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Chapter 10

- Antennas and Antenna Arrays

10-1 Overview

- Antenna structures designed for radiating and

receiving electromagnetic energy effectively in a

prescribed manner. - Antenna array a number of antennas arranged

together to obtain directivity and other

desirable properties. - Procedure for determining radiation

characteristics of an antenna - Determine the magnetic potential A. The phasor

retarded vector magnetic potential

Overview (continued)

- where is the

wavenumber. - Find the magnetic field intensity H from A.
- Find the electric field intensity E from H with

J0 in space. - After knowing E and H, all other radiation

characteristics of the antenna can be determined.

10-2 The Elemental Electric Dipole

- Hertzian dipole a very short (compared to the

operating wavelength), thin, conducting wire of

length

The Elemental Electric Dipole (continued)

- The radiation characteristics of a Hertzian

dipole that carries a time-harmonic current - Step 1 Find the phasor representation of the

retarded vector potential A. - From Eq. (10-1)
- where
- since

The Elemental Electric Dipole (continued)

- the spherical components of

are - Step 2 Find H from A.

The Elemental Electric Dipole (continued)

- Step 3 Find E from H.
- which gives
- where

The Elemental Electric Dipole (continued)

- Far field (radiation field) the field at

distances very far from the antenna (

). - Far fields of a Hertzian dipole
- The other field components can be neglected.

10-3 Antenna Patterns and Directivity

- Radiation pattern of an antenna (antenna pattern)

the graph that describes the relative far-zone

field strength versus direction at a fixed

distance from an antenna. - E-plane pattern the magnitude of the normalized

field strength (with respect to the peak value)

versus for a constant - H-plane pattern the magnitude of the normalized

field strength versus for

Example 10-1

- Plot the E-plane and H-plane radiation patterns

of a Hertzian dipole. - Since in the far zone are

proportional to each other, we need only consider

the normalized magnitude of - E-plane pattern (Fig. 10-2(a)) from Eq. (10-10)
- , which represents a pair of circles.
- H-plane pattern Fig. 10-2(b))

Example 10-1 (continued)

Antenna Patterns and Directivity (continued)

- Radiation intensity, U the time-average power

per unit solid angle (SI unit watt per

steradian (W/sr)). - The total time-average power radiated
- where is the differential solid angle
- Directive gain of an antenna pattern the ratio

of the radiation intensity in the direction

to the average radiation intensity

Antenna Patterns and Directivity (continued)

- The directive gain of an isotropic or

omnidirectional antenna (an antenna that radiates

uniformly in all directions) unity. - Antenna directivity, D the maximum directive

gain of an antenna, the ratio of the maximum

radiation intensity to the average radiation

intensity. - Calculating directivity from far-zone electric

field intensity

Example 10-2

- Find the directive gain and the directivity of a

Hertzian dipole. - The magnitude of the time-average Poynting vector

- From Eqs. (10-9), (10-10), and (10-12),
- Directive gain

Example 10-2 (continued)

- Directivity (the maximum value of )
- which corresponds to

Antenna Patterns and Directivity (continued)

- Power gain (gain), of an antenna referred

to an isotropic source the ratio of its maximum

radiation intensity to the radiation intensity of

a lossless isotropic source with the same power

input. - Ohmic power loss, radiated power, total

input power, - Power gain of an antenna
- Radiation efficiency ratio of the power gain to

the directivity of an antenna

Antenna Patterns and Directivity (continued)

- The efficiency of well-constructed antenna very

close to 100. - Antenna radiation resistance the value of

hypothetical resistance that would dissipate an

amount of power equal to the radiated power

when the current in the resistance is equal to

the maximum current along the antenna. - A high radiation resistance is a desirable

property for an antenna.

Example 10-3

- Find the radiation resistance of a Hertzian

dipole. - Assuming no ohmic losses, the time-average power

radiated by a Herztian dipole - Using the far-zone fields in Eqs. (10-9) and

(10-10) with a current amplitude

Example 10-3 (continued)

- Equating radiation resistance

- If is only about 0.08 , an

extremely small value.

Example 10-4

- Find the radiation efficiency of an isolated

Hertzian dipole made of a metal wire of radius a,

length d, and conductivity - Amplitude of current I, loss resistance of the

wire dipole , radiation resistance - Ohmic power loss
- Radiated power

Example 10-4 (continued)

- Radiation efficiency
- Loss resistance of the metal wire in terms of the

surface resistance - where

Example 10-4 (continued)

- The radiation efficiency of an isolated Hertzian

dipole - Assume that
- We find that

Example 10-4 (continued)

- and
- which is very low.
- Smaller values of and lower

the radiation efficiency.

10-4 Thin Linear Antenna

- Linear dipole antenna centered thin straight

antenna having a length comparable to a

wavelength (Fig. 10-3).

Thin Linear Antenna (continued)

- The current phasor
- The far-field contribution from the differential

current element - In the far zone,

Thin Linear Antenna (continued)

- Using Eqs. (10-32) and (10-34) in Eq. (10-33),
- The integrand containing the product of two even

functions of z, - yields a

nonzero value. - Equation (10-35) then reduces to

Thin Linear Antenna (continued)

- where
- The factor E-plane pattern function

(Fig. 10-4).

10-4.1 The Half-Wave Dipole

- Half-wave dipole length
- With the pattern

function (Fig. 10-4(a)) - A maximum unity at nulls at
- The far-zone field phasors

The Half-Wave Dipole (continued)

- The magnitude of the time-average Poynting vector

- The total power radiated by a half-wave dipole
- The integral in Eq. (10-41) evaluated numerically

1.218. - Hence

The Half-Wave Dipole (continued)

- The radiation resistance of a free-standing

half-wave dipole - Maximum radiation intensity
- The directivity of a half-wave dipoles
- which corresponds to

referring to an omnidirectional radiator.

Example 10-5

- A thin quarter-wavelength vertical antenna over a

perfectly conducting ground is excited by a

time-harmonic source at its base. Find its

radiation pattern, radiation resistance, and

directivity. - The method of images replace the conducting

ground by the image of the vertical antenna (Fig.

10-5(b)). - The quarter-wave vertical antenna in Fig. 10-5(a)

(quarter-wave monopole) the half-wave antenna

in Fig. 10-5(b). - The pattern function applies here for

Example 10-5 (continued)

- The radiation pattern drawn in dashed lines in

Fig. 10-5(b) the upper half of that in Fig.

10-4(a). - The total radiated power is only one-half that

given in Eq. (10-42) - The radiation resistance
- Directivity the same as the directivity of a

half-wave antenna.

Example 10-5 (continued)

10-5 Antenna Arrays

- Antenna arrays A group of several antenna

elements in various configurations (straight

lines, circles, triangles, and so on) with proper

amplitude and phase relations to give certain

desired radiation characteristics.

10-5.1 Two-Element Arrays

- The antennas are excited with a current of the

same amplitude, but the phase in antenna 1 leads

that in antenna 0 by an angle - Far-zone field phasors at point
- where is the pattern function of

the individual antennas, and is an

amplitude function. - The electric field of the two-element array

Two-Element Arrays (continued)

- In the far-zone
- Thus
- where
- The magnitude of the electric field of the array

- where element factor

normalized array factor.

Two-Element Arrays (continued)

- The element factor the magnitude of the pattern

function of the individual radiating elements. - The array factor depends on array geometry as

well as on the relative amplitudes and phases of

the excitations in the elements. - Principle of pattern multiplication the pattern

function of an array of identical elements is

described by the product of the element factor

and the array factor.

Example 10-6

- Plot the H-plane radiation patterns of two

parallel dipoles for the following cases - In the H-plane (Fig. 10-6), the dipole is

omnidirectional, and the normalized pattern

function is equal to the normalized array factor - Thus
- a)

Example 10-6 (continued)

- Broadside array (Fig. 10-7(a)) the pattern has

its maximum at - Their electric fields add in the broadside

direction and cancel each other at - b)
- which has maximum at and vanishes at
- The pattern maximum (Fig. 10-7(b)) in a

direction along the line of array (endfire

array).

Example 10-6 (continued)

Example 10-7

- Discuss radiation pattern of a linear array of

the three isotropic sources spaces apart.

The excitations in the sources are in-phase and

have amplitude ratios 121. - This three-source array two two-element arrays

displaced - (Fig. 10-8).

Example 10-7 (continued)

- By the principle of pattern multiplication we

obtain - The radiation pattern is sketched in Fig. 10-9

(sharper).

Binomial Arrays

- In a binomial array of N elements, the array

factor is a binomial function and

the excitation amplitudes vary according to the

coefficients of a binomial coefficients - To obtain a directive pattern without sidelobes,

d in a binomial array is normally restricted to

be

10-5.2 General Uniform Linear Array

- Uniform linear array an array of more than two

identical antennas equally spaced along a

straight line which is fed with currents of equal

magnitude and a uniform progressive phase shift

along the line (Fig. 10-10).

General Uniform Linear Array (continued)

- The normalized array factor in the xy-plane
- where
- Array factor of an N-element uniform linear array

- Several significant properties

General Uniform Linear Array (continued)

- Main-beam direction. The maximum value of

occurs when or when - which leads to
- Two special cases
- a) Broadside array. Maximum radiation

occurs at a direction perpendicular to the line

of the array - All the elements in a linear broadside

array should be excited in phase. - b) Endfire array. Maximum radiation occurs

at

General Uniform Linear Array (continued)

- Phased arrays antenna arrays equipped

with phase shifters to steer the main beam

electronically. - Sidelobe locations. Sidelobes minor maxima that

occurs approximately when the numerator on the

right side of Eq. (10-60) is a maximum when

or when - The first sidelobe occurs when

General Uniform Linear Array (continued)

- First sidelobe level. The amplitude of the first

sidelobes - down

from the principal maximum which is almost

independent of N as long as N is large. - Tapering excitation amplitudes reduces array

sidelobes.

Example 10-8

- For a five-element uniform linear array with

spacing, find the width of the main beam for

(a) broadside operation, and (b) endfire

operation. - The width of the main beam the region of the

pattern between the first nulls on either side of

the direction of maximum radiation. - The first nulls of the array pattern occur at

that makes (see Eq. 10-60)

Example 10-8 (continued)

- For this example,
- a) Broadside operation.

At first nulls, - b) endfire operation.

At first nulls,

Example 10-8 (continued)

- Width of main beam of endfire array is wider than

that of the corresponding broadside array. - A typical graph of the normalized array factor in

Eq. (10-60) (Fig. 10-11 for N5) a rectangular

plot of - The different transformations in Eqs. (10-64a)

and (10-64b) lead to different array patterns

versus for the same array factor.

Normalized Array Factor of a Five-Element Uniform

Linear Array

10.6 Effective Area and Backscatter Cross Section

- Reciprocity relations for antenna in transmitting

and receiving modes. - The equivalent generator impedance of an antenna

in the receiving mode is equal to the input

impedance of the antenna in the transmitting

mode, and - The directional pattern of an antenna for

reception is identical with that for

transmission. - Approximate Thevenins equivalent circuit at the

receiving end (Fig. 10-12), where the

open-circuit voltage induced, equivalent

generator impedance in the receiving mode,

load impedance.

Effective Area and Backscatter Cross Section

(continued)

10-6.1 Effective Area

- Effective area of a receiving antenna the ratio

of the average power delivered to a matched load

to the time-average power density of the incident

electromagnetic wave at the antenna. - Under matched conditions,
- Neglecting losses, the antenna input impedance in

the transmitting mode - where denotes the radiation resistance.

Effective Area (continued)

- The average power delivered to the matched load

- The time-average power density at the receiving

site - where denote the amplitude of the

electric field intensity at the receiving antenna.

Example 10-9

- Determine the effective area, of an

elemental electric dipole of a length

used to receive an incident plane electromagnetic

wave of wavelength Assume that the dipole

axis makes an angle with the direction of the

incident electromagnetic wave. - The induced open-circuit voltage
- The radiation resistance of the elemental

electric dipole

Example 10-9 (continued)

- The average power delivered to the matched load
- Effective area of Hertzian dipole

Effective Area (continued)

- The directive gain of a Hertzian dipole
- Relation between effective area and directive

gain of an antenna - The above equation holds for any antenna.

10-6.2 Backscatter Cross Section

- Backscatter cross section (radar cross section)

of an object the equivalent area that would

intercept that amount of incident power in order

to produce the same scattered power density at

the receiver site if the object scattered

uniformly (isotropically) in all direction. - Let
- Then

Backscatter Cross Section (continued)

- The backscatter cross section is a measure of the

detectability of the object (target) by radar

(radio detection and ranging) hence the term

radar cross section. - It is a composite measure, depending on the

geometry, orientation, constitutive parameters

and surface conditions of the object, and on the

frequency and polarization of the incident wave

in a complicated way.

10-7 Friis Transmission Formula and Radar Equation

- The average power density at antenna 2 at a

distance r away - where is the total power radiated by

antenna 1 having a directive gain - A received power in a matched load if antenna

2 has an effective area

Friis Transmission Formula and Radar Equation

(continued)

- Friis transmission formula the relation in Eq.

(10-79). - Alternative form of Friis transmission formula
- A radar system uses the same antenna for

transmitting short pulses of time-harmonic

radiation and for receiving the energy scattered

back from a target (Fig. 10-13).

Friis Transmission Formula and Radar Equation

(continued)

Friis Transmission Formula and Radar Equation

(continued)

- The power density at a target at a distance r

away - The received power
- By using Eq. (10-75), the radar equation

Friis Transmission Formula and Radar Equation

(continued)

- Alternative form of radar equation
- Geosynchronous satellites the satellites appear

to be stationary with respect to the earths

surface (geostationary). - The radius of geosynchronous orbit 42,300 (km)

(about 36,000 (km) from the earths surface). - Signals are transmitted from a high-gain antenna

at an earth station toward a satellite, which

receives the signals, amplifies them, and

retransmits them back toward the earth station at

a different frequency.

Example 10-10

- An microwave link is to be established over a

distance of 10 miles at 300 (MHz) by using two

identical parabolic reflectors, each having a

directive gain of 30 (dB). The transmitting

antenna radiates a power of 500 (W). Neglecting

losses, find (a) the power received, and (b) the

magnitude of electric field intensity at the

receiving antenna. - a)
- Using Eq. (10-80), we have

Example 10-10 (continued)

- b) From Eqs. (10-77) and (10-69),
- Thus,

Example 10-11

- Assume that 50 (kW) is fed into the antenna of a

radar system operating at 3 (GHz). The antenna

has an effective area of 4 and a radiation

efficiency of 90. The minimum detectable signal

power (over noise inherent in the receiving

system and from environment) is 1.5 (pW), and the

power reflection coefficient for the antenna on

receiving is 0.05. Determine the maximum usable

range of the radar for detecting a target with a

backscatter cross section of 1 - At

Example 10-11 (continued)

- From Eq. (10-84),
- and

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