Deck 11: Antennas and Radiating Systems

Question 1

Obtain the electric field intensity of a Hertzian dipole by finding both A and V and using Eq. (11-2). Check your result with Eqs. (11-16a, b, c).

Answer 1

Consider the expression for electric field intensity in terms of potential functions A and V.
…… (1)
Where,
is the angular frequency
Consider the expression for electric field intensity in terms of spherical coordinates.
…… (2)
Where,
,
and
are the spherical components of E
Using equation (1), the spherical components of E.
…… (3)
…… (4)
…… (5)
Consider the phasor representation of vector potential.
…… (6)
Consider the spherical components of A
…… (7)
…… (8)
…… (9)
Substitute equation (6) in equations (7), (8)
…… (10)
…… (11)
Consider the expression of scalar potential for the magnetic dipole shown in Figure 1.
…… (12)
Where,
is the electron charge
…… (13)
and
are the distance from the charges
and
to the field
…… (14)
…… (15)
is the permeability of free space
Figure 1
Substitute equations (13), (14), (15) in equations (12)
Since
On simplifying
Substitute
for
…… (16)
Substitute equations (16), (10) in equations (3).
…… (17)
Substitute equations (16), (11) in equations (4).
…… (18)
Substitute equations (16), (9) in equations (5).
…… (19)
Thus, the electric field intensity of a Hertzian dipole is determined and the results are verified.

Question 2

Repeat parts (a) and (b) of Problem for a small rectangular loop of sides L x and L y. Repeat part (c) for f = 1 (MHz), L x = L y = 2b = 1 (m), a = 3 (mm), and compare results.
A time-harmonic uniform current I 0 cos t flows in a small circular loop of radius b ( ) lying in the xy -plane.
a) Find the radiation resistance R r of the magnetic dipole.
b) Obtain an expression for its radiation efficiency n r if the loop is made of copper wire of radius a ,
c) Calculate R r and n f for f = 1 (MHz), b = 50 (cm), and a = 3 (mm).
d) Rework part (c) if the loop has ten closely wound insulated turns.

Answer 2

(a)
Consider the formula for radiation resistance
of the small rectangular loop antenna.
…… (1)
Where,
is denoted as radiation resistance.
Substitute
for
in equation (1).
…… (2)
Hence, the radiation resistance
of small rectangular loop antenna is
(b)
Consider the formula for total input power.
…… (3)
Where,
is denoted as input power,
is denoted as power loss,
is denoted as radiated power.
Consider the formula for power loss.
…… (4)
Where,
is denoted as loss resistance.
Consider the formula for radiated power.
…… (5)
Where,
is denoted as radiation resistance.
Consider the formula for radiation efficiency.
…… (6)
Where,
is denoted as the Radiation efficiency.
Substitute equation (3) in equation (6),
…… (7)
Consider the formula for loss resistance in a copper wire.
…… (8)
Where,
is denoted as surface resistance.
dl is denoted as length of antenna.
a is denoted as radius of the copper wire
Consider the condition
and
in equation (8) rewritten as
…… (9)
Consider the formula for the surface resistance.
…… (10)
Where,
is denoted as conductivity of the copper
.
f is denoted as operating frequency.
Hence, the expression for radiation efficiency is
(c)
Substitute
for
, 1MHz for f and
for
in equation (10).
Convert millimeter to meter.
Consider the formula for
is,
Substitute
for c and 1 MHz for f.
Calculate the value of
Thus, the value of
is
.
Consider the antenna length (4) for dl in square loop.
Substitute
for
,
for a and
for b in equation (8).
Substitute
for
, 1 for
and 1 for
in equation (2).
Substitute
for
and
for
in equation (7).
Hence, the radiation resistance in
and radiation efficiency is
.
Comparison between circular loop and rectangular loop antenna is tabulated in table 1
Table 1
Hence, the comparison of circular loop and rectangular loop antenna is tabulated.

Question 3

For a five-element broadside binomial array:
a) Determine the relative excitation amplitudes in the array elements.
b) Plot the array factor for d = /2.
c) Determine the half-power beamwidth and compare it with that of a five-element uniform array having the same element spacings.

Answer 3

(a)
The excitation amplitudes vary according to the coefficients of binomial expansion
For five elements,
Hence, relative excitation amplitudes in the array elements are
.
(b)
Plot for array factor for
:
In a binomial array of N elements the array factor is binomial function. This is given as
.
It is known that the normalized pattern function is equal to the normalized array factor
.
Write the expression for
.
For
, calculate
.
For five elements, calculate
.
Draw the pattern function for
.
Figure 1
(c)
Consider the following:
Calculate the half-power beam width.
Therefore, the half-power beam width is
.
For uniform array, the normalized array factor is,
Calculate the half-power beam width for 5-element uniform array with
spacing.
Therefore, the half-power beam width for 5-element uniform array is
.
The half-power band width is
where as the half-power beam width for 5-element uniform array with
spacing is
.
Therefore, the half-power band width is less for 5-element uniform array.

Question 4

Communication is to be established between two stations 1.5 (km) apart that operate at 300 (MHz). Each is equipped with a half-wave dipole.
a) If 100 (W) is transmitted from one station, how much power is received by a matched load at the other station
b) Repeat part (a) assuming that both antennas are Hertzian dipoles.

Question 5

Assume that the field in an a × b rectangular aperture in an xy -plane is linearly polarized in the y-direction and that the aperture excitation has a uniform phase and a triangular amplitude distribution

Find (a) the pattern function in the xz -plane, (b) the half-power beamwidth, (c) the location of the first nulls, and (d) the level of the first sidelobes. Compare the results with those obtained in Example 11-13 for uniform field distribution.

Question 6

What is an equiangular spiral Why does it have broadband properties

Question 7

Why are antennas important for wireless communication over long distances

Question 8

What are radiation fields

Question 9

Define radiation resistance of an antenna.

Question 10

What is meant by the normalized array factor of an antenna array How is it different from the pattern function of the individual antennas

Question 11

What is a frequency-scanning array

Question 12

What is a log-periodic antenna

Question 13

A small filamentary rectangular loop of dimensions L x and L y lies in the xy -plane with its center at the origin and sides parallel to the x - and y -axes. The loop carries a current i(t) = I 0 cos t. Assuming L x and L y to be much less than the wavelength, find the instan­taneous expressions for the following quantities at a point in the far zone:
a) vector magnetic potential A ,
b) electric field intensity E ,
c) magnetic field intensity H.
Compare the results in parts (b) and (c) with Eqs. (11-30a) and (11 -30b), respectively.

Question 14

Use the total field expressions in Eqs. (11-15) and (11-16) to find the time-average power radiated by a Hertzian dipole, and compare it with the result in Eq. (11-43) using only the far-zone fields.

Question 15

For a uniform linear array of 12 elements spaced /2 apart:
a) Sketch the normalized array pattern |A( )| in Eq. (11-89) versus .
b) Find the widths of the main beam at half-power points and between the first nulls when the array is operated in the broadside mode.
c) Repeat part (b) for an endfire operation.

Question 16

Show that three satellites equally spaced around the geosynchronous orbit in the equatorial plane would cover almost the entire earth's surface. Explain why the polar regions are not covered. (b) Assuming the main beam of the radiation pattern of the satellite antenna to have the shape of a circular cone that just covers the earth with no spillover, find a relation between the main-lobe beamwidth and the directive gain of the antenna.

Question 17

Do Problem for a uniform-phased cosinusoidal amplitude distribution

and compare your results with those obtained in Example 11-13 for a uniform field distribution.
Problem:
Assume that the field in an a × b rectangular aperture in an xy -plane is linearly polarized in the y-direction and that the aperture excitation has a uniform phase and a triangular amplitude distribution

Find (a) the pattern function in the xz -plane, (b) the half-power beamwidth, (c) the location of the first nulls, and (d) the level of the first sidelobes. Compare the results with those obtained in Example 11-13 for uniform field distribution.

Question 18

Explain the principle of operation of log-periodic dipole arrays.

Question 19

State the procedure for finding the electromagnetic field due to an assumed time-harmonic current distribution on an antenna structure.

Question 20

Define antenna pattern.

Question 21

Discuss how the ratios (a/ ) and (dl/ ) of a Hertzian dipole affect its radiation resis­tance and radiation efficiency.

Question 22

State the principle of pattern multiplication.

Question 23

What are the important consequences of reciprocity relations concerning antennas that operate in the transmitting and receiving modes

Question 24

Give three examples of aperture radiators.

Question 25

A composite antenna consists of an elemental Hertzian electric dipole of length L along the z -axis and an elemental magnetic dipole of area S lying in the xy -plane. Equal time-harmonic currents of amplitude I 0 and angular frequency flow in the dipoles.
a) Verify that the far field of the composite antenna is elliptically polarized.
b) Determine the condition for circular polarization.

Question 26

Sketch the polar radiation pattern versus for a thin dipole antenna of total length 2 h = 1.25 . Determine the width of the main beam between the first nulls.

Question 27

For a uniform linear array with a large number of elements the denominator sin ( /2) in Eq. (11-89) remains small over a large portion of the normalized array pattern near the main beam and can be approximated by ( /2). Use this approximation to deter­mine the directivity of the array of a large uniform array with many elements.

Question 28

The antenna at the earth station of a satellite communication link having a gain of 55 (dB) at 14 (GHz) is aimed at a geostationary satellite 36,500 (km) away. Assume that the antenna on the satellite has a gain of 35 (dB) in transmitting the signal back toward the earth station at 12 (GHz). The minimum usable signal is 8 (pW).
a) Neglecting antenna ohmic and mismatch losses, find the minimum satellite transmitting power required.
b) Find the peak transmitting pulse power needed at the earth station in order to detect the satellite as a passive object, assuming the backscatter cross section of the satellite including its solar panels as 25 (m 2 ) and the minimum detectable return pulse power to be 0.5 (pW).

Question 29

Define effective area of an antenna.

Question 30

For a linearly polarized aperture field with uniform phase, what is the relation between the aperture's field distribution and the pattern function

Question 31

What is a Hertzian dipole

Question 32

Describe the E -plane and H -plane patterns of a Hertzian dipole.

Question 33

Describe the radiation pattern of a half-wave dipole antenna.

Question 34

State the difference between a broadside array and an endfire array.

Question 35

What is the universal constant that is the ratio of the directive gain and the effective area of an antenna

Question 36

What is the directivity of an aperture having an area A and a linearly polarized uniform field distribution at frequency f

Question 37

Assume the spatial distribution of the current on a very thin center-fed half­-wave dipole lying along the z -axis to be I 0 cos z, where = /c - 2 / . Find the charge distribution on the dipole,
(b) Repeat part (a), assuming the current distribution along the dipole to be a triangular function described by

Question 38

Assuming a triangular current distribution on a center-fed /6 dipole (h = /12), find an expression for its effective length. What is its maximum value

Question 39

Using the graph in Fig. 11-15(a) for the normalized array factor of a five-element broadside linear array with d = /2 and amplitude ratios 1:2:3:2:1, plot the polar radiation pattern for d = /4 and = - /2.

Question 40

A transmitting vertical half-wave dipole 60 (m) above the ground radiates 400 (W) at 100 (MHz). Assume the ground to be perfectly conducting.
a) Calculate the power available at a vertical half-wave receiving antenna 50 (km) away at a height 30 (m) above the ground.
b) At a distance 50 (km) from the transmitting antenna, where (at what altitudes) would there be a null field

Question 41

Define backscatter cross section of an object.

Question 42

Describe the manner in which the beamwidth in a principal plane of a rectangular aperture with a uniform field distribution depends on its dimensions.

Question 43

What constitutes an elemental magnetic dipole

Question 44

Define beamwidth of an antenna pattern.

Question 45

What are the radiation resistance and directivity of a half-wave dipole antenna

Question 46

What is a binomial array What are the relative excitation amplitudes of a six-element binomial array

Question 47

Explain the principle of radar.

Question 48

Assume that a linearly polarized constant excitation field exists in a rectangular aperture with width b and a circular aperture with diameter D = b. Compare the main-lobe beamwidths and the first sidelobe levels of their radiation patterns.

Question 49

A 1-(MHz) uniform current flows in a vertical antenna of length 15 (m). The antenna is a center-fed copper rod having a radius of 2 (cm). Find:
a) the radiation resistance,
b) the radiation efficiency, and
c) the maximum electric field intensity at a distance of 20 (km) if the radiated power of the antenna is 1.6 (kW).

Question 50

5 (MHz) uniform plane wave having a peak electric field intensity E 0 is incident on a half-wave dipole at an angle .
a) Find the expression for the open-circuit voltage V oc at the terminals of the dipole.
b) If the dipole is connected to a matched load, what is the maximum power P L delivered to the load
c) Calculate V oc and P L for E 0 = 50 (mV/m) and for = /2 and /4.

Question 51

Letting = exp (j ), we can write the array factor of an equally spaced array as a polynomial, A( ), in , and many characteristics of the array pattern can be estimated by examining the distribution of the zeros of the array polynomial on a unit circle. In general, an N -element linear array has N - 1 zeros, 0m (m = 1,2,..., N - 1), distributed around the unit circle. Find A( ) and locate all 0m on a unit circle for the following linear arrays:
a) a two-element array,
b) a three-element binomial array,
c) a five-element uniform array,
d) a five-element array having amplitude ratios 1:2:3:2:1 (as in Example 11-9).
e) Based on the locations of 0m for the two arrays in parts (c) and (d), explain why the pattern for the array in part (d) has lower sidelobes but a wider beamwidth.

Question 52

The current along an isolated and terminated traveling-wave antenna of length L is given as
I(z) = I 0 e-j z.
a) Find the far-zone vector potential, A(R, ).
b) Determine H (R, ) and E(R, 0) from A(R, ).
c) Sketch the radiation pattern for L = /2.

Question 53

What does the Friis transmission formula say

Question 54

Define the near zone and the far zone of an antenna.

Question 55

Define sidelobe level of an antenna pattern.

Question 56

What is the image of a horizontal dipole over a conducting ground

Question 57

Is the radiation pattern of all linear binomial arrays sidelobeless Explain.

Question 58

Define path gain factor concerning wave propagation near the earth's surface.

Question 59

The amplitude of the time-harmonic current distribution on a center-fed short dipole antenna of length 2 h ( h ) can be approximated by a triangular function

Find (a) the far-zone electric and magnetic field intensities, (b) the radiation resistance, and (c) the directivity.

Question 60

Two elemental dipole antennas, each of length 2 h ( h ), are aligned colinearly along the z -axis with their centers spaced a distance d ( d 2 h ) apart. The excitations in the two antennas are of equal amplitude and equal phase.
a) Write the general expression for the far-zone electric field of this two-element colinear array.
b) Plot the normalized E -plane pattern for d = /2.
c) Repeat part (b) for d = .

Question 61

Obtain the pattern function of a uniformly excited rectangular array of N 1 × N 2 parallel half-wave dipoles. Assume that the dipoles are parallel to the z -axis and their centers are spaced d 1 and d 2 apart in the x - and y -directions, respectively.

Question 62

A turnstile antenna consists of two perpendicular half-wave dipoles, one (antenna A) lying along the x -axis and the other (antenna B) along the y-axis. The output of antenna B, after a 90° phase retardation, is combined with that of antenna A. A right-hand ellipti­cally polarized plane wave E i = E 0 ( a x + a y jp) exp (jkz) is incident on the antennas.
a) Determine the open-circuit voltage at the output terminals of the turnstile antenna. What is its value if p = 1
b) Repeat part (a) for a left-hand elliptically polarized incident wave E i = E 0 ( a x - a y jp) exp (jkz).
c) Repeat part (a) for a linearly polarized incident wave E i = a x E 0 exp (jkz).
(Hint: Find the complex effective length of the turnstile antenna and use Eq. 11-76.)

Question 63

In what essential ways does the radiation pattern of a long traveling-wave antenna differ from that of an unterminated dipole antenna

Question 64

Why are the near-zone fields called quasi-static fields

Question 65

Define radiation intensity.

Question 66

What are the radiation resistance and directivity of a vertical quarter-wave monopole over a conducting ground

Question 67

In the radiation pattern of a uniform linear array of many elements, how many decibels down from the principal maximum are the first sidelobes

Question 68

What is the essential difference between the radiation characteristics of a helical antenna and a dipole antenna

Question 69

The transmitting antenna of a radio navigation system is a vertical metal mast 40 (m) in height insulated from the earth. A 180 (kHz) source sends a current having an amplitude of 100 (A) into the base of the mast. Assuming the current amplitude in the antenna to decrease linearly toward zero at the top of the mast and the earth to be a perfectly conducting plane, determine:
a) the effective length of the antenna,
b) the maximum field intensity at a distance 160 (km) from the antenna,
c) the time-average radiated power,
d) the radiation resistance.

Question 70

A horizontal elemental electric dipole of length dl and carrying a time-harmonic current of amplitude I 0 in the + y-direction is situated at a distance d above a perfectly conducting ground. Find its pattern functions
(a) in the xy-plane,
(b) in the xz-plane, and
(c) in the yz-plane.
(d) Sketch the patterns for parts (a), (b), and (c) for d = /4.

Question 71

Assume that a linearly polarized plane electromagnetic wave is incident on a half­wave dipole, as in Fig. 11-8.
a) Obtain an expression for the effective area, A e ( ).
b) Calculate the maximum value of A e for 100 (MHz).

Question 72

A helical antenna operating in the normal mode has N turns with diameter 2b and interturn spacing s. Both 2b and s are very small in comparison to /N and are adjusted to radiate circularly polarized waves. Find:
a) its directive gain and directivity,
b) its radiation resistance.

Question 73

What are the two different operating modes of a helical antenna Explain.

Question 74

Explain how the magnitude of far fields varies with distance.

Question 75

Define directive gain and directivity of an antenna.

Question 76

Define the effective length of a linear antenna for transmitting. Upon what factors does it depend

Question 77

How can the sidelobes of an equally spaced linear array be made lower than those of a uniform linear array

Question 78

What is a Yagi-Uda antenna

Question 79

Starting from Maxwell's equations, derive the nonhomogeneous wave equations (a) for E , and (b) for H in a simple medium.

Question 80

A time-harmonic uniform current I 0 cos t flows in a small circular loop of radius b ( ) lying in the xy -plane.
a) Find the radiation resistance R r of the magnetic dipole.
b) Obtain an expression for its radiation efficiency n r if the loop is made of copper wire of radius a ,
c) Calculate R r and n f for f = 1 (MHz), b = 50 (cm), and a = 3 (mm).
d) Rework part (c) if the loop has ten closely wound insulated turns.