THE UNIVERSITY OF ZAMBIA
SCHOOL OF ENGINEERING
JUNE 2021
EEE 4021-MID TEST
ENGINEERING ELECTROMAGNETICS
Answer All Questions.
Question 1
Phasor Representation.
(i).
Consider a transmission line conveying a sinusoidal (harmonic) voltage wave in the +z
direction. The transmission line has a characteristic resistance R0 = 50 Ω. At some position along
the line, the voltage phasor is measured to be V = 30∠67°V. Write the time domain expression
for the current measured at that point
[4 marks]
(ii).
Consider the time-domian vector
B  t  = 3cos  t  300  xˆ - 4sin  t  yˆ
1. Find its vector phasor representation B. . Also, transform B back to the time domain, and
verify that it equals B(t). point
[5 marks]
2. If B represents a time-domain field that is a function of position such that
B  t  = 3cos  t z  300  xˆ - 4sin  t z  yˆ
Find the phasor representation
B.
[5 marks]
Travelling Wave.
(iii). The voltage of an electromagnetic wave traveling on a transmission line is given by
v(z,t)= 5ez sin  4 x109 t  20z  (V),where z is the distance in meters from the generator.
1
Find the frequency, wavelength, and phase velocity of the wave;
2. At z = 2 m, the amplitude of the wave was measured to be 2 V; find α
[3 marks]
[3 marks]
Question 2
Lossy Transmission Lines & Parameters.
(i).
At a frequency of 4 MHz a parallel wire transmission line has the following parameters:
R = 0.25 Ω / m
L = 2 μH / m
G = 0
C  = 5.56 pF / m
The line is 100 meters long, terminated in a resistance of 300 Ω. Find the standing wave ratio
and voltage reflection coefficient of the load.
[3 marks]
(ii).
The following characteristics have been measured on a lossy transmission line at
100 MHz:
Z 0 = 50 + j0 Ω
α = 0.01 dB / m
β = 0.8  rad / m
1. Determine the attenuation constant of the line in Np/m. Hint: determine conversion
between dB and Np; where   dB / m 8.68   Np / m .
[1 marks]
2. Determine R’, L’, G’, and C’ for the line.
[4 marks]
(iii) Consider a lossless coaxial transmission line having distributed parameters:
L = 245 nH / m
C  = 200 pF / m
The line is terminated with a resistor RL = 100 Ω as shown above.
The operating frequency is f =1 GHz.
1. Determine the characteristic impedance and phase velocity of the line
. [3marks]
2. Determine the input impedance seen looking into the input terminals of the line at 1 GHz.
[3 marks]
3. Determine the VSWR of the load.
[2 marks]
4. If the transmission line was shortened, determine the length(s) of line that would result in
the same input impedance as part (2).
[4 marks]
Question 3
Gradient of a Scalar Field and Verification of Divergence and Stokes’s Theorems
(i)
A hill can be modeled with the equation H = 10 - x2 - 3y2 where H is the elevation of the
hill. Find the path that a frictionless ball would take in order that it experiences the greatest
2
change of elevation in the shortest change of horizontal position.
Assume that the motion of the ball is unconstrained.
(ii)
Check the divergence theorem using the function vector filed
D = xˆ y 2 + yˆ  2xy + z 2  + zˆ  2xz 
and
[5 marks]
a unit cube at the origin as shown in figure
below:
(iii)
Suppose
D = yˆ  2xz + 3y
2
 + zˆ  4yz 
2
[5 marks]
Check Stokes' theorem for the square
surface shown in figure below
[5 marks]
Electric Potential/Voltage
(iv)
Evaluate the energy required to move a charge q against a vector field
E = xy xˆ - 2xyˆ
along the path AB on quarter circle as depicted above, considering the
equation of the quarter circle defined by
x 2 + y 2 = 9 where the radius r = 3 and (0  x  3, 0  y  3)
Hint: solve this problem Cartesian coordinates preferably easier than in cylindrical coordinates..
[5 marks]
END OF TEST
3

a2 - y2
Formulas Sheet
1
1
y
dy = y a 2 - y 2 + a 2tan -1
2
2
a2 - y2
2
2
x
a
x
dx = 
1 2 2 3/2
a - x 
3
4
5