8. Antennas and Radiating Systems · Antennas and Radiating Systems 1 Electromagnetic Field Theory...
Transcript of 8. Antennas and Radiating Systems · Antennas and Radiating Systems 1 Electromagnetic Field Theory...
Prof. Rakhesh Singh Kshetrimayum
8. Antennas and Radiating Systems
Prof. Rakhesh Singh Kshetrimayum
4/24/20181 Electromagnetic Field Theory by R. S. Kshetrimayum
8.1 Introduction We use mobile phones everyday
Mobile phone converts our voice into electrical signal using microphone
This signal is modulated and radiated to free space
by antennas as EM waves
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by antennas as EM waves
which is picked up by the base station antennas
We generally use transmission line like tv cables
for transferring EM energy from one point to another within a circuit
8.1 Introduction This mode of energy transfer is called guided wave propagation
It basically means that wave inside transmission line like coaxial cable is guided inside it and
will not come out from it into free space
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will not come out from it into free space
Hence antenna is also called as mode transformer which
transforms guided-wave field into a radiated wave field for transmitting antenna and vice versa
8.1 Introduction
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Fig. Antenna as mode transformer
8.1 Introduction An important property of antenna is its ability to transmit power in a preferred direction like in microwave towers
where we align the transmitting antenna and receiving antenna
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antenna
for line of sight (LOS) communication
8.1 Introduction
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Fig. Microwave tower: LOS communication
8.1 Introduction Radiation pattern shows how power is radiated from the antenna in 3-dimension
Unlike the previous case, ideally base station (BS) and mobile station (MS) antennas should radiate equally in all directions
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directions
as well as they can pick up signals from all directions
Such isotropic antennas do not exist in practice
Omnidirectional directional antennas are used for such cases
8.1 Introduction
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Fig. Mobile communication
8.1 Introduction
Antennas
Radiation fundamentals
Antenna pattern and parameters
Types of antennas
Antenna arrays
When does a charge radiate?
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Fig. 8.1 Antennas (cover antenna pattern and parameters after types of antennas)
charge radiate?
Wave equation for potential functions
Solution of wave equation for potential functions
Hertz dipole
Dipole antenna
Loop antenna
8.2 Radiation fundamentals8.2.1 When does a charge radiates?
accelerating/decelerating charges or time-varying currents in a conductor radiate EM waves
r
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Fig. 8.2 A giant sphere of radius r with a source of EM wave at its origin
Source
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,
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( ) ( )∫∫ •×=•= ∗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
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away 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 (for EM waves we need both E and H)
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
power remains constant in that case
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power remains constant in that case
irrespective of the distance from the source
8.2 Radiation fundamentals From Coloumb’s law, electrostatic fields decrease as 1/r2,
whereas Biot Savart’s law also states that magnetic fields decrease as 1/r2
So the total decrease in the Poynting vector is
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So the total decrease in the Poynting vector is proportional to 1/r4
Hence power decreases as 1/r2
It dies out after some distance from the source
implies no radiation
8.2 Radiation fundamentalsCASE 3: A time varying current or acceleration (or deceleration) of charge will radiate
To create radiation
there must be a time varying current or
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there must be a time varying current or
acceleration (or deceleration) of charge
8.2 Radiation fundamentals
Basic radiation equation:
where L=length of current carrying element, m
dt
dvQ
dt
diL =
dt
diL
dt
dvQ+
Fig. Fundamental law of radiation
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L=length of current carrying element, m =time changing current, As-1(units)
Q=charge, C
=acceleration of charge, ms-2
dt
di
dt
dv
8.2 Radiation fundamentals For this case (we will show this later),
a time varying field (both E and H) is produced
which varies as 1/r
whose field direction is along and θ φ
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Hence the direction of Poynting vector is radiallyoutward
Since Poynting vector varies as 1/r2, total power is always constant
It can go to infinite distance
8.2 Radiation fundamentals Two conditions for EM waves:
1) Fields produced by the EM source should have components varying as 1/r
2) Field direction should not be radial but transversal so that the power flow or Poynting vector should be radial
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the power flow or Poynting vector should be radial
It can be shown that for an infinitesimally small current carrying element (Hertz dipole) which is the building block for antenna, it indeed produces such fields when supplied with time varying currents
8.2 Radiation fundamentals But it is a difficult process to find such fields directly from current density and calculations are highly complex
A major simplification is possible when we find the magnetic vector potential first and
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we find the magnetic vector potential first and find the fields from it
It is similar to find electric field from electric potential than directly finding electric field
This way it is easier
8.2 Radiation fundamentals
Jr
HErr
,
Ar
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Fig. Two ways to find fields
In order to find magnetic vector potential,
we need to write down the wave equation for magnetic vector potential first
8.2 Radiation fundamentals8.2.2 Wave equation for potential functions
One of the Maxwell’s divergence equation
Hence, we can write0=•∇ B
r
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It means that we can find magnetic flux density
from the curl of magnetic vector potential
ABrr
×∇=
8.2 Radiation fundamentals8.2.2 Wave equation for potential functions
Putting this in the following Maxwell’s curl equation
t
BE
∂
∂−=×∇
rr ( )
t
AE
∂
×∇∂−=×∇⇒
rr
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which can be rewritten as
For time varying fields,
t∂ t∂
0A
Et
∂⇒∇× + =
∂
rr
t
AVE
∂
∂−−∇=
rr
8.2 Radiation fundamentals Putting this in the following Maxwell’s divergence equation
ε
ρ=•∇ E
r
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ε
ρ−=
∇+
∂
∂•∇ V
t
Ar
2AV
t
ρ
ε
∂⇒∇ • + ∇ = −
∂
r
8.2 Radiation fundamentals Applying Lorentz Gauge condition
Applying above condition
0=∂
∂+•∇
t
VA µεr
∂ ∂ ∂r
r r
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22
2
V AA J V
t t tµε µ µε
∂ ∂ ∂ ⇒∇ − − ∇ = − + ∇
∂ ∂ ∂
rr r
22
2
AA J
tµε µ
∂⇒∇ − = −
∂
rr r
8.2 Radiation fundamentals
Another Maxwell’s curl equation
t
EJB
∂
∂+=×∇
rrr
µεµ
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Simplifies to
( )t
Vt
A
JAAA∂
∇+
∂
∂∂
−=∇−•∇∇=×∇×∇
r
rrrrµεµ2
8.2 Radiation fundamentals Applying Lorentz Gauge condition
Applying above condition
0=∂
∂+•∇
t
VA µεr
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( )ε
ρ−=∇+
∂
•∇∂V
t
A 2
r
22
2
VV
t
ρµε
ε
∂⇒∇ − = −
∂
8.2 Radiation fundamentals8.2.2 Wave equation for potential functions
From Maxwell’s equations for time varying fields,
we have derived the two wave equations for potential functions
magnetic vector and
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magnetic vector and
electric potentials
Why find potential functions instead of fields?
Jt
AA
rr
rµµε −=
∂
∂−∇
2
22
22
2;
VV
t
ρµε
ε
∂∇ − = −
∂
8.2 Radiation fundamentals8.2.3 Solution of wave equation for potential functions
For time harmonic functions of potentials,
JAArrr
µβ −=+∇ 22
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where
To solve the above equation, we can apply Green’s function technique
JAA µβ −=+∇
µεωβ =
8.2 Radiation fundamentals8.2.3 Solution of wave equation for potential functions
Green’s function G is the solution of the above equation with the R.H.S equal to a delta function
( )spaceGG δβ =+∇ 22
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Once we obtain the Green’s function,
we can obtain the solution for any arbitrary current source by applying the convolution theorem
( )spaceGG δβ =+∇ 22
( ) ( ) ( ) ( )rGrrArJrrrrrr
→→− δµ ;
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
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function) with the input current
( ) ( ) ( )( ) ( ) '
V
'
rrjβ
''dv
rr
erJ
4π
µrJµrGrA
'
∫−
=−∗=
−−
rrrrrrrr
rr
8.2 Radiation fundamentals Notation:
The prime coordinates denote the source variables
unprimed coordinates denote the observation points
The modulus sign in is to make sure that rr
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is positive
since the distance in spherical coordinates is always positive
'rrrr
−
8.2 Radiation fundamentalsDigression:
LTI system
For such system with an impulse response h(t) and input signal x(t),
the output signal is given by y(t)=h(t)*x(t)
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the output signal is given by y(t)=h(t)*x(t)
Note that is LTI system,
we consider x(t), h(t) and y(t) are functions of time
In magnetic vector calculation,
are functions of space ( ) ( ) ( )rA,r,GrJµ
rrrrr−
8.2 Radiation fundamentals
Sl. No. System LTI Magnetic vector potential calculation
Table: Analogy of LTI and Magnetic vector potential calculation
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calculation
1 x(t)
2 h(t)
3 y(t)
( )rJµrr
−
( )rGr
( )rArr
8.2 Radiation fundamentals For radiation problems,
the most appropriate coordinate system is spherical since the wave moves out radially in all directions
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Fig. An antenna radiating equally in all directions
8.2 Radiation fundamentals
It has also symmetry along θ and φ directions
0G G
θ φ
∂ ∂= =
∂ ∂
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Hence, the above equation reduces to
( )rGr
Gr
rrδβ =+
∂
∂
∂
∂ 22
2
1
8.2 Radiation fundamentals
Putting Ψ= G r,
For r not equal to 0 (field should not be obtained at the
( )rrr
δβ =Ψ+Ψ∂
∂ 2
2
2
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source itself),
Therefore,
02
2
2
=Ψ+Ψ∂
∂β
r
rjrjBeAe
ββ +− +=Ψ
8.2 Radiation fundamentals Since the radiation moves radially in positive r direction negative r direction is not physically feasible for a source of a field, we get,
rjAe
β−=Ψ AeG
rjβ−
=
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we can find the constant A and hence
which is magnetic vector potential produced by a delta source
Ae=Ψr
G =
r
eGA
rj
ππ
β
4;
4
1 −
−=−=
8.4 Kinds of antennas8.4.1 Hertz dipole Electrically small antennas are small relative to wavelength
whereas electrically large antennas are large relative to wavelength
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wavelength Hertz dipole is not of much practical use
but it is the basic building block of any kind of antennas
An infinitesimally small current element is called a Hertz dipole
8.4 Kinds of antennas8.4.1 Hertz dipole
Let us find the fields of a small current carrying element of length dl
The procedure involves
Determining the current on the antenna
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Determining the current on the antenna
Then compute
EHAJrrrr
→→→
8.4 Kinds of antennas The infinitesimal time-varying current in the Hertz dipole is assumed as
where ω is the angular frequency of the current
Since the current is assumed along the z-direction,
( ) zeItItj ˆ0
ω=
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Since the current is assumed 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 Note that for this case
For infinitesimally small current element at the origin
( )' '
0
j tJ r dv I dle
ω=r r
'r r r− =r r r
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r r r− =
( ) ( ) ( )( ) ( ) '
V
'
rrjβ
''dv
rr
erJ
4π
µrJµrGrA
'
∫−
=−∗=
−−
rrrrrrrr
rr
zr
edleIzAA
tjrj
zˆ
4ˆ 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
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0
j tI e
ω
8.4 Kinds of antennas
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
= − = − − −
r
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Coordinate transformation
cos ; sin ; 0r z z
A A A A Aθ φθ θ⇒ = = − =
r
θ
8.4 Kinds of antennas
Using the symmetry of the problem (no variation in ),
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
φ
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
0J =r
ωεj
HE
rr ×∇
=
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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
Note that Er has only 1/r2 and 1/r3 variation with r
( )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
( )
+
∂−=
∂−= e
rj
rrjHr
rrjE φθ β
πωεθ
θωε 4sin
sin2
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
Which wave is this?
Points to be noted:
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Fields can be classified into three types
Radiation fields (spatial variation 1/r)
Induction fields (spatial variation 1/r2) and
Electrostatic fields (spatial variation 1/r3)
8.4 Kinds of antennas1) Field variation with frequency since
Electrostatic fields (1/r3) are also inversely proportional to the frequency ( )
µεωβ =
β
ω
1
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Induction field (1/r2) is independent of frequency ( )
Radiation field (1/r) is proportional to frequency ( )ω
β 2
ω
β
8.4 Kinds of antennas2) Field variation with r
For small values of r,
electrostatic field is the dominant term and
Induction field is the
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transition from electrostatic field from radiation fields
For large values of r,
radiation field is the dominant term
8.4 Kinds of antennas Near fields:
We can show that Hertz dipole has reactive near field
2
0
4
sin
r
edleIH
rjtj
nf π
θβω
φ
−
=
− sin jedleIrjtj βθβω
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−=−
32
0
4
sin
r
j
r
edleIE
rjtj
nf
β
πεω
θβω
θ
+=−
32
0 1
4
cos2
rr
j
j
edleIE
rjtj
rnf
β
πεω
θβω
8.4 Kinds of antennas It looks like the static magnetic field produced by a current carrying element using Biot Savart’s law
2
0
4
sin
r
edleIH
rjtj
nf π
θβω
φ
−
=
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It resembles the electric field produced by an electric dipole
−=−
3
0
4
sin
r
jedleIE
rjtj
nf πεω
θβω
θ
=−
3
0 1
4
cos2
rj
edleIE
rjtj
rnf πεω
θβω
8.4 Kinds of antennas Note that 1/r5 term in Poynting vector is purely reactive
Note that the 1/r4 term in the Poynting vector is
due to induction fields
which will die out after some distance from the source
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Far fields:
r
edleIjH
rjtj
ff π
θβ βω
φ4
sin0
−
=
r
edleIjE
rjtj
ff πεω
θβ βω
θ4
sin0
2 −
=
8.4 Kinds of antennas Note that the 1/r2 term in the Poynting vector
gives total power over a giant sphere always constant
Hence such waves from the radiation fields
will go to infinite distance in free space
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They also satisfy two conditions for EM waves:
Transversal fields (Eθ and Hφ gives Poynting vector along radial direction)
Fields vary as 1/r (Poynting vector varies as 1/r2)
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=
λ − 2sin jjedleIrjtj ββθβω
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For r< λ/2=,
1/r3 term dominates
For r>> λ/2=,
the 1/r term dominates
−+=−
32
2
0
4
sin
r
j
rr
jedleIE
rjtj
r
ββ
πεω
θβω
8.4 Kinds of antennas 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
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for an ideal infinitesimally small Hertz dipole
electrostatic fields dominate
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 e
E jr
ω
θ
θ
πεω≈ −
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The magnitude of the near field is
34r
rπεω 34E j
rθ
πεω≈ −
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
( ) θθθ 22 sincos4 +=F
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Maximum field is along
θ=00, θ=1800 and
minimum is along
θ=900, θ=2700 (see Fig. 8.6(a))
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Fig. 8.6 (a) Near field pattern plot of a Hertz dipole located at the origin and oriented along z-axis (maximum radiation 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 Fresnel near field region that’s why we have chosen an r>> λ/2=), radiation field is the dominant term
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radiation field is the dominant term
In other words kr>>1, we have,
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 HφIt is interesting to note that the ratio of electric field and
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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
No radiation along z-axis unlike near field case
They are maximum along θ=π /2
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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
*1ReS E H r= $
2 3
0 sin1 I dlr
θ β =
$
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum64
Antenna power flows radially outward
Power density is not same in all directions
The net real power is only due to
the radiations fields (i.e. jβ2/r and jβ/r) of electric and magnetic fields
*Re2
avgS E H rθ φ= $ 0
2 4r
rπ ωε=
$
8.4 Kinds of antennas Total radiated power:
The total radiated power from a Hertz dipole
2 sinavg
W S r d dθ θ φ= ∫∫
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum65
∫∫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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum66
the frequency and
the length
of the Hertz dipole increases
8.4 Kinds of antennas Radiation resistance of a Hertz Dipole:
Hertz dipole can be equivalently modeled as a radiation resistance
Since W=1/2 I02 Rrad
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implies that Rrad = 80π2
2
2 2
040dl
W Iπλ
∴ =
2dl
λ
Fig. Equivalence of Hertz dipole and radiation resistance
8.4 Kinds of antennas
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
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(c))
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Fig. 8.6 (c) A typical 3-D radiation pattern of a Hertz dipole in the far field
8.4 Kinds of antennas Two principal planes radiation patterns (2-D) are normally plotted
E-plane (vertical cut)
H-plane (horizontal cut) radiations patterns
are sufficient to describe the radiation pattern of a Hertz
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum70
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
E-plane (xz-lane) radiation pattern looks like 8 shape
φ φ
8.4 Kinds of antennas
00=θ00=φ
090=φ0270=φ
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum71
Fig. H-plane and E-plane radiation patterns of Hertz dipole
0270=θ0180=θ
090=θ
0180=φ
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 φ
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum72
Note that θ is also known as elevation angle and as azimuth angle
E-plane pattern for a dipole is also known as elevation pattern
H-plane pattern as azimuthal pattern
φ
8.3 Antenna pattern and parameters
Which parameters are used for specifying an antenna?
What parameters differentiate one antenna from others?
How do one characterize an antenna?
Antenna has two types of characteristics from
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum73
Field concepts
Circuit concepts
8.3 Antenna pattern and parameters
An antenna can launch free space wave in a desired direction
This directional characteristics of an antenna can be interpreted from its radiation characteristics
We also know that an antenna is connected to a
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum74
We also know that an antenna is connected to a transmission line
and it converts guided wave to free space wave
Hence it can be thought as a load to the transmission line
One can find its equivalent circuit (input characteristics)
8.3 Antenna pattern and parameters
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum75
Fig. Antenna characteristics: (a) Radiation characteristics (b) input characteristics
8.3 Antenna pattern and parameters
Radiation characteristics
Radiation pattern
Directivity
Gain
Polarization, etc.
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Polarization, etc.
8.3 Antenna pattern and parameters
Input characteristics
Input impedance
Bandwidth
Reflection coefficient
Voltage standing wave ratio, etc.
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Voltage standing wave ratio, etc.
8.3 Antenna pattern and parameters
Radiation characteristics
Radiation intensity:
The most fundamental parameter is its radiation intensity
Based on this parameter,
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Based on this parameter,
gain and
directivity
of the antenna will be defined
8.3 Antenna pattern and parameters
rdA
dΩ 2
r
dAd =Ω
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Fig. Solid angle (Ratio of the area subtended by the solid angle to the squared radius)
2r
d =Ω
8.3 Antenna pattern and parameters
Radiation intensity is defined as power crossing per unit solid angle
Power crossing over the area dA is ( )dAS φθ ,
( ) ( ) ( ) ( )SrWrSdAS
U /,,
, 2φθφθ
φθ ==
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum80
Radiation intensity can be calculated by
multiplying the Poynting vector by r2
( ) ( ) ( ) ( )SrWrSd
dASU /,
,, 2φθ
φθφθ =
Ω=
8.3 Antenna pattern and parameters
For example
Hertz dipole
Note that U(θ,φ) is independent of r
( ) ( )0
2
222
0
22
32
sin,,,
ηπ
θβφθφθ
dlIrrSU ==
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Normalized radiation intensity is a dimensionless quantity
For Hertz dipole
( ) ( )( )φθ
φθφθ
,
,,
maxU
UU n =
( ) θφθ 2sin, =nU
8.3 Antenna pattern and parameters
It is usually expressed in dB
For Hertz dipole
( ) ( )θφθ 2
10 sinlog10, =dBnU
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The power or radiation intensity pattern is
the angular distribution of antenna’s radiated power
per unit solid angle
8.3 Antenna pattern and parameters
One could also find the total radiated power of an antenna as
2
4 0 0
( , ) ( , ) sinradP U d U d d
π π
π θ φ
θ φ θ φ θ θ φΩ= = =
= Ω =∫ ∫ ∫
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum83
Average radiation intensity is defined as
4 0 0π θ φΩ= = =
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
8.3.1 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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum84
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
Dipole AntennaOmni-directional radiation pattern
8.3 Antenna pattern and parameters
Horn Antenna Directional radiation pattern
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
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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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum88
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
For example,
Hertz dipole
How to decide E- and H- planes?
Current is flowing in z-direction
( ), ,P r θ φ
0
j tI e
ω
r
θ
φ
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Current is flowing in z-direction
Magnetic vector potential follows the current direction
Magnetic field will be along φ-direction (H-plane is in x-y plane)
Electric field will be along θ-direction (E-plane is in x-z plane)
8.3 Antenna pattern and parameters
00=θ00=φ
090=φ0270=φ
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum90
Fig. H-plane and E-plane radiation patterns of Hertz dipole
0270=θ0180=θ
090=θ
0180=φ
8.3 Antenna pattern and parameters
Besides 2-D and 3-D polar plots,
radiation pattern may be plotted as rectangular plots
In this case,
horizontal axis is in degrees
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vertical axis is normalized radiated power in dB
8.3 Antenna pattern and parameters
0 dB
-3 dB
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Fig. 8.3 (c) Typical radiation pattern of an antenna (rectangular plots)
maxθ
θ00 500-500 1000-1000
-15 dB
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)
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum93
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
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
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of the antenna
It is the radiation lobe in the direction of maximum radiation
8.3 Antenna pattern and parameters
θmax is the direction in which maximum radiation occurs
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
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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
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the maximum electric field
Its shows how sharp is the beam
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
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of HPBW for narrow beams and broad beams
HPBW is a better parameter for specifying the effective beam width
It gives an idea of the main beam shape
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
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antenna radiation pattern
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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum99
Fig. Microwave tower: LOS communication
8.3 Antenna pattern and parameters
Back lobe Sometimes antenna may have back lobe radiation
It is specified by front-to-back ratio
It is basically the ratio of the peak of the main lobe
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over the peak of the back lobe
It gives an idea abut the directivity of an antenna
Like side lobe, back lobe is also an unwanted radiation
8.3 Antenna pattern and parameters
Horn Antenna Directional radiation pattern
8.3 Antenna pattern and parameters
At what distance from the antenna, we may assume far field region
Region surrounding an antenna may be divided into three regions:
Reactive near field
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Reactive near field
Radiating near field
Far field
3
max0 0.62D
r< ≤
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Fig. 8.3 (a) Antenna field regions
max10 0.62
nf
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)
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum104
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
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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
<
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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)
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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)
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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
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to the radiation intensity averaged over all directions 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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum110
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
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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
≅ =
× ×
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum112
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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum113
remaining part is lost in the imperfect conductors as well 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
( ) radrad
4 ( , )e4 ( , ), e ( , )
input rad
UUG D
P P
π θ φπ θ φθ φ θ φ= = =
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum114
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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum115
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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum116
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
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(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
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directions
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
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and 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
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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) CPπ
δ = ±
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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
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This implies that the total radiated electric field vector of the antenna traces out a circle as time progresses from 0 to 2 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), then the antenna polarization is RHCP (LHCP)
For
the total electric field vector traces out an ellipse and
( ) ( ), , 0,E E andθ φθ φ θ φ α π≠ ≠ πδ ,0≠
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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 axial ratio (AR)
For instance, AR=0 dB for CP and AR= ∞ dB for LP
8.3 Antenna pattern and parameters
Antenna polarization:
To sum up,
The polarization of a radiated wave in the far field region of an antenna defines antenna polarization
Assume a transmitting antenna transmitting electric field of
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum124
Assume a transmitting antenna transmitting electric field of maximum magnitude Em in the far field region of an antenna
where is the unit vector which gives polarization for the transmitting antenna
tm
teEE ˆ=
r
te
8.3 Antenna pattern and parameters
It is the complex vector representation of field normalized to unity
Assume that there is a receiving antenna in the far field region of this antenna
The field of some magnitude Em that would give maximum
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum125
The field of some magnitude Em that would give maximum received power at the received antenna terminal is
where is the unit vector which gives polarization for the receiving antenna
r
mreEE ˆ=
r
re
8.3 Antenna pattern and parameters
Polarization efficiency is defined as
Receiving LP wave with an LP receiving antenna: Let us consider a case where the transmitting antenna is
2*ˆˆ.. rt eeEP •=
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Let us consider a case where the transmitting antenna is placed at an angle θt w.r.t z-axis
and the receiving antenna is placed at an angle θr w.r.t z-axis
Both transmitting and receiving antennas are LP antennas
8.3 Antenna pattern and parameters
tθrθ
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8.3 Antenna pattern and parameters
Therefore, polarization unit vectors for the transmitting and receiving antennas are
Polarization efficiency can be calculated as
yzeyze rrrtttˆsinˆcosˆ;ˆsinˆcosˆ θθθθ +=+=
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum128
( )rtrt eeEP θθ −=•= 22
* cosˆˆ..rt θθ −
8.3 Antenna pattern and parameters
θt-θ
rP.E. Illustration
0 1
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π/2 0
8.3 Antenna pattern and parameters
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Fig. Receiving a CP wave with a CP receiving antenna
8.3 Antenna pattern and parameters
Receiving a CP wave with a CP receiving antenna
Consider a RHCP receiving antenna is kept in the far field region of RHCP transmitting antenna
Assume wave propagation along positive y-axis
Hence the polarization unit vectors for transmitting and
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum131
Hence the polarization unit vectors for transmitting and receiving antennas are
( ) ( )zjxezjxe rtˆˆ
2
1ˆ;ˆˆ
2
1ˆ −=−=
8.3 Antenna pattern and parameters
Polarization efficiency
For instance, the receiving antenna is LHCP
1ˆˆ..2
* =•= rt eeEP
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Polarization efficiency
( ) ( )zjxezjxe rtˆˆ
2
1ˆ;ˆˆ
2
1ˆ +=−=
0ˆˆ..2
* =•= rt eeEP
8.3 Antenna pattern and parameters
It is like two persons shaking hands
Right to right hand shake is matched but right to left hand shake is not matched
Receiving an LP wave by a CP receiving antenna:
Consider an LP wave transmitting antenna which is oriented
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum133
Consider an LP wave transmitting antenna which is oriented at an angle θt with the z-axis
Receiving antenna is a RHCP antenna
Hence
( )zjxezxe rtttˆˆ
2
1ˆ;ˆsinˆcosˆ −=+= θθ
8.3 Antenna pattern and parameters
tθ
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum134
Fig. Receiving an LP wave by a CP receiving antenna
8.3 Antenna pattern and parameters
Therefore
Polarization efficiency is
2
1sincos
2
1ˆˆ..
22* =+=•= ttrt jeeEP θθ
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum135
Hence there is a 3-dB signal loss
The same case when one use a RHCP/LHCP transmitting antenna and a LP receiving antenna
2sincos
2ˆˆ.. =+=•= ttrt jeeEP θθ
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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum136
another 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
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the cross-polarization for this case is vertical 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 eβ−= +
r
8.3 Antenna pattern and parameters
where the co- and cross-polarization unit vectors satisfy the orthonormality condition
Therefore,
* * * *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• = • = • = • =
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum138
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 cosco LP LP cr LP LP
u 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
− += =
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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−+ − += =
+ +
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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
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Fig. Antenna measurement set up
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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum143
Another antenna is usually fixed and it is connected in 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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum144
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
Let us consider Hertz dipole which is a vertically polarized antenna
For E-plane co-polar measurements,
source and test antenna are oriented vertically
so that the main beam of the test antenna (receiving
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum145
so that the main beam of the test antenna (receiving antenna) is directed towards y-axis
Source antenna is kept stationary and
we rotate the test antenna in the x-z plane
and record the electric fields variation over θ
8.3 Antenna pattern and parameters
Fig. E-plane co-polar measurements
Test antenna
z
Rotate around
x‐z plane
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum146
x
ySource
antenna
8.3 Antenna pattern and parameters
For E-plane cross-polar measurements,
source is oriented horizontally and test antenna is oriented vertically
Source antenna is kept stationary and
we rotate the test antenna in the x-z plane
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we rotate the test antenna in the x-z plane
and record the electric fields variation over θ
8.3 Antenna pattern and parameters
Fig. E-plane cross-polar measurements
Test antenna
z
Rotate in the
x‐z plane
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x
ySource
antenna
8.3 Antenna pattern and parameters
For H-plane co-polar measurements,
source and test antenna are oriented vertically
Source antenna is kept stationary and
we rotate the test antenna in the x-y plane
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum149
and record the electric fields variation over φ
8.3 Antenna pattern and parameters
Fig. H-plane co-polar measurements
z
Rotate around
x‐y plane
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Test antenna
x
ySource
antenna
8.3 Antenna pattern and parameters
For H-plane cross-polar measurements,
source is oriented horizontally and test antenna is oriented vertically
Source antenna is kept stationary and
we rotate the test antenna in the x-y plane
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we rotate the test antenna in the x-y plane
and record the electric fields variation over φ
8.3 Antenna pattern and parameters
Fig. H-plane cross-polar measurements
z
Rotate around
x‐y plane
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Test antenna
x
y
Source
antenna
Cross-polar
Co-polar
Fig. Typical E-plane pattern4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum153
Cross-polar
Co-polar
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum154
polar
Fig. Typical H-plane pattern
8.3 Antenna pattern and parameters
Let us look at the antenna input characteristics
Antenna is considered as a load to the transmission line
Hence one can find out the equivalent circuit of an antenna
Antenna is a resonator
It can be equivalently modelled as a series RLC circuit
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It can be equivalently modelled as a series RLC circuit
8.3 Antenna pattern and parameters
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum156
Fig. Equivalent circuit 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,
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum157
r L
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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum158
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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum159
as 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
Small antennas are a necessity for portable devices For example, We need small antennas for mobile devices Smaller antenna is an attractive feature But the fundamental question remains
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum160
But the fundamental question remains Can we keep on reducing antenna size?
Is there some limit beyond which we can reduce the antenna size?
Consider a smallest sphere which enclose a small antenna as shown in next slide
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum161
Fig. Smallest sphere which can enclose a small antenna
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 J. L. Chu’s relation
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum162
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:
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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
CASE STUDY of a small antenna
for a Hertz dipole of
very small length 0.01λ,
Qmin =32283
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum164
It has very high Q and
hence a very narrow FBW (0.000031) (very narrow)
Gmax=0.0638 or -12dB (very small no practical use)
Note that in the above calculations r=0.005λ has been used
8.3 Antenna pattern and parameters
Very small antenna like Hertz dipole has no practical use For practical use, we need to have a sufficiently large antennaAntenna size, quality factor,
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum165
quality factor, bandwidth and radiation efficiency is interlinked
There is no complete freedom to optimize each one of them independently
8.3 Antenna pattern and parameters
RECAP:
Antenna acts as an interface between a guided wave and a
free-space wave
One of the most important characteristics of an antenna is its
directional property
Ability to concentrate radiated power in a certain directionAbility to concentrate radiated power in a certain direction
Or receive power from a preferred direction
This directional property is characterized in Radiation pattern
From reciprocity theorem, we can show that
the pattern characteristics of an antenna are the same in the
transmit and
the receive modes
8.3 Antenna pattern and parameters
Antenna equivalent circuit can be thought of as
Radiation resistance
Loss resistance
Reactive parts
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So the feed line also has a characteristic impedance Z0
usually of 50 Ohm
At the input of the antenna, the impedance seen by the feed line can be assumed as ZL
8.3 Antenna pattern and parameters
Then the reflection coefficient and
VSWR may be calculated as
1
1;
0
0
Γ−
Γ+=
+
−=Γ VSWR
ZZ
ZZ
L
L
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2:
1,10
10
≤
∞≤≤≤Γ≤
Γ−+
VSWRBW
VSWR
ZZ L
%100×−
=c
lh
f
ffBW
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum169
fl fh
fc
( )dBΓ
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
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It consists of a conductor of length 2L
fed by a voltage or current source at its center
We will first find the current distribution on such antennas
Fig. 8.7 (a) Dipole of length 2L
(Dipole can be θ
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum171
(Dipole can be assumed
to composed of many Hertz dipoles)
8.4 Kinds of antennas Let us assume a transmission line loaded with a load ZL
Since the line impedance and load impedance may be different
Any wave incident from the line to the load will be reflected
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum172
reflected
Hence at any location along the line,
there will be both reflected and incident wave which may be expressed as
( ) zjzj eVeVzV ββ +−−+ += 00
8.4 Kinds of antennas
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum173
Fig. A transmission line loaded with an impedance of ZL
8.4 Kinds of antennas Corresponding electric current wave along the line may be expressed as
( ) ( )0
0
0
0
0
0
0
0
0
0 ;ZZ
ZZ
V
Vee
Z
Ve
Z
Ve
Z
VzI
L
Lzjzjzjzj
+
−==ΓΓ−=−=
+
−+−
++
−−
+ββββ
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum174
The characteristic impedance of the line is defined as
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+ +
− += − = −
−
−
+
+
−==0
0
0
00
I
V
I
VZ
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,
( )0 00 0( ) 2 sin( ( )) sin ; 2
V VI z j L z I L z I j
Z Zβ β
+ +
= − − = − = −
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum175
How to plot the current distribution of dipoles of various length?
( )0 0
0 0Z Z
2
λ
2
λ
o.c.
Cut at any point before 8
λ
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum176
Fig. Current distribution of a 2L<<λ dipole
8
L
L
2
λ
2
λ
4
λ
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Fig. Current distribution of a λ/2 dipole
4
4
λ
4
λ
2
λ
2
λ
o.c.
Cut at 2
λ
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum178
Fig. Current distribution of a λ dipole
2
λ
2
λ
2
λ
2
λ
o.c.
Cut at
4
3λ
2
λ
2
λ
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum179
Fig. Current distribution of a 3λ/2 dipole
2
2
λ2
λ
8.4 Kinds of antennasHow to find the electric field?
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)
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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
−
= =
Fig. 8.7 (a) Dipole of length 2L
(Dipole can be θ
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum181
(Dipole can be assumed
to composed of many Hertz dipoles)
zcosθ
8.4 Kinds of antennas Since the observation point P is at a very far distance, the lines OP and QP are parallel and therefore
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≅
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since the dipole size is quite small in comparison to the distance of the observation point P from the origin
Hence
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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum183
or infinitesimal dipole is the building block for many antennas),
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)
( )( )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
− − −⇒ ≅ =
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In the previous equation, the term under bracket is F(θ) and it is the variation of electric field as a function of θand
it is the E-plane radiation pattern
8.4 Kinds of antennas In the H-plane like that of Hertz dipole,
Eθ is a constant and it is not a function of
hence it is a circle
The E–plane radiation pattern of the dipole
φ
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varies with the length of the dipole as depicted in Fig. 8.8
8.4 Kinds of antennas Note that according to Fig. 8.7 (a),
we have considered the total dipole length is 2L
Fig. 8.8 E-plane radiation pattern for dipole of length
(a) 2L=2×λ/4= λ/2
λ λ
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(b) 2L=2×λ=2λ
(c) 2L=2×2λ=4λ
( )( )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
− − −⇒ ≅ =
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum187
(a) 2L=2×λ/4= λ/2
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(b) 2L=2×λ=2λ
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(c) 2L=2×2λ=4λ
8.4 Kinds of antennasPoints to be noted:
1. Input impedance of the dipole (z=0, center of dipole)
For dipole of length 2L, where L =odd multiples of
0 sin
in inin
in
V VZ
I I Lβ= =
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum190
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= = inZ⇒ = ∞
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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum191
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 monotonically
8.4 Kinds of antennas3. 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
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8.4 Kinds of antennas
cos( cos ) cos0
sin
null
null
L Lβ θ β
θ
−=
nullθcos⇒ 1mλ
π= ± ±
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o 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→ →
− ≅ − + − − +
( ) ( ) ( ) ( ) ( ) ( )0, 0,
4 42 24 22 1 cos sin 1 cossin sin10
2! 4! sin 2! 4!
L LL L
Lim Limθ π θ π
β θ β θ θβ θ β θ
θ→ →
− + = − × = − =
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum194
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
( )dF
d
θ
θ
maxθ
8.4 Kinds of antennas What is the radiation resistance of a dipole antenna?
Radiated power can be obtained from the radiation intensity as
2
0
2
I
PR rad
r =
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum195
as
( ) φθθφθπ
φ
π
θ
ddUPrad sin,
2
0 0
∫ ∫= =
=
8.4 Kinds of antennas Radiation intensity can be obtained from the Poynting vector as
Therefore,
( ) ( ) ( )θ
θπ
πθ
πφθφθ
2
2
2
0
22
0
2
sin
cos2
cos3030
,,
=== IFISrU
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum196
Therefore,
∫=
=π
θ
θθ
θπ
π0
2
2
2
0sin
cos2
cos30
dIPrad
8.4 Kinds of antennasMonopole 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))
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum197
Monopole is usually fed from a coaxial cable (see Fig. 8.7 (b))
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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum198
(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))
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum199
(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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum200
cases
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 antennas8.4.3 Loop antenna
Loop antennas could be of various shapes: circular,
triangular,
square,
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum201
square,
elliptical, etc.
They are widely used in applications up to 3GHz
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
4/24/2018Electromagnetic Field Theory by R. S. Kshetrimayum202
are practically useless
Electrically small loop antennas could be analyzed assuming that it is equivalently represented as a Hertz dipole
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
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the beam or radiation pattern is fixed
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,
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circular, planar, etc.
(b) element spacing
(c) element excitation amplitude
(d) element excitation phase
(e) patterns of array elements
φ
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 φ
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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
the excitation amplitude and phase of the elements
φ
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
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assuming no coupling 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= = −
2 cos cossinˆd dj r j jjI dl e
β α β αβ θ θθβθ
− + − + = +
r
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2 cos cos2 2 2 20 sinˆ
4
j r j j
total
jI dl eE e e
r
β θ θθβθ
πεω
− + − +
= +
r
2
0 sinˆ 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
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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
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straight line along z-axis at positions 0, d, 2d, …, (N-2) d and (N-1) d respectively
8.5 Antenna Arrays Current of equal amplitudes
but with phase difference of 0, α, 2α, … , (N-2) α and (N-1) α
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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
Nψ
0I ⟨
I α⟨
( 1)
2
NI α
−⟨
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For small values of
( )sin( )
2
sin( )2
NAF
ψ
ψ=
( )sin( )
2
2
N
N
AF
ψ
ψ=
0I ⟨
I α⟨−
( 1)
2
NI α
−⟨−
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
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other N-1 side lobes
in one period of the AF 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
( )sin( )
1 21
sin2
N
N
AFN
ψ
ψ
=
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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=
( )sin( )
1 21
sin2
N
N
AFN
ψ
ψ
=
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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
== + = ⇒ =
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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
== + = ⇒ = −
8.5 Antenna Arrays
Draw the polar plot of radiation pattern for the following
uniform linear array (ULA)
of N isotropic radiating antennas spaced λ/2
apart for the following cases: θ
I α⟨
( 1)
2
NI α
−⟨
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(a) Broadside array (Maximum field is at θ=90˚)
(b) End fire array (Maximum field is at θ=0˚)
(c) Maximum field is at θ=60˚ and
(d) Null at θ=60˚
( )sin( )
1 21
sin2
N
N
AFN
ψ
ψ
=
0I ⟨
I α⟨−
( 1)
2
NI α
−⟨−
8.5 Antenna Arrays 0
0
90cos 0 0d
θψ β θ α α
== + = ⇒ =
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(a) Broadside array (Maximum field is at θ=90˚)
8.5 Antenna Arrays 00cos 0d d
θψ β θ α α β
== + = ⇒ = −
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(b) End fire array (Maximum field is at θ=0˚)
8.5 Antenna Arrays
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(c) Maximum field is at θ=60˚
00cos 0 cosd d
θ θψ β θ α α β θ
== + = ⇒ = −
8.5 Antenna Arrays
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(d) Null at θ=60˚L,3,2,1,cos
2=−= nd
N
nnullθβ
πα