Department of Electronics and Telecommunication Engineering

Chapter-1 Numericals
1.The normalized radiation intensity of an antenna is represented by
U (θ) = cos2 (θ ) cos2 (3θ ), (0 ≤ θ ≤ 90◦ , 0 ◦ ≤ φ ≤ 360◦ ) Find the a. half-power beam width
HPBW (in radians and degrees) b. first-null beam width FNBW (in radians and degrees)
2. A lossless resonant half-wavelength dipole antenna, with input impedance of 73 ohms, is
connected to a transmission line whose characteristic impedance is 50 ohms. Assuming that
the pattern of the antenna is given approximately by U = B0 sin3 θ find the maximum absolute
gain of this antenna.
3. A uniform plane wave is incident upon a very short lossless dipole (l << λ). Find the
maximum effective area assuming that the radiation resistance of the dipole is R r = 80(πl/λ)2,
and the incident field is linearly polarized along the axis of the dipole.
4. A hypothetical isotropic antenna is radiating in free-space. At a distance of 100 m from the
antenna, the total electric field (Eθ ) is measured to be 5 V/m. Find the (a) power density (Wrad)
(b) power radiated (Prad).
5. The maximum radiation intensity of a 90% efficiency antenna is 200 mW/unit solid angle.
Find the directivity and gain (dimensionless and in dB) when the (a) input power is 125.66 mW
(b) radiated power is 125.66 mW.
6. The power radiated by a lossless antenna is 10 watts. The directional characteristics of the
antenna are represented by the radiation intensity of U = Bo cos3 θ (watts/unit solid angle) (0
≤ θ ≤ π/2, 0 ≤ φ ≤ 2π ) For each, find the (a) maximum power density (in watts/square meter)
at a distance of 1,000 m (assume far-field distance). Specify the angle where this occurs. (b)
exact and approximate beam solid angle . (c) directivity, exact and approximate, of the antenna
(dimensionless and in dB). (d) gain, exact and approximate, of the antenna (dimensionless and
in dB).
7. In target-search ground-mapping radars it is desirable to have echo power received from a
target, of constant cross section, to be independent of its range. For one such application, the
desirable radiation intensity of the antenna is given by
U (θ , φ) = 1 0◦ ≤ θ < 20◦
= 0.342 csc (θ ) 20◦ ≤ θ < 60◦
= 0 60◦ ≤ θ ≤ 180◦
For 0 ◦ ≤ φ ≤ 360◦ Find the directivity (in dB) using the exact formula.
8. The normalized radiation intensity of a given antenna is given by
(a) U = sin θ sin φ (b) U = sin θ sin2 φ (c) U = sin θ sin3 φ (d) U = sin2 θ sin φ
 (e) U = sin2 θ sin2 φ (f) U = sin2 θ sin3 φ The intensity exists only in the 0 ≤ θ ≤ π, 0 ≤ φ ≤ π
region, and it is zero elsewhere. Find the (a) exact directivity (dimensionless and in dB). (b)
azimuthal and elevation plane half-power beam widths (in degrees).
9.

10. A λ/2 dipole, with a total loss resistance of 1 ohm, is connected to a generator whose internal
impedance is 50 + j25 ohms. Assuming that the peak voltage of the generator is 2 V and the
impedance of the dipole, excluding the loss resistance, is 73 + j42.5 ohms, find the power (a)
supplied by the source (real) (b) radiated by the antenna (c) dissipated by the antenna
11. An antenna with a radiation resistance of 48 ohms, a loss resistance of 2 ohms, and a
reactance of 50 ohms is connected to a generator with open-circuit voltage of 10 V and internal
impedance of 50 ohms via a λ/4-long transmission line with characteristic impedance of 100
ohms. (a) Draw the equivalent circuit (b) Determine the power supplied by the generator (c)
Determine the power radiated by the antenna.
12. The E-field pattern of an antenna, independent of φ, varies as follows:
E=1 0◦ ≤ θ ≤ 45◦
=0 45◦ < θ ≤ 90◦
= 1/2 90◦ < θ ≤ 180◦
(a) What is the directivity of this antenna? (b) What is the radiation resistance of the antenna
at 200 m from it if the field is equal to 10 V/m (rms) for θ = 0◦ at that distance and the terminal
current is 5 A (rms)?
13. An antenna has a maximum effective aperture of 2.147 m2 at its operating frequency of 100
MHz. It has no conduction or dielectric losses. The input impedance of the antenna itself is 75
ohms, and it is connected to a 50-ohm transmission line. Find the directivity of the antenna
system (“system” meaning includes any effects of connection to the transmission line). Assume
no polarization losses.
14. The radiation intensity of an antenna can be approximated by U (θ , φ) = cos4(θ ) 0◦ ≤ θ <
90◦ 0 90◦ ≤ θ ≤ 180◦ with 0◦ ≤ φ ≤ 360◦ Determine the maximum effective aperture (in m 2) of
the antenna if its frequency of operation is f = 10 GHz.
15. A communication satellite is in stationary (synchronous) orbit about the earth (assume
altitude of 22,300 statute miles). Its transmitter generates 8.0 W. Assume the transmitting
antenna is isotropic. Its signal is received by the 210- ft diameter tracking paraboloidal antenna
on the earth at the NASA tracking station at Goldstone, California. Also assume no resistive
losses in either antenna, perfect polarization match, and perfect impedance match at both
antennas. At a frequency of 2 GHz, determine the: (a) power density (in watts/m2) incident on
the receiving antenna. (b) power received by the ground-based antenna whose gain is 60 dB.
16. A lossless (ecd = 1) antenna is operating at 100 MHz and its maximum effective aperture
is 0.7162 m2 at this frequency. The input impedance of this antenna is 75 ohms, and it is
attached to a 50-ohm transmission line. Find the directivity (dimensionless) of this antenna if
it is polarization-matched.
17. Two X-band (8.2–12.4 GHz) rectangular horns, with aperture dimensions of 5.5 cm and
7.4 cm and each with a gain of 16.3 dB (over isotropic) at 10 GHz, are used as transmitting and
receiving antennas. Assuming that the input power is 200 mW, the VSWR of each is 1.1, the
conduction-dielectric efficiency is 100%, and the antennas are polarization-matched, find the
maximum received power when the horns are separated in air by (a) 5 m (b) 50 m (c) 500 m
18. A series of microwave repeater links operating at 10 GHz are used to relay television signals
into a valley that is surrounded by steep mountain ranges. Each repeater consists of a receiver,
transmitter, antennas, and associated equipment. The transmitting and receiving antennas are
identical horns, each having gain over isotropic of 15 dB. The repeaters are separated indistance
by 10 km. For acceptable signal-to-noise ratio, the power received at each repeater must be
greater than10 nW. Loss due to polarization mismatch is not expected to exceed 3 dB. Assume
matched loads and free-space propagation conditions. Determine the minimum transmitter
power that should be used.
19. A base station cellular communication systems lossless antenna has a maximum gain of 16
dB (above isotropic) at 1,900 MHz. Assuming the input power to the antenna is 8 watts, what
is the maximum radiated power density (in watts/cm2) at a distance of 100 meters?
20.