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ECE 5333/7333 FINAL EXAMINATION
Electrical and Computer Engineering Department
Southern Methodist University
Available 11:30 AM, Dec 14, 2020 Due 2:30 PM, December 15, 2020
Open notes, Computer usage including Mathematica. All questions of equal value. 5333 Students may
choose to drop 1 problem. If you choose to drop a problem, mark the problem as DROPPED.
1. The vector potential is given by
A D
4
I0ı`
ejkr
r
Oz;
where I0ı` is a small differential of current (short dipole) located at the origin (x D 0, y D 0, z D 0), and
r Dpx2 C y2 C z2 is the distance from the origin to the observation point.
(a.) Calculate the x component of the magnetic field, Hx , on the y axis.
(b.) Calculate the x component of the magnetic field Hx , on the y axis in the radiation zone (far-field where
x ).
(c.) Calculate the z component of the electric field Ez , on the y axis in the radiation zone.
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2. The electric and magnetic fields for a short dipole of length `, on the z axis with current directed along Oz
are given by
H' D I0 `k
2
4
ejkr

j
kr
C 1
.kr/2

sin ;
Er D 2I0 `k
2
4
0e
jkr

1
.kr/2
j
.kr/3

cos ;
E D I0 `k
2
4
0e
jkr

j
kr
C 1
.kr/2
j
.kr/3

sin ;
where the free-space wave impedance is 0 D 120 .
(a.) Write the far-field approximations for the fields,
(b.) Determine the ratio of E and H' in the far-field region,
(c.) Calculate the radial component of the Poynting vector S in the far-field region,
(d.) Compute the maximum value of the Poynting vector and specify the location of the maximum,
(e.) Compute the radiation resistance of the short dipole assuming current at the dipole terminals is I0.
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3. The radiation pattern of an antenna is determined from the radial component of the Poynting vector, Or
S.r; ; '/, in the radiation zone (far-field region: using only the r2 terms and dropping the r3 and higher
terms),
F.; '/ D Or S
Smax
;
where Smax is the maximum value of the Poynting vector.
(a.) Sketch the radiation pattern of the short dipole as a function when ' D 0, that is, F.; 0/,
(b.) Sketch the radiation pattern of the short dipole as a function ' for D =2, that is, F.=2; '/.
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4. Assume an antenna has a radiation pattern given by
F.; '/ D

1I for 60ı < < 120ı and 180ı < ' < 180ı,
0I elsewhere.
(a.) Sketch the polar radiation pattern F.; 0/ for 0 < ; 90ı,
(b.) Sketch the polar radiation pattern F.90ı; '/,
(c.) Calculate the beam efficiency about 80ı < < 100ı and 180ı < ' < 180ı,
(d.) Plot the beam efficiency about D 90ı as a function of ı , that is, for 90ı ˙ ı and 45ı < ı < 45ı.
You may ignore the azimuth angle '.
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5. The linear array of eight isotropic sources, shown below, is located along the z axis. The element spacing is
d and the current distribution is uniform with I0 to each element. Assume all element currents have identi-
cal phases.
θ
z
x
y
d
(a.) Calculate the array pattern in the x z plane as a function of the polar angle .
(b.) If the spacing is d D 3=2 determine the pattern BW (beam width) at D =2. (Assume BW is the
angular distance between the two pattern zeros that occur about the main beam. )
(c.) Determine the number of side lobes in the interval 0 < < 90ı.
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6. An isotropic array contains 3 elements with a uniform spacing as illustrated in the figure below.
θ
z
x
To observation
point.
d d
ϕ
The radiation pattern is given by
f .u/ D
1X
mD1
ame
jkud ;
where u D sin', and an represents the current in the
nth element. The radiation power as a function of u
is given by
P D
1X
mD1
1X
nD1
ama

ne
jkud.mn/
where an is the complex conjugate of an. The total
power radiated is
P D
1X
mD1
1X
nD1
ama

n
Z 1
1
ejkud.mn/du
The power in the region u0 < u < u0 (u0 D sin'0)
is given by
PB D
1X
mD1
1X
nD1
ama

n
Z u0
u0
ejkud.mn/du:
The beam efficiency is
D PB
P
Determine the spacing d= that produces the optimum beam efficiency when u0 D 0:1.
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7. An isotropic array contains 2N C 1 elements with a uniform spacing as illustrated in the figure below.
θ
x
To observation
point.
d d
ϕ
0 1 N−1−N
z
(a.) Assuming d D =2 calculate the maximum beam efficiency for N D 1; 3; 5; 7. Assume the total beam
angle is 10ı so that u0 D sin 5=180.
(b.) Sketch the computed values of for different array sizes.

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