Tutorial problem sheet - University of Birmingham

Tutorial Problems
Introduction to Antennas & Electromagnetic Wave
Propagation
Dr. Costas Constantinou
School of Electronic, Electrical & Computer Engineering
University of Birmingham
Question 1
A current element I 0 dl is defined to be an antenna which has a constant in-phase current
I 0 e jωt flowing along its length dl ≪ λ ≡ 2πc/ω. Likewise, a Hertzian dipole I 0 dl is defined
to be an antenna which has a current I 0 e jωt at its terminals (in its centre) falling linearly to
zero at its two ends a distance dl apart.
Show that:
(a) the fields of the current element are twice as great as those of the Hertzian dipole;
(b) the radiation resistance of the current element is four times as great as that of the
Hertzian dipole;
(c) the power gain of the two antennas is equal.
Comment on how you can generalise (c) to any electrically small antenna.
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Question 2
A half-wave dipole driven in its balanced mode has a sinusoidal current Ie jωt at its terminals
and its electric field vector in the far field is given by,
E θ = IZ 0cos ( π
cosθ) 2
2πr sinθ
e j(ωt−kr)
where Z 0 is the impedance of free space, r is the distance from the antenna and θ is the angle
defining the point (r,θ) measured from the axis of the dipole.
(a) Calculate the radiation resistance of a vertical quarter wavelength monopole situated
on an infinite perfectly conducting ground plane.
(b) Comment on the feed mechanisms required by the dipole and ground-based monopole
antennas.
Note: It is given that ∫ π/2
0
cos 2 ( π 2 cosθ)
sinθ
dθ ≃ 0.609.
Question 3
(a) Twoparallel,verticalλ/2dipolesarespacedadistanceλ/2apart. Themutualimpedance
Z 12 between the dipoles at this separation is (−13−j30)Ω and the self-impedance of
each dipole is (73 + j42)Ω. Given that the currents in the two dipoles are equal in
magnitude and phase by symmetry arguments, find the power gain of this two-dipole
array, if the maximum far-field radiation magnetic field of each dipole at range r is
|H φ | = I
2πr .
(b) The power gain G of any antenna can be written as, G = 4πAe
λ 2 , where A e is the effective
receiving area andλisthewavelength ofoperation. Findanexpression intermsofopen
circuit voltage and radiation resistance for the maximum power which can be delivered
to a matched load connected to the above two-dipole array, when it is irradiated by
a vertically polarised uniform plane wave of electric field magnitude E i . State clearly
any assumptions you make.
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Note: The gain of an isolated λ/2 dipole is G ≃ 1.64.
Question 4
It is given that for a vertical current element I 0 dl at the earth’s surface the radiation field at
a distant point also at the earth’s surface is given by,
|E z | = 60kI 0dl
A s
r
where k is the free-space wavenumber and A s is the attenuation constant due to lossy ground.
Assuming that the radiation resistance of the current element is the same as in free space for
equal lengths of element, show that,
|E z | =
√
180P
r
A s
where P is the radiated power in Watts and r is the range in metres.
Question 5
The electrical characteristics of the D-layer of the ionosphere through which a 5 MHz radio
beam passes are such that it can be represented approximately by an effective permittivity
ǫ 0 and conductivity σ = ǫ 0 νωp/ω 2 2 where ǫ 0 = 8.854×10 −12 Fm −1 , ν = 10 6 collisions/m 3 /s
and ω p = 3.214 × 10 12 rad 2 s −2 . Show that if the conduction current is much less than
the displacement current the propagation attenuation constant α is approximately equal to
σZ 0 /2, where Z 0 is the impedance of free space. Hence find the attenuation of the above
signal produced in the ionosphere by a vertical path of upward length 30 km.
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