Antennas and Antenna Arrays

1
Chapter 10

• Antennas and Antenna Arrays

2
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

3
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.

4
10-2 The Elemental Electric Dipole

• Hertzian dipole a very short (compared to the
  operating wavelength), thin, conducting wire of
  length

5
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

6
The Elemental Electric Dipole (continued)

• the spherical components of
  are
• Step 2 Find H from A.

7
The Elemental Electric Dipole (continued)

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

8
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.

9
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

10
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))

11
Example 10-1 (continued)
12
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

13
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

14
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

15
Example 10-2 (continued)

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

16
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

17
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.

18
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

19
Example 10-3 (continued)

• Equating radiation resistance
  
• If is only about 0.08 , an
  extremely small value.

20
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

21
Example 10-4 (continued)

• Radiation efficiency
• Loss resistance of the metal wire in terms of the
  surface resistance
• where

22
Example 10-4 (continued)

• The radiation efficiency of an isolated Hertzian
  dipole
• Assume that
• We find that

23
Example 10-4 (continued)

• and
• which is very low.
• Smaller values of and lower
  the radiation efficiency.

24
10-4 Thin Linear Antenna

• Linear dipole antenna centered thin straight
  antenna having a length comparable to a
  wavelength (Fig. 10-3).

25
Thin Linear Antenna (continued)

• The current phasor
• The far-field contribution from the differential
  current element
• In the far zone,

26
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

27
Thin Linear Antenna (continued)

• where
• The factor E-plane pattern function
  (Fig. 10-4).

28
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

29
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

30
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.

31
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

32
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.

33
Example 10-5 (continued)
34
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.

35
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
  

36
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.

37
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.

38
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)

39
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).

40
Example 10-6 (continued)
41
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).

42
Example 10-7 (continued)

• By the principle of pattern multiplication we
  obtain
• The radiation pattern is sketched in Fig. 10-9
  (sharper).

43
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

44
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).

45
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

46
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

47
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

48
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.

49
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)

50
Example 10-8 (continued)

• For this example,
• a) Broadside operation.
  At first nulls,
• b) endfire operation.
  At first nulls,

51
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.

52
Normalized Array Factor of a Five-Element Uniform
Linear Array
53
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.

54
Effective Area and Backscatter Cross Section
(continued)
55
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.

56
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.

57
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

58
Example 10-9 (continued)

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

59
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.

60
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

61
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.

62
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

63
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).

64
Friis Transmission Formula and Radar Equation
(continued)
65
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

66
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.

67
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

68
Example 10-10 (continued)

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

69
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

70
Example 10-11 (continued)

• From Eq. (10-84),
• and