clearly identify significant
features on the chart. A transmission line with characteristic impedance 50  is terminated by a
load of impedance Zr = 40 + j50 .
(a)
What is the SWR? (5 pts)Solution: The normalized zr is 0.8 + j 1.0. Find this
point on the Smith Chart, draw the constant SWR circle, read off the SWR=3.0
(b)
What is the phase of r? (5 pts)72 degrees, from the Smith Chart.
(c)
What is the normalized admittance at the load? (5 pts) 0.5-j0.6, from the Smith
Chart.
(d)
What is the normalized admittance at d = 12.2 toward the generator from the
load? (5 pts) 0.5+j0.6, from the Smith Chart.
(e)
What is the phase of  at d = 12.2 toward the generator from the load? (10 pts) 72 degrees, from the Smith Chart. Be sure to read the phase of Gamma from the
impedance side of the constant SWR circle.
(f)
What is the shortest distance from the load at which a short-circuited stub could
be attached to achieve an impedance match? (10 pts) 0.265 wavelengths, from the
Smith Chart.
(g)
What would the normalized input admittance of the stub be? (10 pts) -j1.1
/4
Zs
ZR
Z o = 50 
Vs
z = -d
z=0
5. For the above figure, let Vs = 1 volt, Zs = 30 , and ZR = 40 .
(a)
What is the impedance at the input of the line (z = -d) ? (10 pts)
Z in 
(b)
What is the phasor current through Zs? (10 pts)
I in 
(c)
Z o2 ( 50 )2

 62.5 
ZR
40
1
1

 0.01081 A
Z S  Z in 30  62.5
What is the phasor current through ZR ? (20 pts)
Ts=5/8, s=-2/8, R= -1/9, ,
V+ = -j0.608 V, 
 j 0.608
1   0.111 
50 
IR=-j13.51 mA
(d)
What is the time-average power delivered to the load ? (10 pts)
1
1
P= Re Vin I in*   Re 62.5  0.01081 0.01081
2
2
P=3.65 mW