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Dr. Rakhesh Singh Kshetrimayum
8. Antennas and Radiating Systems
Dr. Rakhesh Singh Kshetrimayum
7/6/20131 Electromagnetic Field Theory by R. S. Kshetrimayum
8.1 Introduction Antenna is a device used for radiating and receiving EM waves
Any wireless communication can’t happen without antennas.
Antennas have many applications like in mobile communications (all mobile phones has in-built antennas)
wireless local areas networks (your laptop connecting wireless
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum2
wireless local areas networks (your laptop connecting wireless internet also has antennas)
television (old TV antennas are Yagi-Uda antennas, now, generally disc antennas are employed for direct to home (DTH) TVs)
satellite communications (usually have large parabolic antennas or microstrip antenna arrays)
rockets and missiles (microstrip antenna arrays)
8.1 Introduction
Antennas
Radiation fundamentals
Antenna pattern and parameters
Types of antennas
Antenna arrays
When does a charge radiate?
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum3
Fig. 8.1 Antennas
charge radiate?
Wave equation for potential functions
Solution of wave equation for potential functions
Hertz dipole
Dipole antenna
Loop antenna
8.2 Radiation fundamentals 8.2.1 When does a charge radiates? accelerating/decelerating charges radiate EM waves
r
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum4
Fig. 8.2 A giant sphere of radius r with a source of EM wave at its origin
Source
r
8.2 Radiation fundamentals Consider a giant sphere of radius r which encloses the source of EM waves at the origin (Fig. 8.2)
The total power passing out of the spherical surface is given by Poynting theorem,
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum5
( ) ( )∫∫ •×=•= ∗sdHEsdSrP avgtotal
rrrrrRe
( )lim
totalP P r
r=
→∝
8.2 Radiation fundamentals This is the energy per unit time that is radiated into infinity and it never comes back to the source
The signature of radiation is irreversible flow of energy away from the source
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum6
from the source
Let us analyze the following three cases:
CASE 1: A stationary charge will not radiate. no flow of charge =>no current=>no magnetic field=>no radiation
8.2 Radiation fundamentalsCASE 2: A charge moving with constant velocity will not radiate.
The area of the giant sphere of Fig. 8.2 is 4> r2
So for the radiation to occur Poynting vector must decrease no faster than 1/r2
from Coloumb’s law, electrostatic fields decrease as 1/r2,
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum7
from Coloumb’s law, electrostatic fields decrease as 1/r ,
whereas Biot Savart’s law states that magnetic fields decrease as 1/r2
So the total decrease in the Poynting vector is proportional to 1/r4 no radiation
CASE 3: A time varying current or acceleration (or deceleration) of charge will radiate.
8.2 Radiation fundamentals Basic radiation equation:
where L=length of current element, m
=time changing current, As-1(units)
dt
dvQ
dt
diL =
di
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum8
=time changing current, As-1(units)
Q=charge, C
=acceleration of charge, ms-2
dt
di
dt
dv
8.2 Radiation fundamentals
To create radiation there must be a time varying current or
acceleration (or deceleration) of charge
Static charges=>no radiation
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum9
Static charges=>no radiation
If the charge motion is time varying with acceleration or deceleration then there will be radiation even if the wire is straight
8.2 Radiation fundamentals
Charges moving with uniform velocity: no radiation if the wire is straight and infinite in extent and
radiation if the wire is curved,
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum10
curved,
bent,
discontinuous,
terminated or
truncated (these will either accelerate or decelerate the charge)
8.2 Radiation fundamentals 8.2.2 Wave equation for potential functions
From Maxwell’s equations for time varying fields, we can derive the two wave equations for potential functions (magnetic vector and electric potentials)
rr
r ∂ 2 2V ρ∂
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum11
Jt
AA
rr
rµµε −=
∂
∂−∇
2
22
22
2;
VV
t
ρµε
ε
∂∇ − = −
∂
8.2 Radiation fundamentals 8.2.3 Solution of wave equation for potential functions
For time harmonic functions of potentials,
where
JAArrr
µβ −=+∇ 22
LCβ ω=
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum12
where
To solve the above equation, we can apply Green’s function technique
Green’s function G is the solution of the above equation with the R.H.S equal to a delta function
LCβ ω=
( )spaceGG δβ =+∇ 22
8.2 Radiation fundamentals Once we obtain the Green’s function,
we can obtain the solution for any arbitrary current source by applying the convolution theorem
For radiation problems, the most appropriate coordinate system is spherical
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum13
the most appropriate coordinate system is spherical since the wave travels out radially in all directions
It has also symmetry along θ and directions
Hence, the above equation reduces to
0G G
θ φ
∂ ∂= =
∂ ∂
( )rGr
Gr
rrδβ =+
∂
∂
∂
∂ 22
2
1
8.2 Radiation fundamentals Putting Ψ= G r,
For r not equal to 0,
( )rrr
δβ =Ψ+Ψ∂
∂ 2
2
2
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum14
Therefore,
02
2
2
=Ψ+Ψ∂
∂β
r
rjrjBeAe
ββ +− +=Ψ
8.2 Radiation fundamentals Since the radiation travels radially in positive r direction
negative r direction is not physically feasible for a source of a field, we get,
rjAe
β−=Ψ
rjβ−
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum15
From example 8.2, we can find the constant A and hence
r
AeG
rjβ−
=
r
eGA
rj
ππ
β
4;
4
1 −
−=−=
8.2 Radiation fundamentals Since the medium surrounding the source is linear,
we can obtain the potential for any arbitrary current input by the convolution of the impulse function (Green’s function) with the input current
( )'rrj −−
r
rrβ
µ
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum16
( ) '
'
'0
4dv
rr
erJA
V
rrj
∫−
=
−−
rr
rrβ
π
µ
8.2 Radiation fundamentals The prime coordinates denote the source variables
unprimed coordinates denote the observation points
The modulus sign in is to make sure that
is positive
since the distance in spherical coordinates is always positive
'rr −
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum17
since the distance in spherical coordinates is always positive
8.3 Antenna pattern and parameters
8.3.1 What is antenna radiation pattern?
The radiation pattern of an antenna is a 3-D graphical representation of the radiation properties of the antenna as a function of position (usually in spherical coordinates)
If we imagine an antenna is placed at the origin of a spherical
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum18
If we imagine an antenna is placed at the origin of a spherical coordinate system, its radiation pattern is given by measurement of the magnitude of the electric field over a surface of a sphere of radius r
8.3 Antenna pattern and parameters
For a fixed r, electric field is only a function of θ and
Two types of patterns are generally used: (a) field pattern (normalized or versus spherical coordinate position) and
φ
( ),E θ φr
Er
Hr
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum19
coordinate position) and
(b) power pattern (normalized power versus spherical coordinate position).
8.3 Antenna pattern and parameters
3-D radiation patterns are difficult to draw and visualize in a 2-D plane like pages of this book
Usually they are drawn in two principal 2-D planes which are orthogonal to each other
Generally, xz- and xy- plane are the two orthogonal principal
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum20
Generally, xz- and xy- plane are the two orthogonal principal planes
E-plane (H-plane) is the plane in which there are maximum electric (magnetic) fields for a linearly polarized antenna
8.3 Antenna pattern and parameters
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum21
Fig. 8.3 (c) Typical radiation pattern of an antenna
maxθ
θ
8.3 Antenna pattern and parameters
A typical antenna radiation pattern looks like as in Fig. 8.3 (c)
It could be a polar plot as well
An antenna usually has either one of the following patterns: (a) isotropic (uniform radiation in all directions, it is not possible to realize this practically)
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum22
possible to realize this practically)
(b) directional (more efficient radiation in one direction than another)
(c) omnidirectional (uniform radiation in one plane)
8.3 Antenna pattern and parameters
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum23
Fig. 8.3 (a) Antenna field regions
3
max10 0.62nf
Dr
λ< ≤
3 2
max max2
20.62 nf
D Dr
λ λ< ≤
2
max2ff
Dr
λ<
8.3 Antenna pattern and parameters
The antenna field regions could be divided broadly into three regions (see Fig. 8.3 (a)):
Reactive near field region:
This is the region immediately surrounding the antenna where the reactive field (stored energy-standing waves)
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum24
where the reactive field (stored energy-standing waves) dominates
Reactive near field region is for a radius of
where Dmax is the maximum antenna dimension
3
max10 0.62
nf
Dr
λ< ≤
8.3 Antenna pattern and parameters
Radiating near field (Fresnel) region:
The region in between the reactive near field and the far-field (the radiation fields are dominant)
the field distribution is dependent on the distance from the antenna
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum25
antenna
Radiating near field (Fresnel) region is usually for a radius of
3 2
max max2
20.62 nf
D Dr
λ λ< ≤
8.3 Antenna pattern and parameters
Far field (Fraunhofer) region:
This is the region farthest from the antenna where the field distribution is essentially independent of the distance from the antenna (propagating waves)
Fraunhofer far field region is usually for a radius of2
max2ff
Dr
<
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum26
Fraunhofer far field region is usually for a radius of
In the far field region, the spherical wavefront radiated from a source antenna can be approximated as plane wavefront
The phase error in approximating this is π/8
maxffr
λ<
8.3 Antenna pattern and parameters
Fig. 8.3 (b) Illustration of
far field region
(antenna under test: AUT)
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum27
8.3 Antenna pattern and parameters
We can calculate the distance rff by equating the maximum error (which is at the edges of the AUT of maximum dimension Dmax) in the distance r by approximating spherical wavefront to plane wavefront to λ/16 (Exercise 8.1)
8.3.2 Direction of the main beam (θmax)
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum28
8.3.2 Direction of the main beam (θmax)
A radiation lobe is a clear peak in the radiation intensity surrounded by regions of weaker radiation intensity
8.3 Antenna pattern and parameters
Main beam is the biggest lobe in the radiation pattern of the antenna
It is the radiation lobe in the direction of maximum radiation
θmax is the direction in which maximum radiation occurs
Any lobe other than the main lobe is called as minor lobe
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum29
Any lobe other than the main lobe is called as minor lobe
The radiation lobe opposite to the main lobe is also termed as back lobe (This will be more appropriate for polar plot of radiation pattern)
8.3 Antenna pattern and parameters
8.3.3 Half power beam width (HPBW)
It is the angular separation between the half of the maximum power radiation in the main beam
At these points, the radiation electric field reduces by of the maximum electric field
1
2
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum30
of the maximum electric field
Half power is also equal to -3-dB
We also call HPBW as -3-dB beamwidth
They are measured in the E-plane and H-plane radiation patterns of the antenna
8.3 Antenna pattern and parameters
8.3.4 Beam width between first nulls (BWFN)
It is the angular separation between the first two nulls on either side of the main beam
For same values of BWFN, we can have different values of HPBW for narrow beams and broad beams
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum31
HPBW for narrow beams and broad beams
HPBW is a better parameter for specifying the effective beam width
8.3 Antenna pattern and parameters
8.3.5 Side lobe level (SLL)
The side lobes are the lobes other than the main beam and it shows the direction of the unwanted radiation in the antenna radiation pattern
The amplitude of the maximum side lobe in comparison to
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum32
The amplitude of the maximum side lobe in comparison to the main beam maximum amplitude of the electric field is called as side lobe level (SLL)
It is normally expressed in dB and a SLL of -30 dB or less is considered to be good for a communication system
8.3 Antenna pattern and parameters
8.3.6 Radiation intensity
Radiation intensity U(θ, ) is defined as U(θ, )= Power along direction (θ, )/Solid angle (dΩ)
φ φ
φ
2
( , ) ( , ) sinP U d U d d
π π
θ φ θ φ θ θ φ= Ω =∫ ∫ ∫
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum33
4 0 0
( , ) ( , ) sinradP U d U d dπ θ φ
θ φ θ φ θ θ φΩ= = =
= Ω =∫ ∫ ∫
2
0 0
1( , )sin
4 4
radavg
PU U d d
π π
θ φ
θ φ θ θ φπ π
= =
= = ∫ ∫
8.3 Antenna pattern and parameters
8.3.7 Directivity
The directivity of an antenna is defined as the ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions which equivalent to the radiation intensity of an isotropic
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum34
which equivalent to the radiation intensity of an isotropic antenna
2
0 0
( , ) 4 ( , )( , )
( , )sinavg
U UD
UU d d
π π
θ φ
θ φ π θ φθ φ
θ φ θ θ φ= =
= =
∫ ∫
8.3 Antenna pattern and parameters
D (θ, ) is maximum at θmax and minimum along θnull
is also known as beam solid angle
φ
( ) max maxmax 2
0 0
4 ( , ) 4,
( , )sin A
U UD
U d d
π π
θ φ
π θ φ πθ φ
θ φ θ θ φ= =
= = =Ω
∫ ∫avgU
Ω
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum35
is also known as beam solid angle
It is also defined as the solid angle through which all the antenna power would flow if the radiation intensity was for all angles in
AΩ
max ( , )U θ φ
AΩ
8.3 Antenna pattern and parameters
Given an antenna with one narrow major beam, negligible radiation in its minor lobes
where and are the half-power beam widths in
HPBW HPBW
rad rad
A θ φΩ ≈ ×
θ φ
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum36
where and are the half-power beam widths in radians which are perpendicular to each other
For narrow beam width antennas
It can be shown that the maximum directivity is given by
HPBWθHPBWφ
( ), 1HPBW HPBW
θ φ <<
max
4
HPBW HPBW
rad radD
π
θ φ≅
×
8.3 Antenna pattern and parameters
If the beam widths are in degrees, we have
8.3.8 Gain
2
max deg deg deg deg
1804
41,253
HPBW HPBW HPBW HPBW
D
ππ
θ φ θ φ
≅ =
× ×
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum37
8.3.8 Gain
In defining directivity, we have assumed that the antenna is lossless
But, antennas are made of conductors and dielectrics
8.3 Antenna pattern and parameters
It has same in-built losses accompanied with the conductors and dielectrics,
Thereby, the power input to the antenna is partly radiated and remaining part is lost in the imperfect conductors as well as in dielectrics
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum38
as in dielectrics
The gain of an antenna in a given direction is defined as the ratio of the intensity in a given direction to the radiation intensity that would be obtained if the power accepted by the antenna were radiated isotropically
8.3 Antenna pattern and parameters
Note that definitions of the antenna directivity and gain are essentially the same except for the power terms used in the definitions
( ) radrad
4 ( , )e4 ( , ), e ( , )
input rad
UUG D
P P
π θ φπ θ φθ φ θ φ= = =
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum39
essentially the same except for the power terms used in the definitions
Directivity is the ratio of the antenna radiated power density at a distant point to the total antenna radiated power radiated isotropically
Gain is the ratio of the antenna radiated power density at a distant point to the total antenna input power radiated isotropically
8.3 Antenna pattern and parameters
The antenna gain is usually measured based on Friistransmission formula and it requires two identical antennas
One of the identical antennas is the radiating antenna, and the other one is the receiving antenna
Assuming that the antennas are well matched in terms of
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum40
Assuming that the antennas are well matched in terms of impedance and polarization, the Friis transmission equation is
2
10 10
1 420log 10log
4 2
r rt r t r
t t
P PRG G G G G G
P R P
λ π
π λ
= = = ∴ = +
Q
8.3 Antenna pattern and parameters
Friis transmission equation states that the ratio of the received power at the receiving antenna and transmitted power at the transmitting antenna is: directly proportional to both gains of the transmitting (Gt) and receiving (Gr) antennas
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum41
receiving (Gr) antennas
inversely proportional to square of the distance between the transmitting and receiving antennas (1/R2) and
directly proportional to the square of the wavelength of the signal transmitted (λ2)
8.3 Antenna pattern and parameters
Assumptions made are:
(a) antennas are placed in the far-field regions
(b) there is free space direct line of sight propagation between the two antennas
(c) there are no interferences from other sources and no
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum42
(c) there are no interferences from other sources and no multipaths between the transmitting and receiving antennas due to reflection, refraction and diffraction
8.3 Antenna pattern and parameters
8.3.9 Polarization
Let us consider antenna is placed at the origin of a spherical coordinate system and wave is propagating radially outward in all directions
In the far field region of an antenna,
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum43
In the far field region of an antenna,
( ) ( ) ( )ˆ ˆ, , ,E E Eθ φθ φ θ φ θ θ φ φ= +r
8.3 Antenna pattern and parameters
Putting the time dependence, we have,
where δ is the phase difference between the elevation and azimuthal components of the electric field
( ) ( ) ( ) ( ) ( ) ( ) ( )ˆ ˆ ˆ ˆ, , , cos , cos , , , ,E t E t E t E t E tθ φ θ φθ φ θ φ ω θ θ φ ω δ φ θ φ θ θ φ φ= + + = +r
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum44
azimuthal components of the electric field
The figure traced out by the tip of the radiated electric field vector as a function of time for a fixed position of space can be defined as antenna polarization
8.3 Antenna pattern and parameters
a) LP
When δ=0, the two transversal electric field components are in time phase
The total electric field vector
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum45
makes an angle with the -axis
( ) ( ) ( ) ( ) ( )( )ˆ ˆ, , , cos , cosE t E t E tθ φθ φ θ φ ω θ θ φ ω φ= +r
LPθ θ
( )( )
( )( )
1 1, , , ,
tan tan, , , ,
LP
E t E t
E t E t
φ φ
θ θ
θ φ θ φθ
θ φ θ φ− −
= =
8.3 Antenna pattern and parameters
The tip of the total radiated electric field vector traces out a line
Therefore, the antenna’s polarization is LP
b) CP
When , the two transversal electric field components πδ = ±
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum46
When , the two transversal electric field components are out of phase in time
and if the two transversal electric field components are of equal amplitude
2
πδ = ±
( ) ( ) ( )0, , ,E E Eθ φθ φ θ φ θ φ= =
8.3 Antenna pattern and parameters
The total electric field vector
makes an angle with the -axis
( ) ( ) ( ) ( ) ( )( )ˆ ˆ, , , cos , sinE t E t E tθ φθ φ θ φ ω θ θ φ ω φ= +r
CPθ θ
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum47
This implies that the total radiated electric field vector of the antenna traces out a circle as time progresses from 0 to and so on
( )( )
( )1 1sin
tan tan tancos
CP
tt t
t
ωφ ω ω
ω− −
= = =
π
ω
8.3 Antenna pattern and parameters
If the vector rotates counterclockwise (clockwise) as the wave approaches towards the reader of this book, then the antenna polarization is RHCP (LHCP)
For ( ) ( ), , 0,E E andθ φθ φ θ φ α π≠ ≠
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the total electric field vector traces out an ellipse and hence it is elliptically polarized (EP)
The ratio of the major and minor axes of the ellipse is called AR
For instance, AR=0 dB for CP and AR= ∞ dB for LP
8.3 Antenna pattern and parameters
Let us try to understand two terms (co- and cross-polarization) which are important for the antenna radiation pattern
Co-polarization means you measure the antenna with another antenna oriented in the same polarization with the antenna
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum49
antenna oriented in the same polarization with the antenna under test (AUT)
Cross-polarization means that you measure the antenna with antenna oriented perpendicular w.r.t. the main polarization
8.3 Antenna pattern and parameters
Cross-polarization is the polarization orthogonal to the polarization under consideration
For example, if the field of an antenna is horizontally polarized, the cross-polarization for this case is vertical polarization
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum50
polarization
If the polarization is RHCP, the cross-polarization is LHCP
Let us put this into mathematical expressions:
We may write the total electric field propagating along z-axis as
( )ˆ ˆ j z
co co cr crE E u E u e
β−= +r
8.3 Antenna pattern and parameters
where the co- and cross-polarization unit vectors satisfy the orthonormality condition
Therefore, the co- and cross-polarization components of the
* * * *ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ1, 1, 0, 0co co cr cr co cr cr co
u u u u u u u u• = • = • = • =
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum51
Therefore, the co- and cross-polarization components of the electric fields can be obtained as
* *ˆ ˆ,co co cr crE E u E E u= • = •r r
8.3 Antenna pattern and parameters
a) LP
For a general LP wave, we can write,
For a x-directed LP wave, =0, hence,ˆ ˆ ˆ ˆ ˆ ˆcos sin , sin cos
co LP LP cr LP LPu x y u x yφ φ φ φ= + = −
LPφ
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For a y-directed LP wave, =900, hence,
* *ˆ ˆ ˆ ˆ ˆ ˆ, ; ,co cr co co x cr cr y
u x u y E E u E E E u E= = − = • = = • = −r r
LPφ
* *ˆ ˆ ˆ ˆ ˆ ˆ, ; ,co cr co co y cr cr xu y u x E E u E E E u E= = = • = = • =r r
8.3 Antenna pattern and parameters
b) CP
For a RHCP wave, we can write,
ˆ ˆ ˆ ˆˆ ˆ,
2 2co cr
x jy x jyu u
− += =
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum53
For a LHCP wave, co- and cross-polarization unit vectors and components of the electric field will interchange
* *ˆ ˆ,2 2
x y x y
co co cr cr
E jE E jEE E u E E u
+ −= • = = • =r r
8.3 Antenna pattern and parameters
c) EP
For a EP wave, we can write,
2 2
ˆ ˆ ˆ ˆˆ ˆ,
1 1
EP EPj j
co cr
x Ae y Ae x yu u
A A
φ φ−+ − += =
+ +
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum54
In order to determine the far-field radiation pattern of an AUT, two antennas are required
The one being tested (AUT) is normally free to rotate and it is connected in receiving mode
8.3 Antenna pattern and parameters
Note that AUT as a receiving antenna measurement will generate the same radiation pattern to that of AUT used as a transmitting antenna (from reciprocity theorem)
Another antenna is usually fixed and it is connected in transmitting mode
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transmitting mode
The AUT is rotated by a positioner and it can rotate 1-, 2-and 3-degrees of freedom of rotation
8.3 Antenna pattern and parameters
The AUT is rotated in usually two principal planes (elevation and azimuthal)
The received field strength is measured by a spectrum analyzer or power meter which will be used to generate the antenna radiation pattern in
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum56
which will be used to generate the antenna radiation pattern in two principal planes also known as E- and H- planes
The antenna radiation patterns in these two principal planes can be used to generate the 3-D radiation pattern of an antenna
8.3 Antenna pattern and parameters
8.3.10 Quality factor and bandwidth
The equivalent circuit of a resonant antenna can be approximated by a series RLC resonant circuit
where R=Rr+RL are the radiation and loss resistances,
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum57
L is the inductance and
C is the capacitance of the antenna
8.3 Antenna pattern and parameters
For a resonant antenna like dipoles, the FBW is related to the radiation efficiency and quality factor Q (FBW=1/Q)
The quality factor of an antenna is defined as 2πf0 (f0 is the resonant frequency) times the energy stored over the
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum58
the resonant frequency) times the energy stored over the power radiated and Ohmic losses
( )
( ) ( )
( )
2 2
2
0 00
20
0
1 1 1
4 4 2 2 12
1 2
2
1
2
r L r Lr L
rlossless rad
r r L
I L If L f L
Q fR R f R R C
I R R
RQ e
f R C R R
π ππ
π
π
+
= = =+ ++
= = •+
8.3 Antenna pattern and parameters
where Qlossless is the quality factor when the antenna is lossless (RL=0) and
erad is the antenna radiation efficiency.
Note that the radiation efficiency of an antenna is defined as the ratio of the power delivered to the radiation resistance Rr
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum59
the ratio of the power delivered to the radiation resistance Rr
to the power delivered to Rr and RL
( ) ( )
2
2
1
21
2
rr
rad
r Lr L
I RR
eR R
I R R
= =++
8.3 Antenna pattern and parameters
There is a very important concept on designing electrically small antennas
When kr<1 (electrically small antennas), the quality factor Q of a small antenna can found from the Chu’s relation
( )2
+
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where k is the wave number and r is the radius of the smallest sphere enclosing the antenna
( )
( ) ( )
2
3 2
1 2
1rad
krQ e
kr kr
+= •
+
8.3 Antenna pattern and parameters
The above relation gives the relationship between the antenna size, efficiency and quality factor.
This expression can be reduced further for smallest Q for a LP very small antenna (kr<<1) as follows:
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum61
Harrington gave also a practical upper limit to the gain of a small antenna for a reasonable BW as
( )min 3
1 1Q
krkr= +
( ) ( )2
max 2G kr kr= +
8.3 Antenna pattern and parameters
For example, for a Hertz dipole of very small length 0.01λ, Qmin =32283
It has very high Q and hence a very narrow FBW (0.000031)
Gmax=0.0638 or -12dB
Note that in the above calculations r=0.005λ has been used
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum62
Note that in the above calculations r=0.005λ has been used
But the gain of the antenna is also negative
Antenna size, quality factor, bandwidth and radiation efficiency is interlinked
There is no complete freedom to optimize each one of them independently
8.4 Kinds of antennas 8.4.1 Hertz dipole
An infinitesimally small current element is called a Hertz dipole
Electrically small antennas are small relative to wavelength
Whereas electrically large antennas are large relative to
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum63
Whereas electrically large antennas are large relative to wavelength
Hertz dipole is not of much practical use but it is the basic building block of any kind of antennas
8.4 Kinds of antennas The infinitesimal time-varying current in the Hertz dipole is
where ω is the angular frequency of the current
Since the current is along the z-direction, the magnetic
( ) zeItI tj ˆ0ω=
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum64
Since the current is along the z-direction, the magnetic vector potential at the observation point P is along z-direction
zr
edleIzAA
tjrj
z ˆ4
ˆ 00
π
µ ωβ−
==r
8.4 Kinds of antennas Fig. 8.5 Hertz dipole
located at the origin and
oriented along z-axis ( ), ,P r θ φ
r
θ
φ
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum65
0
j tI e
ω
8.4 Kinds of antennas Note that for this case
For infinitesimally small current element at the origin
using the coordinate transformation from the Rectangular to Spherical coordinate systems
( )' '
0
j tJ r dv I dle
ω=r r
'r r r− =r r r
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Spherical coordinate systems
sin cos sin sin cos sin cos sin sin cos 0
cos cos cos sin sin cos cos cos sin sin 0
sin cos 0 sin cos 0
xr
y
zz
AA
A A
AA A
θ
φ
θ φ θ φ θ θ φ θ φ θ
θ φ θ φ θ θ φ θ φ θ
φ φ φ φ
= − = − − −
cos ; sin ; 0r z zA A A A Aθ φθ θ⇒ = = − =
8.4 Kinds of antennas
Using the symmetry of the problem (no variation in ), we
0
AH
µ
∇×=
rr
Q2
0
ˆ ˆˆ sin
1
sin
sinr
r r r
r r
A rA r Aθ φ
θ θφ
µ θ θ φ
θ
∂ ∂ ∂ = ∂ ∂ ∂
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Using the symmetry of the problem (no variation in ), we have,
2
0
ˆ ˆˆ sin
10
sin
cos sin 0z z
r r r
Hr r
A rA
θ θφ
µ θ θ φ
θ θ
∂ ∂ ∂ = ≡ ∂ ∂ ∂
−
r
8.4 Kinds of antennas
( ) ( )2
0
sin0; 0; sin cos
sinr z z
rH H H rA A
r rθ φ
θθ θ
µ θ θ
∂ ∂ ⇒ = = = − −
∂ ∂
( )0 0ˆ ˆcos sin
sin sin4 4
j t j tj r j rj r j rI dle I dlee e
H e j er r r r r
ω ωβ ββ β
φ
φ φθ θθ β θ
π θ π
+ +− −− −
∂ ∂ ⇒ = − − = +
∂ ∂
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The Hertz dipole has only component of the magnetic field, i.e., the magnetic field circulates the dipole
0
2
ˆ sin 1
4
j t j rI dle e j
r r
ω βφ θ β
π
+ − = +
φ
8.4 Kinds of antennas The electric field for ( in free space, we don’t have any conduction current flowing) can be obtained as
Using the symmetry of the problem (no variation in ) like
0J =r
ωεj
HE
rr ×∇
=
φ
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum69
Using the symmetry of the problem (no variation in ) like before, we have,
φ
( ) ( )2 2
ˆ ˆˆ sin
1 1 ˆˆ0 sin sinsin sin
0 0 sin
r r r
E r H r r H rj r r j r r
r H
φ φ
φ
θ θφ
θ θ θωε θ θ φ ωε θ θ
θ
∂ ∂ ∂ ∂ ∂ = ≡ = − ∂ ∂ ∂ ∂ ∂
r
8.4 Kinds of antennas
( )20 0
2 2 2 2
1 1sin 2sin cos
sin 4 sin 4
j t j r j t j r
r
I dle e I dle ej jE r r
j r r r j r r r
ω β ω ββ βθ θ θ
ωε θ π θ ωε θ π
− −∂ = + = +
∂
( )
+∂
∂−=
∂
∂−= − rj
tj
er
jrrj
dleIHr
rrj
rE
βω
φθ βπωε
θθ
θωε
1
4
sinsin
sin
0
2
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We see that electric field is in the (r, θ) plane whereas the magnetic field has component only
rj θωε sin
2
0
2 3
sin
4
j tj rI dle j j
er r r
ωβθ β β
πε ω ω ω−
= + −
φ
8.4 Kinds of antennas Therefore, the electric field and magnetic field are perpendicular to each other
Points to be noted:
1) Fields can be classified into three categories
(a) Radiation fields (spatial variation 1/r)
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum71
(a) Radiation fields (spatial variation 1/r)
(b) Induction fields (spatial variation 1/r2) and
(c) Electrostatic fields (spatial variation 1/r3)
8.4 Kinds of antennas 2) Electrostatic fields are also inversely proportional to the frequency
3) Induction field is independent of frequency since
4) Radiation field is proportional to frequency
5) For small values of r, electrostatic field is the dominant
LCβ ω=
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum72
5) For small values of r, electrostatic field is the dominant term and for large values of r, radiation field is the dominant term
8.4 Kinds of antennas We can also observe that the three types of fields are equal in magnitude when
β2/r= β/r2=1/r3 => r=1/β= λ/2>
For r< λ/2>, 1/r3 term dominates
For r>> λ/2>, the 1/r term dominates
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum73
For r>> λ/2>, the 1/r term dominates
Near field region:
For r<< λ/2> (in fact the near field region distance r= λ/2> is for D<<λ for an ideal infinitesimally small Hertz dipole and the near field region distance for is for D>>λ), electrostatic fields dominate
3
0.62D
rλ
=
8.4 Kinds of antennas as r<< λ/2>
1→− rje
βQ
0
3
cos2 ;
4
j t
r
I dl eE j
r
ωθ
πεω≈ − 0
3
sin
4
j tI dl eE j
r
ω
θ
θ
πεω≈ −
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The magnitude of the near field is
4 rπεω 4 rπεω
2 2 2 20
34cos sin
4r
I dlE E E
rθ θ θ
πεω= + = +
8.4 Kinds of antennas A polar plot of the near field can be generated by writing a MATLAB program for plotting
Maximum field is along θ=00, θ=1800 and minimum is along
( ) θθθ 22 sincos4 +=F
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum75
Maximum field is along θ=00, θ=1800 and minimum is along θ=900, θ=2700 (see Fig. 8.6(a))
8.4 Kinds of antennas
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum76
Fig. 8.6 (a) Near field pattern plot of a Hertz dipole located at the origin and oriented along z-axis
8.4 Kinds of antennas Far field region:
For r>> λ/2> (in between reactive near field and Fraunhofer far field region, there exists the Fresenel near field region that’s why we have chosen an r>> λ/2>), radiation field is the dominant term
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum77
radiation field is the dominant term
2
0 0sin sin;
4 4
j t j r j t j rI dl e e I dl e e
E j H jr r
ω β ω β
θ φ
θ β θ β
πεω π
− −
= =
8.4 Kinds of antennas The electric fields and magnetic fields are in phase with each other
They are 90˚ out of phase with the current due to the (j) term in the expressions of Eθ and Hφ
It is interesting to note that the ratio of electric field and
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum78
It is interesting to note that the ratio of electric field and magnetic field is constant
E
H
θ
φ
ω µεβ µη
ωε ωε ε= = = =
8.4 Kinds of antennas Hence, the fields have sinusoidal variations with θ
They are zero along θ=0
They are maximum along θ=π /2 (see Fig. 8.6 (b))
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8.4 Kinds of antennas
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Fig. 8.6 (b) E-plane radiation pattern of a Hertz dipole in far field (H-plane radiation will look like a circle)
8.4 Kinds of antennas Power flow:
( ) * *1 1 ˆ ˆˆRe Re2 2
avg rS E H E r E Hθ ϕθ φ= × = + ×r r
1 $2 3sin1 I dl θ β $
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The net real power is only due to the radiations fields (i.e. jβ2/r and jβ/r) of electric and magnetic fields
*1Re
2avgS E H rθ φ= $
3
0 sin1
2 4
I dlr
r
θ β
π ωε
=
$
8.4 Kinds of antennas In the far field, electric field is along direction
magnetic field is along direction
Both of them are perpendicular to the power flow (Poyntingvector) which is along direction
Total radiated power:
θ
φ
r$
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum82
Total radiated power:
The total radiated power from a Hertz dipole
2 sinavgW S r d dθ θ φ= ∫∫2
2 2
040dl
W Iπλ
∴ =
8.4 Kinds of antennas Power radiated by the Hertz dipole is proportional to
the square of the dipole length and
inversely proportional to the dipole wavelength
It implies more and more power is radiated as the frequency and
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum83
the frequency and
the length
of the Hertz dipole increases
Radiation resistance of a Hertz Dipole:
Hertz dipole can be equivalently modeled as a radiation resistance
8.4 Kinds of antennasSince W=1/2 I0
2 Rrad
implies that Rrad = 80π2
Radiation pattern of a Hertz Dipole:
2
2 2
040dl
W Iπλ
∴ =
2dl
λ
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum84
Radiation pattern of a Hertz Dipole:
F(θ, )=sin θ for a Hertz dipole
The 3D plot of sin θ looks like an apple (see Figure 8.6 (c))
8.4 Kinds of antennas
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum85
Fig. 8.6 (c) A typical 3-D radiation pattern of a Hertz dipole in the far field
8.4 Kinds of antennas To get the 3-D plot from the 2-D plot you need to rotate the E-plane pattern along the H-plane pattern
For this case it will give the shape of an apple
Note that θ is also known as elevation angle and as azimuth angle
φ
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum86
azimuth angle
E-plane pattern for a dipole is also known as elevation pattern
H-plane pattern as azimuthal pattern
8.4 Kinds of antennas As mentioned before, it is easier to visualize 2-D plots than 3-D on a 2-D plane like pages of this book
Two principal planes radiation patterns are normally plotted E-plane (vertical: all planes containing z-axis like xz-plane, yz-plane)
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum87
plane)
H-plane (horizontal) radiations patterns
are sufficient to describe the radiation pattern of a Hertz dipole
H-plane (xy-plane) radiation pattern is in the form of circle of radius 1 since F(θ, ) is independent of φ φ
8.4 Kinds of antennas Since the ratio of electric and magnetic field amplitudes in the far field region of an antenna is 377 Ohm
We may also plot the power pattern instead of field pattern from the relation p(θ, )=sin2 θ
Polarization of Hertz dipole:
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum88
In the far-field region of a Hertz dipole, only θ component of the electric field the electric field is LP along direction
That means electric field is perpendicular to the line from the center of the dipole to the field observation point (see Fig. 8.5) and it lies in the plane containing this line and the dipole axis
θ
8.4 Kinds of antennas 8.4.2 Dipole antenna
The next extension of a Hertz dipole is a linear antenna or a dipole of finite length as depicted in Fig. 8.7 (a)
It consists of a conductor of length 2L fed by a voltage or current source at its center
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum89
current source at its center
The current distribution can be obtained by assuming an o.c. transmission line
For o.c. transmission line
( )0 0
0 0
( ) 2 sinj z j zV VI z e e j z
Z Z
β β β+ +
− += − = −
8.4 Kinds of antennas
θ
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum90
Fig. 8.7 (a) Dipole of length
2L (Dipole can be assumed
to composed of many Hertz dipoles)
8.4 Kinds of antennas The current is zero at z= L (since at the ends, there is no path for the current to flow), so, we can write,
The electric field due to the current element dz (it has the
( )0 00 0
0 0
( ) 2 sin( ( )) sin ; 2V V
I z j L z I L z I jZ Z
β β+ +
= − − = − = −
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum91
The electric field due to the current element dz (it has the same expression of the Hertz dipole of the previous section except that now we have a length of dz and current of I(z) ) at far away observation point or in the far field can be written as
12
1 0
sin ( );
4
j R dEj I z dzedE dH
R
βθ
θ φ
β θ
πεω η
−
= =
8.4 Kinds of antennas Since the observation point P is at a very far distance, the lines OP and QP are parallel
Note that for the amplitude, we can approximate since the dipole size is quite small in comparison to the
1 cosR R z θ∴ ≅ −
1
1 1
R R≅
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum92
since the dipole size is quite small in comparison to the distance of the observation point P from the origin
2 cossin ( )
4
j R j Zj I z dze e
dER
β β θ
θ
β θ
πεω
−
≅
8.4 Kinds of antennas Since we have assumed that the dipole of length 2L is composed of many Hertz dipoles as depicted in Fig. 8.7 (a) (this is one of the reasons why we say that Hertz dipole or infinitesimal dipole is the building block for many antennas), we can write the total radiated electric field as
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum93
we can write the total radiated electric field as
2 cos2 cos0sin sin( ( ))sin ( )
4 4
j R j ZL L Lj R j Z
z L z L z L
j I L z e ej I z e eE dE dz dz
R R
β β θβ β θ
θ θ
β θ ββ θ
πεω πεω
−−
=− =− =−
−= = =∫ ∫ ∫
8.4 Kinds of antennas It can be shown that (see textbook for derivations)
In the previous equation, the term under bracket is F(θ) and it is the variation of electric field as a function of θ and it is
( )( )0 0
cos cos cos60 60
sin
j R j RL Le eE j I j I F
R R
β β
θ
β θ βθ
θ
− − −⇒ ≅ =
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum94
it is the variation of electric field as a function of θ and it is the E-plane radiation pattern
8.4 Kinds of antennas In the H-plane, Eθ is a constant and it is not a function of hence it is a circle
The E–plane radiation pattern of the dipole varies with the length of the dipole as depicted in Fig. 8.8
Note that according to Fig. 8.7 (a), we have considered the total
φ
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum95
Note that according to Fig. 8.7 (a), we have considered the total dipole length is 2L) and the H-plane radiation is always a circle
Fig. 8.8 E-plane radiation pattern for dipole of length
(a) 2L=2×λ/4= λ/2
(b) 2L=2×λ=2λ
(c) 2L=2×2λ=4λ
8.4 Kinds of antennas
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum96
(a) 2L=2×λ/4= λ/2
8.4 Kinds of antennas
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum97
(b) 2L=2×λ=2λ
8.4 Kinds of antennas
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(c) 2L=2×2λ=4λ
8.4 Kinds of antennas Points to be noted:
1. Input impedance of the dipole (z=0)
For dipole of length 2L, where L =odd multiples of
0 sin
in inin
in
V VZ
I I Lβ= =
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum99
For dipole of length 2L, where L =odd multiples of
=1,
For dipole of length 2L, where L =even multiples of .
, , sin4 2
mL L
λ πβ β=
0
inin
VZ
I=
, , sin 04
L m Lλ
β π β= =in
Z⇒ = ∞
8.4 Kinds of antennas That’s why it is preferable to have dipoles of length odd multiples of λ/2, otherwise it is difficult to have a source with infinite impedance
2. Since increasing the dipole length more and more current is available for radiation, the total power radiated increases
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum100
is available for radiation, the total power radiated increases monotonically
3. The electric field has only component and hence it is linearly polarized
4. The radiation pattern have nulls and it can be calculated by equating F(θ)=0
$θ
8.4 Kinds of antennas
cos( cos ) cos0
sin
null
null
L Lβ θ β
θ
−=
nullθcos⇒ 1mλ
π= ± ±
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For m=0,
But, in denominator is also zero
So let us take the limit of F(θ) as θ→0, and see
πθ θ ,0cos ,1 =±= nullnull
sinnull
θ
8.4 Kinds of antennas
( ) ( ) ( ) ( ) ( ) ( )
0, 0,
2 4 2 4cos cos cos cos cos1
1 1sin sin 2! 4! 2! 4!
L L L L L L
Lim Limθ π θ π
β θ β β θ β θ β β
θ θ→ →
− ≅ − + − − +
( ) ( ) ( ) ( ) ( ) ( )4 42 24 22 1 cos sin 1 cossin sin10
2! 4! sin 2! 4!
L LL L
Lim Lim
β θ β θ θβ θ β θ
θ
− + = − × = − =
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum102
5. To find θ for maximum radiation, we have to find the solution of
=0
We can also take the mean of the first two nulls to approximate
0, 0,
02! 4! sin 2! 4!Lim Lim
θ π θ πθ
→ →
( )dF
d
θ
θ
maxθ
8.4 Kinds of antennas Monopole antennas:
A monopole is a dipole that has been divided in half at its center feed point and fed against a ground plane
Monopole is usually fed from a coaxial cable (see Fig. 8.7 (b))
A monopole of length L placed above a perfectly conducting
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum103
A monopole of length L placed above a perfectly conducting and infinite ground plane will have the same field distribution to that of a dipole of length 2L without the ground plane
8.4 Kinds of antennas
Monopole
Ground plane
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(b) Monopole of length L over a ground plane
Ground plane
Coaxial cable
Dipole in free space
Image of monopole
8.4 Kinds of antennas This is because an image of the monopole will be formed inside the ground plane (similar to the method of images in chapter 2)
The monopole looks like a dipole in free space (see Fig. 8.7 (b))
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum105
(b))
Since this monopole is of length L only, it will radiate only half of the total radiated power of a dipole of length 2L
Hence, the radiation resistance of a monopole is half that of a dipole
8.4 Kinds of antennas Similarly, directivity of the monopole is twice that of a dipole
Since the field distributions are the same for a monopole and dipole, the maximum radiation intensity will be also same for both cases
But for monopole, the total radiated power is half that of a
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum106
But for monopole, the total radiated power is half that of a dipole
Hence, the directivity of a monopole above a conducting ground plane is twice that of dipole in free space
8.4 Kinds of antennas 8.4.3 Loop antenna
Loop antennas could be of various shapes: circular, triangular, square, elliptical, etc.
They are widely used in applications up to 3GHz
Loop antennas can be classified into two:
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum107
Loop antennas can be classified into two: electrically small (circumference < 0.1 λ) and
electrically large (circumference approximately equals to λ)
8.4 Kinds of antennas Electrically small loop antennas have very small radiation resistance
They have very low radiation and are practically useless
Electrically small loop antennas could be analyzed assuming that it is equivalently represented as a Hertz dipole
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum108
that it is equivalently represented as a Hertz dipole
Let us consider electrically large circular loop of constant current
8.4 Kinds of antennas Fig. 8.9 Loop antenna
θrr
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum109
φ
'φ
'P
'rr
ψ
Rr
8.4 Kinds of antennas We can express magnetic vector potential (see textbook) as
( )( )
( )
01 1
2
1 120
( ) ( sin ) ( sin )4
1( ) ( 1) ( ) ( sin ) ( sin )
2 ! !
j r
m m
n n
n n nm nm
I aeA j J a J a
r
zJ z z J z J z J a J a
m n m
β
φ
µθ π β θ β θ
π
β θ β θ
−
∞
+=
= − −
−= ∴ − = − ⇒ − = −
+∑Q
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum110
We can express electric field as
( )0
0 1
2 ! !
( sin )( )
2
m
j r
m n m
j I ae J aA
r
β
φ
µ β θθ
=
−
+
∴ =
( )AE j j Aω
ωµε
∇ ∇ •∴ = − −
rrr
8.4 Kinds of antennas Note that magnetic vector potential has only component which is a function of θ variable only
φ
0 1
0;
( sin )0 0; 0
2
j r
r r r
A
I ae J aA A E j A E j A E j A
r
β
θ θ θ φ φ
ωµ β θω ω ω
−
∴∇ • =
= = ⇒ = − = = − = ∴ = − =
r
Q
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0 0; 02
r r rA A E j A E j A E j A
rθ θ θ φ φω ω ω= = ⇒ = − = = − = ∴ = − =Q
( )01 sin ; 0
2
j r
r
E I aeH J a H H
r
βφ
θ φ
ωµβ θ
η η
−−= − = = =
8.4 Kinds of antennas Poynting vector for a wave traveling (radiating) in radiallyoutward direction should have direction along positive radial direction
Therefore must be negative
Fig. 8.10 shows the far-field radiation pattern of the loop
Hθ
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Fig. 8.10 shows the far-field radiation pattern of the loop antenna
It can be observed that the radiation field has higher magnitude with the larger radius of the loop antenna
For larger radiation power we need a loop antenna of larger radius
8.4 Kinds of antennas
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum113
Fig. 8.10 Plot of for various values of angle θ (far-field radiation patterns) (dotted line a=0.1λ, dashed line a=0.2λ, solid line a=0.3λ)
8.4 Kinds of antennas For small loops
( ) θβθβθβθββ sin2
1...sin
16
1sin
2
1)sin(;
3
1 31 aaaaJa ≅+−=<
( ) ( )0 01 1sin ; sin
j r j rI a I ae eE a H a
β β
φ θ
ωµ ωµβ θ β θ
η
− −
∴ = = −
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Note that for the dipole polarization was along direction
But for small loop antennas it is along the direction
( ) ( )sin ; sin2 2 2 2
E a H ar r
φ θβ θ β θη
∴ = = −
θ
φ
8.5 Antenna Arrays
One of the disadvantages of single antenna is that it has fixed radiation pattern
That means once we have designed and constructed an antenna, the beam or radiation pattern is fixed
If we want to tune the radiation pattern, we need to apply
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum115
If we want to tune the radiation pattern, we need to apply the technique of antenna arrays
Antenna array is a configuration of multiple antennas (elements) arranged to achieve a given radiation pattern
8.5 Antenna Arrays
There are several array design variables which can be changed to achieve the overall array pattern design
Some of the array design variables are:
(a) array shape (linear, circular, planar, etc.)
(b) element spacing
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum116
(b) element spacing
(c) element excitation amplitude
(d) element excitation phase
(e) patterns of array elements
8.5 Antenna Arrays
Given an antenna array of identical elements, the radiation pattern of the antenna array may be found according to the pattern multiplication principle
It basically means that array pattern is equal to
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum117
It basically means that array pattern is equal to the product of the
pattern of the individual array element into
array factor (a function dependent only on the geometry of the array and the excitation amplitude and phase of the elements)
8.5 Antenna Arrays
8.5.1 Two element array
Let us investigate an array of two infinitesimal dipoles positioned along the z axis as shown in Fig. 8.10 (a)
The field radiated by the two elements, assuming no coupling between the elements is equal to the sum of the two fields
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum118
between the elements is equal to the sum of the two fields
where the two antennas are excited with current
1 1 2 22 2
1 21 2
1 2
sin sinˆ ˆ4 4
j r j j r j
total
jI dl e e jI dl e eE E E
r r
β δ β δθ β θ βθ θ
πεω πεω
− −
= + = +r r r
1 1 2 2I and Iδ δ< <
8.5 Antenna Arrays
1rr
rr
1θ
θ
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Fig. 8.10 (a) Two Hertz dipoles
2rr2θ
8.5 Antenna Arrays
For 1 2 oI I I= =
1 2,2 2
α αδ δ= = −
2sind dj r
jI dl eβ α β αβ θ θθβ
− + − + r
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2 cos cos2 2 2 20 sinˆ
4
d dj r j j
total
jI dl eE e e
r
β α β αβ θ θθβθ
πεω
− + − +
= +
r
2
0sinˆ 2cos cos4 2 2
j rjI dl e d
r
βθβ β αθ θ
πεω
− = +
8.5 Antenna Arrays
Hence the total field of the array is equal to
the field of single element positioned at the origin
multiplied by a factor which is called as the array factor
Array factor is given
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum121
Normalized array factor is
( )1
2cos cos2
AF dβ θ α
= +
( )2
1cos cos
2AF dβ θ α
= +
8.5 Antenna Arrays
8.5.2 N element uniform linear array (ULA)
This idea of two element array can be extended to N element array of uniform amplitude and spacing
Let us assume that N Hertz dipoles are placed along a straight line along z-axis at positions 0, d, 2d, …, (N-2) d
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum122
straight line along z-axis at positions 0, d, 2d, …, (N-2) d and (N-1) d respectively
Current of equal amplitudes but with phase difference of 0, α, 2α, … , (N-2) α and (N-1) α are excited to the corresponding dipoles at 0, d, 2d, …, (N-2) d and (N-1) d respectively
8.5 Antenna Arraysz
2d
(N-1)d
.
.
.
2I α⟨
( 1)I N α⟨ −
0I ⟨
I α⟨
( 1)
2
NI α
−⟨
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Fig. 8.10 (b) ULA 1 (c) ULA 2 (assume N is an odd number)
d
2d
00I ⟨
I α⟨
2I α⟨0I ⟨
I α⟨−
( 1)
2
NI α
−⟨−
8.5 Antenna Arrays
Then the array factor for the N element ULA of Fig. 8.10 (b) will become
( ) ( ) ( )cos 2 cos ( 1) cos1 .....
j d j d j N dAF e e e
β θ α β θ α β θ α+ + − +∴ = + + + +
( )( )ψ−1 e
jNN
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( )( ) αθβψψ
ψαθβ +=
−
−=⇒=⇒ ∑
=
+− cos;1
1
1
cos1d
e
eAFeAF
j
jN
N
N
n
dnj
1
1
jN
j
e
e
ψ
ψ−
=−
( ) ( )1 2 2( )2
1 1( ) ( )2 2
N Nj jN
j
j j
e ee
e e
ψ ψψ
ψ ψ
−−
−
−=
−
1( )
2
sin( )2
sin( )2
Nj
N
eψ
ψ
ψ
−
=
8.5 Antenna Arrays
If the reference point is at the physical center of the array as depicted in Fig. 8.10 (c), the array factor is
( )sin( )
2
sin( )2
N
N
AF
ψ
ψ=
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For small values of
sin( )2
ψ
( )sin( )
2
2
N
N
AF
ψ
ψ=
8.5 Antenna Arrays
The maximum value of AF is for and its value is N
Apply L’ Hospital rule since it is of the form
To normalize the array factor so that the maximum value is equal to unity, we get,
0ψ =
0
0sin
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( )sin( )
1 21
sin2
N
N
AFN
ψ
ψ
=
sin( )2
2
N
N
ψ
ψ≅
8.5 Antenna Arrays
This is the normalized array factor for ULA
As N increases, the main lobe narrows
The number of lobes is equal to N (one main lobe and other N-1 side lobes) in one period of the AF
The side lobes widths are of 2>/N and main lobes are two
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The side lobes widths are of 2>/N and main lobes are two times wider than the side lobes
The SLL decreases with increasing N
This can be verified from Fig. 8.11 (see textbook)
8.5 Antenna Arrays
Null of the array
To find the null of the array,
( )sin( ) 0 cos2
Ndψ ψ β θ α= = +Q
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( )
1
2cos cos
2 2
1 2cos 1,2,3,......
n
N N nn d n d
N
nn
d N
πψ π β θ α π β θ α
πθ α
β−
⇒ = ± ⇒ + = ± ⇒ = − ±
⇒ = − ± =
8.5 Antenna Arrays
Maximum values
It attains the maximum values for 0ψ =
( )1
cos2 2
m
dθ θ
ψβ θ α
=
= + 0=
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum129
8.5.3 Broadside array
We know that when
mθ θ=
1cosmd
αθ
β−
−
⇒ =
8.5 Antenna Arrays
the maximum radiation occurs
It is desired that maximum occurs at θ=90˚
c o s 0dψ β θ α= + =
0
0
90cos 0 0d
θψ β θ α α
== + = ⇒ =
7/6/2013Electromagnetic Field Theory by R. S. Kshetrimayum130
8.5.4 Endfire array
We know that when
the maximum radiation occurs
090cos 0 0d
θψ β θ α α
== + = ⇒ =
cos 0dψ β θ α= + =
8.5 Antenna Arrays
It is desired that maximum occurs at θ=0˚,
8.5.5 Phase scanning array
We know that when
00cos 0d d
θψ β θ α α β
== + = ⇒ = −
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We know that when
the maximum radiation occurs
It is desired that maximum occurs at θ=θ0
cos 0dψ β θ α= + =
00cos 0 cosd d
θ θψ β θ α α β θ
== + = ⇒ = −