8 slides

Dr. Rakhesh Singh Kshetrimayum
8. Antennas and Radiating Systems
Dr. Rakhesh Singh Kshetrimayum
7/6/20131 Electromagnetic FieldTheory 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 FieldTheory by R. S. Kshetrimayum2
wireless local areas networks (your laptop connecting wireless
internet also has antennas)
television (oldTV antennas areYagi-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?
Fig. 8.1Antennas
charge radiate?
Wave equation for
potential functions
Solution of wave equation
for potential functions
Hertz dipole
Dipole antenna
Loop antenna

8.2 Radiation fundamentals
8.2.1When does a charge radiates?
accelerating/decelerating charges radiate EM waves
r
Fig. 8.2 A giant sphere of radius r with a source of EM wave at
its origin
Source
r

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,
( ) ( )∫∫ •×=•= ∗
sdHEsdSrP avgtotal
rrrrr
Re
( )
lim
totalP P r
r
=
→∝

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
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

CASE 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,
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.

Basic radiation equation:
where L=length of current element, m
=time changing current,As-1(units)
dt
dv
Q
dt
di
L =
di
=time changing current,As-1(units)
Q=charge, C
=acceleration of charge, ms-2
dt
di
dt
dv

To create radiation
there must be a time varying current or
acceleration (or deceleration) of charge
Static charges=>no radiation
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

Charges moving with uniform velocity:
no radiation if the wire is straight and infinite in extent and
radiation if the wire is
curved,
curved,
bent,
discontinuous,
terminated or
truncated (these will either accelerate or decelerate the charge)

8.2.2Wave 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)
r
r
r ∂2 2
V ρ∂
J
t
A
A
r
r
r
µµε −=
∂
∂
−∇ 2
2
2
2
2
2
;
V
V
t
ρ
µε
ε
∂
∇ − = −
∂

8.2.3 Solution of wave equation for potential functions
For time harmonic functions of potentials,
where
JAA
rrr
µβ −=+∇ 22
LCβ ω=
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

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
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
0
G G
θ φ
∂ ∂
= =
∂ ∂
( )rG
r
G
r
rr
δβ =+





∂
∂
∂
∂ 22
2
1

Putting Ψ= G r,
For r not equal to 0,
( )rr
r
δβ =Ψ+Ψ
∂
∂ 2
2
2
Therefore,
02
2
2
=Ψ+Ψ
∂
∂
β
r
rjrj
BeAe ββ +−
+=Ψ

Since the radiation travels radially in positive r direction
negative r direction is not physically feasible for a source of a
field, we get,
rj
Ae β−
=Ψ
rjβ−
From example 8.2, we can find the constant A and hence
r
Ae
G
rjβ−
=
r
e
GA
rj
ππ
β
4
;
4
1 −
−=−=

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
rr
β
µ
( ) '
'
'0
4
dv
rr
e
rJA
V
rrj
∫ −
=
−−
rr
rr
β
π
µ

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 −
since the distance in spherical coordinates is always positive

8.3 Antenna pattern and parameters
8.3.1What 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
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

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 φ
r
E
r
H
r
coordinate position) and
(b) power pattern (normalized power versus spherical
coordinate position).

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
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

Fig. 8.3 (c)Typical radiation pattern of an antenna
maxθ
θ

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)
possible to realize this practically)
(b) directional (more efficient radiation in one direction than
another)
(c) omnidirectional (uniform radiation in one plane)

Fig. 8.3 (a) Antenna field regions
3
max
10 0.62nf
D
r
λ
< ≤
3 2
max max
2
2
0.62 nf
D D
r
λ λ
< ≤
2
max2
ff
D
r
λ
<

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)
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
max
10 0.62nf
D
r
λ
< ≤

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
antenna
Radiating near field (Fresnel) region is usually for a radius of
3 2
max max
2
2
0.62 nf
D D
r
λ λ
< ≤

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 of
2
max2
ff
D
r
 
< 
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
max
ffr
λ
< 
 

Fig. 8.3 (b) Illustration of
far field region
(antenna under test:AUT)

We can calculate the distance rff by equating the maximum
error (which is at the edges of theAUT 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)
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

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
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)

8.3.3 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
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

8.3.4 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
HPBW for narrow beams and broad beams
HPBW is a better parameter for specifying the effective
beam width

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
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

8.3.6 Radiation intensity
Radiation intensity U(θ, ) is defined as U(θ, )= Power
along direction (θ, )/Solid angle (d )
φ φ
φ
2
( , ) ( , )sinP U d U d d
π π
θ φ θ φ θ θ φ= Ω =∫ ∫ ∫
4 0 0
( , ) ( , )sinradP U d U d d
π θ φ
θ φ θ φ θ θ φ
Ω= = =
= Ω =∫ ∫ ∫
2
0 0
1
( , )sin
4 4
rad
avg
P
U U d d
π π
θ φ
θ φ θ θ φ
π π = =
= = ∫ ∫

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
which equivalent to the radiation intensity of an isotropic
antenna
2
0 0
( , ) 4 ( , )
( , )
( , )sinavg
U U
D
U
U d d
π π
θ φ
θ φ π θ φ
θ φ
θ φ θ θ φ
= =
= =
∫ ∫

D (θ, ) is maximum at θmax and minimum along θnull
is also known as beam solid angle
φ
( ) max max
max 2
0 0
4 ( , ) 4
,
( , )sin A
U U
D
U d d
π π
θ φ
π θ φ π
θ φ
θ φ θ θ φ
= =
= = =
Ω
∫ ∫
avgU
Ω
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

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
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
HPBWθ HPBWφ
( ), 1HPBW HPBWθ φ <<
max
4
HPBW HPBW
rad rad
D
π
θ φ
≅
×

If the beam widths are in degrees, we have
8.3.8 Gain
2
max deg deg deg deg
180
4
41,253
HPBW HPBW HPBW HPBW
D
π
π
θ φ θ φ
 
 
 ≅ =
× ×
8.3.8 Gain
In defining directivity, we have assumed that the antenna
is lossless
But, antennas are made of conductors and dielectrics

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
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

Note that definitions of the antenna directivity and gain are
essentially the same except for the power terms used in the definitions
( ) rad
rad
4 ( , )e4 ( , )
, e ( , )
input rad
UU
G D
P P
π θ φπ θ φ
θ φ θ φ= = =
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

The antenna gain is usually measured based on Friis
transmission 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
Assuming that the antennas are well matched in terms of
impedance and polarization, the Friis transmission equation is
2
10 10
1 4
20log 10log
4 2
r r
t r t r
t t
P PR
G G G G G G
P R P
λ π
π λ
     
= = = ∴ = +     
      
Q

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
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)

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
(c) there are no interferences from other sources and no
multipaths between the transmitting and receiving antennas
due to reflection, refraction and diffraction

8.3.9 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,
In the far field region of an antenna,
( ) ( ) ( )ˆ ˆ, , ,E E Eθ φθ φ θ φ θ θ φ φ= +
r

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
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

a) LP
When δ=0, the two transversal electric field components are
in time phase
The total electric field vector
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 φ
θ
θ φ θ φ
− −
   
= =   
      

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 componentsπ
δ = ±
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
π
δ = ±
( ) ( ) ( )0, , ,E E Eθ φθ φ θ φ θ φ= =

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
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 tan
cos
CP
t
t t
t
ω
φ ω ω
ω
− −
 
 = = =   
  
π
ω

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≠ ≠
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

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
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

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
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

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 cou u u u u u u u• = • = • = • =
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

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 cosco LP LP cr LP LPu x y u x yφ φ φ φ= + = −
LPφ
For a y-directed LP wave, =900, hence,
* *
ˆ ˆ ˆ ˆ ˆ ˆ, ; ,co cr co co x cr cr yu 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

b) CP
For a RHCP wave, we can write,
ˆ ˆ ˆ ˆ
ˆ ˆ,
2 2
co cr
x jy x jy
u u
− +
= =
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 jE
E E u E E u
+ −
= • = = • =
r r

c) EP
For a EP wave, we can write,
2 2
ˆ ˆ ˆ ˆ
ˆ ˆ,
1 1
EP EPj j
co cr
x Ae y Ae x y
u u
A A
φ φ−
+ − +
= =
+ +
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

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
transmitting mode
TheAUT is rotated by a positioner and it can rotate 1-, 2-
and 3-degrees of freedom of rotation

TheAUT 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
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

8.3.10 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,
L is the inductance and
C is the capacitance of the antenna

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
the resonant frequency) times the energy stored over the
power radiated and Ohmic losses
( )
( ) ( )
( )
2 2
2
0 0
0
2
0
0
1 1 1
4 4 2 2 1
2
1 2
2
1
2
r L r L
r L
r
lossless rad
r r L
I L I
f L f L
Q f
R R f R R CI R R
R
Q e
f R C R R
π π
π
π
π
+
= = =
+ ++
= = •
+

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
the ratio of the power delivered to the radiation resistance Rr
to the power delivered to Rr and RL
( ) ( )
2
2
1
2
1
2
r
r
rad
r L
r L
I R
R
e
R RI R R
= =
++

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
+
where k is the wave number and r is the radius of the smallest
sphere enclosing the antenna
( )
( ) ( ){ }
2
3 2
1 2
1
rad
kr
Q e
kr kr
+
= •
+

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:
Harrington gave also a practical upper limit to the gain of a
small antenna for a reasonable BW as
( )
min 3
1 1
Q
krkr
= +
( ) ( )
2
max 2G kr kr= +

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
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
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

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
ˆ0
ω
=
Since the current is along the z-direction, the magnetic
vector potential at the observation point P is along z-
direction
z
r
edleI
zAA
tjrj
z ˆ
4
ˆ 00
π
µ ωβ−
==
r

Fig. 8.5 Hertz dipole
located at the origin and
oriented along z-axis
( ), ,P r θ φ
ˆr
ˆθ
ˆφ
0
j t
I e ω

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 t
J r dv I dle ω
=
r r
'
r r r− =
r r r
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⇒ = = − =

Using the symmetry of the problem (no variation in ), we
0
A
H
µ
∇×
=
r
r
Q 2
0
ˆ ˆˆ sin
1
sin
sinr
r r r
r r
A rA r Aθ φ
θ θφ
µ θ θ φ
θ
 
 
∂ ∂ ∂ =
 ∂ ∂ ∂
 
  
Using the symmetry of the problem (no variation in ), we
have,
2
0
ˆ ˆˆ sin
1
0
sin
cos sin 0z z
r r r
H
r r
A rA
θ θφ
µ θ θ φ
θ θ
 
 
∂ ∂ ∂ = ≡
 ∂ ∂ ∂
 
−  
r

( ) ( )2
0
sin
0; 0; sin cos
sin
r z z
r
H H H rA A
r r
θ φ
θ
θ θ
µ θ θ
∂ ∂ 
⇒ = = = − − 
∂ ∂ 
( )0 0
ˆ ˆcos sin
sin sin
4 4
j t j tj r j r
j r j rI dle I dlee e
H e j e
r r r r r
ω ωβ β
β β
φ
φ φθ θ
θ β θ
π θ π
+ +− −
− −
    ∂ ∂ 
⇒ = − − = +    
∂ ∂     
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 r
I dle e j
r r
ω β
φ θ β
π
+ −
  
= +  
  
φ

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
H
E
r
r ×∇
=
φ
Using the symmetry of the problem (no variation in ) like
before, we have,
φ
( ) ( )2 2
ˆ ˆˆ sin
1 1 ˆˆ0 sin sin
sin sin
0 0 sin
r r r
E r H r r H r
j r r j r r
r H
φ φ
φ
θ θφ
θ θ θ
ωε θ θ φ ωε θ θ
θ
 
 
∂ ∂ ∂ ∂ ∂  = ≡ = −  ∂ ∂ ∂ ∂ ∂ 
 
  
r

( )20 0
2 2 2 2
1 1
sin 2sin cos
sin 4 sin 4
j t j r j t j r
r
I dle e I dle ej j
E r r
j r r r j r r r
ω β ω β
β β
θ θ θ
ωε θ π θ ωε θ π
− −
∂    
= + = +    
∂    
( ) 











+
∂
∂
−=
∂
∂
−= − rj
tj
e
r
j
rrj
dleI
Hr
rrj
r
E β
ω
φθ β
πωε
θ
θ
θωε
1
4
sin
sin
sin
0
2
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 t
j rI dle j j
e
r r r
ω
βθ β β
πε ω ω ω
−  
= + − 
 
φ

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)
(a) Radiation fields (spatial variation 1/r)
(b) Induction fields (spatial variation 1/r2) and
(c) Electrostatic fields (spatial variation 1/r3)

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=
5) For small values of r, electrostatic field is the dominant
term and for large values of r, radiation field is the dominant
term

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
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
0.62
D
r
λ
=

as r<< λ/2
1→− rje βQ
0
3
cos
2 ;
4
j t
r
I dl e
E j
r
ω
θ
πεω
≈ − 0
3
sin
4
j t
I dl e
E j
r
ω
θ
θ
πεω
≈ −
The magnitude of the near field is
4 rπεω 4 rπεω
2 2 2 20
3
4cos sin
4
r
I dl
E E E
r
θ θ θ
πεω
= + = +

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
Maximum field is along θ=00, θ=1800 and minimum is along
θ=900, θ=2700 (see Fig. 8.6(a))

Fig. 8.6 (a) Near field pattern plot of a Hertz dipole located at the
origin and oriented along z-axis

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
radiation field is the dominant term
2
0 0sin sin
;
4 4
j t j r j t j r
I dl e e I dl e e
E j H j
r r
ω β ω β
θ φ
θ β θ β
πεω π
− −
= =

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
It is interesting to note that the ratio of electric field and
magnetic field is constant
E
H
θ
φ
ω µεβ µ
η
ωε ωε ε
= = = =

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))

Fig. 8.6 (b) E-plane radiation pattern of a Hertz dipole in far field
(H-plane radiation will look like a circle)

Power flow:
{ } ( ){ }* *1 1 ˆ ˆˆRe Re
2 2
avg rS E H E r E Hθ ϕθ φ= × = + ×
r r
{ }1 $
2 3
sin1 I dl θ β  $
The net real power is only due to the radiations fields (i.e.
jβ2/r and jβ/r) of electric and magnetic fields
{ }*1
Re
2
avgS E H rθ φ= $
3
0 sin1
2 4
I dl
r
r
θ β
π ωε
 
=  
 
$

In the far field, electric field is along direction
magnetic field is along direction
Both of them are perpendicular to the power flow (Poynting
vector) which is along direction
Total radiated power:
ˆθ
ˆφ
r$
Total radiated power:
The total radiated power from a Hertz dipole
2
sinavgW S r d dθ θ φ= ∫∫
2
2 2
040
dl
W Iπ
λ
 
∴ =  
 

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
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

SinceW=1/2 I0
2 Rrad
implies that Rrad = 80π2
Radiation pattern of a Hertz Dipole:
2
2 2
040
dl
W Iπ
λ
 
∴ =  
  2
dl
λ
 
 
 
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))
 

Fig. 8.6 (c)A typical 3-D radiation pattern of a Hertz dipole in the far field

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
azimuth angle
E-plane pattern for a dipole is also known as elevation
pattern
H-plane pattern as azimuthal pattern

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)
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 ofφ φ

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:
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
ˆθ

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
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 V
I z e e j z
Z Z
β β
β
+ +
− +
= − = −

θ
Fig. 8.7 (a) Dipole of length
2L (Dipole can be assumed
to composed of many Hertz dipoles)

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 0
0 0
0 0
( ) 2 sin( ( )) sin ; 2
V V
I z j L z I L z I j
Z Z
β β
+ +
 = − − = − = − 
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 dze
dE dH
R
β
θ
θ φ
β θ
πεω η
−
= =

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
≅
since the dipole size is quite small in comparison to the
distance of the observation point P from the origin
2 cos
sin ( )
4
j R j Z
j I z dze e
dE
R
β β θ
θ
β θ
πεω
−
≅

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
we can write the total radiated electric field as
2 cos2 cos
0sin 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 e
E dE dz dz
R R
β β θβ β θ
θ θ
β θ ββ θ
πεω πεω
−−
=− =− =−
−
= = =∫ ∫ ∫

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 cos
60 60
sin
j R j R
L Le e
E j I j I F
R R
β β
θ
β θ β
θ
θ
− −
 −
⇒ ≅ = 
 
it is the variation of electric field as a function of θ and it is
the E-plane radiation pattern

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
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

(a) 2L=2×λ/4= λ/2

(b) 2L=2×λ=2λ

(c) 2L=2×2λ=4λ

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 in
in
in
V V
Z
I I Lβ
= =
For dipole of length 2L, where L =odd multiples of
=1,
For dipole of length 2L, where L =even multiples of .
, ,sin
4 2
m
L L
λ π
β β=
0
in
in
V
Z
I
=
, ,sin 0
4
L m L
λ
β π β= = inZ⇒ = ∞

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
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

cos( cos ) cos
0
sin
null
null
L Lβ θ β
θ
−
=
nullθcos⇒ 1
mλ
π
= ± ±
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
sin nullθ

( ) ( ) ( ) ( ) ( ) ( )
0, 0,
2 4 2 4
cos cos cos cos cos1
1 1
sin 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 sin1
0
2! 4! sin 2! 4!
L LL L
Lim Lim
β θ β θ θβ θ β θ
θ
   − +   
= − × = − =   
   
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,
0
2! 4! sin 2! 4!Lim Limθ π θ π
θ→ →
   
      
( )dF
d
θ
θ
maxθ

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
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

Monopole
Ground plane
(b) Monopole of length L over a ground plane
Ground plane
Coaxial cable
Dipole in free space
Image of monopole

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))
(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

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
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

8.4.3 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:
Loop antennas can be classified into two:
electrically small (circumference < 0.1 λ) and
electrically large (circumference approximately equals to λ)

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
that it is equivalently represented as a Hertz dipole
Let us consider electrically large circular loop of constant
current

Fig. 8.9 Loop antenna
θ
r
r
φ
'
φ
'
P
'
r
r
ψ
R
r

We can express magnetic vector potential (see textbook) as
{ }
( )
( )
( )
0
1 1
2
1 12
0
( ) ( sin ) ( sin )
4
1
( ) ( 1) ( ) ( sin ) ( sin )
2 ! !
j r
m m
n n
n n nm n
m
I ae
A j J a J a
r
z
J z z J z J z J a J a
m n m
β
φ
µ
θ π β θ β θ
π
β θ β θ
−
∞
+
=
= − −
−
= ∴ − = − ⇒ − = −
+
∑Q
We can express electric field as
( )0
0 1
2 ! !
( sin )
( )
2
m
j r
m n m
j I ae J a
A
r
β
φ
µ β θ
θ
=
−
+
∴ =
( )A
E j j Aω
ωµε
∇ ∇•
∴ = − −
r
rr

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 a
A A E j A E j A E j A
r
β
θ θ θ φ φ
ωµ β θ
ω ω ω
−
∴∇• =
= = ⇒ = − = = − = ∴ = − =
r
Q
0 0; 0
2
r r rA A E j A E j A E j A
r
θ θ θ φ φω ω ω= = ⇒ = − = = − = ∴ = − =Q
( )0
1 sin ; 0
2
j r
r
E I ae
H J a H H
r
β
φ
θ φ
ωµ
β θ
η η
−
−
= − = = =

Poynting vector for a wave traveling (radiating) in radially
outward 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
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

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)

For small loops
( ) θβθβθβθββ sin
2
1
...sin
16
1
sin
2
1
)sin(;
3
1 3
1 aaaaJa ≅+−=<
( ) ( )0 01 1
sin ; sin
j r j r
I a I ae e
E a H a
β β
φ θ
ωµ ωµ
β θ β θ
η
− −
∴ = = −
Note that for the dipole polarization was along direction
But for small loop antennas it is along the direction
( ) ( )sin ; sin
2 2 2 2
E a H a
r r
φ θβ θ β θ
η
∴ = = −
ˆθ
ˆφ

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
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
(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
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
8.5.1Two 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
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 2
1 2
1 2
sin sinˆ ˆ
4 4
j r j j r j
total
jI dl e e jI dl e e
E E E
r r
β δ β δ
θ β θ β
θ θ
πεω πεω
− −
= + = +
r r r
1 1 2 2I and Iδ δ< <

8.5 Antenna Arrays
1r
r
r
r
1θ
θ
Fig. 8.10 (a)Two Hertz dipoles
2r
r2θ

8.5 Antenna Arrays
For 1 2 oI I I= =
1 2,
2 2
α α
δ δ= = −
2
sin
d dj r
jI dl e
β α β αβ θ θθβ
   − + − + r
2 cos cos
2 2 2 20 sinˆ
4
d dj r j j
total
jI dl e
E e e
r
β α β αβ θ θθβ
θ
πεω
   − + − +   
   
 
= + 
 
 
r
2
0 sinˆ 2cos cos
4 2 2
j r
jI dl e d
r
β
θβ β α
θ 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
Normalized array factor is
( )
1
2cos cos
2
AF dβ θ α
 
= + 
 
( )2
1
cos cos
2
AF 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
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 Arrays
z
2d
(N-1)d
.
.
.
2I α〈
( 1)I N α〈 −
0I〈
I α〈
( 1)
2
N
I α
−
〈
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
N
I 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) cos
1 .....
j d j d j N d
AF e e e
β θ α β θ α β θ α+ + − +
∴ = + + + +
( )( )
ψ−1 e jNN
( )( ) αθβψ
ψ
ψ
αθβ +=
−
−
=⇒=⇒ ∑
=
+− cos;
1
1
1
cos1 d
e
e
AFeAF
j
jN
N
N
n
dnj
1
1
jN
j
e
e
ψ
ψ
−
=
−
( ) ( )1 2 2( )
2
1 1
( ) ( )
2 2
N N
j jN
j
j j
e e
e
e e
ψ ψ
ψ
ψ ψ
−−
−
 
 
 
  
−
=
−
1
( )
2
sin( )
2
sin( )
2
N
j
N
e
ψ
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
=
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
( )
sin( )
1 2
1
sin
2
N
N
AF
N
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
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 cos
2
N
dψ ψ β θ α= = +Q
( ){ }
1
2
cos cos
2 2
1 2
cos 1,2,3,......n
N N n
n d n d
N
n
n
d N
π
ψ π β θ α π β θ α
π
θ α
β
−
⇒ = ± ⇒ + = ± ⇒ = − ±
  
⇒ = − ± =  
  

8.5 Antenna Arrays
Maximum values
It attains the maximum values for 0ψ =
( )
1
cos
2 2 m
d
θ θ
ψ
β θ α
=
= + 0=
8.5.3 Broadside array
We know that when
mθ θ=
1
cosm
d
α
θ
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
90
cos 0 0d θ
ψ β θ α α=
= + = ⇒ =
8.5.4 Endfire array
We know that when
0
90
cos 0 0d θ
ψ 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
0
0
cos 0d dθ
ψ β θ α α β=
= + = ⇒ = −
We know that when
It is desired that maximum occurs at θ=θ0
cos 0dψ β θ α= + =
0
0cos 0 cosd dθ θ
ψ β θ α α β θ=
= + = ⇒ = −

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