Problem set-2 Solutions
ECE 357
Winter 2015
1.
At a frequency of 4 MHz a parallel wire transmission line has the following parameters: R =
0.025 /m, L = 2 H/m, G = 0, C = 5.56 pF/m. The line is 100 meters long, terminated in a
resistance of 300 . Find the standing wave ratio and voltage reflection coefficient of the load.
Solution:
2.
A 100 km telephone line has a series resistance of 4 /km, an inductance of 3 mH/km, a leakage
conductance of 1 S/km, and a shunt capacitance of 0.015 F/km, at an angular frequency  =
5000 rad/s. At the sending end there is a generator supplying 100 volts peak, at 5000 radians
per second, in series with a resistance of 300 . The load at the receiving end consists of a 200
 resistor. Find the voltage and current as functions of z, and calculate their values at the
midpoint of the line, if the load current IL = −0.115 + j0.0268 A.
Solution:
3.
The following characteristics have been measured on a lossy transmission line at 100 MHz:
Z 0  50  j 0 
  0.01 dB/m
  0.8 rad/m
(a) Determine the attenuation constant of the line in Np/m. Hint: to determine the conversion
between dB and Np, consider determining the attenuation constant  for a line that is 1m
long that has exactly 1 dB of loss (1 dB/m).
(b) Determine R, L, G, and C for the line.
Solution:
4.
A transmission line operating at 125 MHz has Z0 = 40 ,  = 0.02 Np/m, and  = 0.75 rad/m. Find
the line parameters R, L, G, and C.
5.
Consider a lossless coaxial transmission line having distributed parameters L = 245 nH/m and C =
200 pF/m. The line is terminated with a resistor RL = 100 Ω as shown. The operating frequency is
f =1 GHz.
a) Determine the characteristic impedance and phase velocity of the line.
b) Determine the input impedance seen looking into the input terminals of the line at 1 GHz.
c) Determine the VSWR of the load.
d) If the transmission line was shortened, determine the length(s) of line that would result in the
same input impedance as part (b).
Solution:
6.
Cheng P.9-20.
Solution:
7.
The attenuation constant for a 50 distortionless line is
0.1 nF
0.01 dB
m and its capacitance is
m.
a) What are the values of the line resistance, conductance and inductance per meter?
b) What is the value of the phase velocity of the propagating waves?
c) What are the ratios of the magnitude of the propagating voltage waves at 1 km and 5 km with
respect to the starting point at z  0 ?
Solution:
For a distortionless line, we know the following:
R G

L C
(1)
Z 0  R0  jX 0 
    j 
a) Using (2),
L
 j0
C
(2)
C
R  jL (3)
L
L  Z 0 C  50 2  0.110 9
2
 m
L  2.5  10 7 H
Using (3)

C
L
0.01
2.5  10 7
R  R 


L
C 8.686
0.1  10 9
 m
R  0.0576 
Using (1):
R G
RC 0.0576  0.1  10 9
 G 

L C
L
2.5  10 7
G  23.02Sie 
m

b) We know that for a distortionless line:
Vp 
1
LC

1
7
2.5  10  0.1 10 9
 s   0.667c
V p  2  10 8 m
where c is the speed of light in vacuum.

V0 e z e  jz
V z 

 e z
c)
V z  0 V0  e  0 e  j 0
(Use correct units for  )

0.01
 1.151  10 3  Np 
 m
8.686
Then,
3
V z  1000m
 e 1.15110 1000  32%
V z  0
3
V z  5000m 
 e 1.15110 5000  0.32%
V z  0
200 pF 
0.5 H 
m
m . It is excited with




 and inductance of
A lossless TL has capacitance of
sinusoidal source of frequency 1kHz . The magnitude of the voltage measured across a 35
8.
load is 100V . Find the following:
a) The line characteristic impedance 100V 
b) The voltage reflection coefficient at load 
c) The phase velocity,
d) The wavelength 
Vp

e) The forward and backward traveling waves amplitudes,
f)
The line propagation constant  .
Solution:
 m
H m
C  200  10 12 F
H  0.5  10 6
f  1 10 3 Hz 
RL  35
V ( z'  0)  V ( z'  l )  VL  100volts
Vo ,Vo

a) For lossless line R= R0 
b)  
L

C
0.5  10 6
 50
200  10 12
RL  R0 35  50

 0.1765
R0  RL 35  50
c) V p 
 


1
1



 1 108  0.33c m
s

6

12
  LC
LC
0.5  10  200  10
c is the speed of light in vacuum.
d)  
Vp
f

1 108
 10 5 m
3
1 10
e) Apply Cheng’s (9-97a), (9.97-b), with VL=100 V and IL= VL / RL =100/35 A to
V  121.43 exp  jl and V  21.43 exp  jl  .

0

0
f)  
2



2
 6.2832  10 5 rad
5
m
10

find: