EMG6343 :Part I: Antennas
•Radiation fields of elemental dipoles.
•Antenna patterns and antenna parameters:
beamwidth, directivity, gain, sidelobes, linear
polarization, circular polarization, radiation
resistance, equivalent circuit of receiving
antenna, effective length, capture area.
• Friis transmission formula.
•Reciprocity theorem.
•Radiation by dynamic currents and charges,
retarded potentials, isotropic source.
•Half-wave dipole, loop antenna
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Chap 1: Antenna Basic
1
Text and Reference
• D.K. Cheng, “Field and waves electromagnetic”, 2nd Ed,
1989, Addison-Wesley.
• Matthew N. O. Sadiku, “Elements of Electromagnetics”,
Saunders College Publishing.
• Kraus & Fleisch, “Electromagnetics – with applications”,
McGraw Hill
• Kraus & Marhefka, “Antennas – for all applications”,
McGraw Hill
• C. A. Balanis, “Antenna theory – analysis and design”,
Wiley.
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Chap 1: Antenna Basic
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Definition of Antenna
• That part of a transmitting or receiving system
that is designed to radiate or to receive
electromagnetic waves.
• A transducer between a guided wave
propagating in a transmission line and an
electromagnetic wave propagating in an
unbounded medium (usually free space), or
vice versa.
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Chap 1: Antenna Basic
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Types of radiating structures
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Chap 1: Antenna Basic
4
Wire antenna (half wave dipole)
The most basic form of antenna,
and most popular
Transmission line
/2
Radiation pattern
of a 2 thin wire
dipole – omnidirectional
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E-plane
Chap 1: Antenna Basic
Total length
of the
radiating
element is
halfwavelength
H-plane
5
Input impedance of a half-wave dipole is
complex with;
Real part having a
radiation
resistance of
&
~73 + the ohmic
loss of the wire
Imaginary part is
~42, but
decreases to 0 if
length of dipole is
reduced by 4%.
Thus, by trimming the dipole by 4%, the
input impedance is purely real, and
~73, making it possible to match to a
75 transmission line.
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Chap 1: Antenna Basic
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Quarter wave monopole
• A quarter-wave
monopole antenna
excited by a source at its
base exhibits the same
radiation pattern in the
region above the ground
plane as a half-wave
dipole in free space
• Use image theory for
analysis
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Chap 1: Antenna Basic
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Dipole of other lengths
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Chap 1: Antenna Basic
8
Short Electric Dipole
What is a “short” dipole?
L
An infinitesimal linear wire positioned
symmetrically at the origin of the
coordinate system and oriented along
the z-axis.
Why do we need to study short dipole?
Any linear antenna may be considered as consisting of a large
number of very short conductors connected in series.
It is the oldest, simplest, cheapest and in many cases, the most
versatile.
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Chap 1: Antenna Basic
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What makes a short dipole?
Length L is very short compared to wavelength (L<<).
Carries uniform current I along the entire length L. To
allow such uniform current, we attach plates at the ends
of the dipole as capacitive load. However, we assume
the plates are small that their radiation is negligible.
The dipole may be energized by balanced transmission
line. However, it is assumed that the transmission line
does not radiate.
+q
L
I
The diameter d of the dipole is small compared to its
length (d<<L).
Thus a short dipole consist simple of a
thin conductor of length L with a uniform
current I and point charges q at the ends.
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Chap 1: Antenna Basic
dq
I
dt
-q
10
Retarded current
What is retardation effect?
The electromagnetic waves have finite propagation times. Thus, if a
current is flowing in the short dipole, the effect of the current is not felt
instantaneously at the point P, but only after an interval equal to the time
required for the disturbance to propagate over the distance r.
The effect observed a distant point P from a given source at any instant
t is due to a current flowing at an earlier time which is,
  r 
I  I m cos   t - 
  c 
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Chap 1: Antenna Basic
(1.1)
11
Retarded magnetic vector potential
General solution to wave equation is given as,
2

A
 2 A   2   J
t
For a short dipole with current I, and length L,
retarded magnetic vector potential is,
LI m cos(t  r )
Az 
4r
What can we do with Az?
From curl of Az, get H
 J 
A
dv '

V
'
4
r
(1.2)
 L / 2 I 
Az 
dz


L
/
2
4
r
(1.3)
If r>>L and >>L, we
neglect the phase
differences of the field
contributions. Take the
integrand as constant.
Obtain E from H, can use Ampere’s law
Deduce the far field pattern, radiated power, radiation resistance,
directivity and all other antenna parameters!
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Get H from A
 LI m e j t  r  
Az  Re 

4

r


or
LI m  jr
Az 
e
4r
Express Az into Ar, A and A,
Ar  Az cos  ,
A   Az sin  ,
A  0
  A sin   A  
1
H    A 

ar



r sin  

 
1
1  1 Ar  rA  


a


r  sin  
r 
1   rA  Ar  


a


r  r
 
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Chap 1: Antenna Basic
(1.4)
13
1   A sin   A 
Hr 

0


 r sin   
 
1
1 1  1 Ar  rA 
H 

0


 r  sin  
r 
1 1   rA  Ar  1  






H 



rA
sin


A
cos

z
z

 r  r
  r  r


Solving and simplifying further, we obtain,
I m L sin e  jr  j 1 
H 
 2

4
 r r 
(1.5)
Thus, the magnetic fields from the dipole have only one component.
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Chap 1: Antenna Basic
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Get E from H
In phasor form,
Er 
E
1
j
I m L cos( ) e  jr
2
E  0
E 
 H
(1.6)
 

1
r2
j r 3
(1.7a)
(1.7b)
I m L sin( ) e  jr
4
   
j
r

r
2
1
j r
3
(1.7c)
Thus, the electric field of a short dipole has two components.
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Chap 1: Antenna Basic
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Field regions
No abrupt changes in
the field
configurations are
noted as the
boundaries are
crossed – but there
are distinct
differences between
the fields
Radiating near
field (Fresnel)
region
antenna
Reactive near
field region
Far field (Fraunhofer) region
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Chap 1: Antenna Basic
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Reactive near field region (r<<1)
“That portion of the near field region immediately surrounding the
antenna wherein the reactive field predominates”
Outer boundary R  0.62 D 3 
I m L sin e  jr
H 
4r 2
Er 
I m L cos( ) e  jr
E 
I m L sin( ) e  jr
j 2 r 3
j 4 r
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3
For short dipole boundary is  / 2
(1.8a) Fields of a short dipole can be
approximated by these expressions.
(1.8b)
(1.8c)
The E field components are in time
phase, but they are in time phase
quadrature with the H field. Thus,
there is no time average power flow.
Chap 1: Antenna Basic
17
Radiating near field region (r>1)
“That region of the field of an antenna between the reactive near
field region and the far field region wherein radiation fields
predominate and wherein the angular field distribution is dependant
upon the distance from the antenna”
If the antenna has a maximum dimension that is not large
compared to the
this region may not exist.
R  2 Dwavelength,
/
2
Inner boundary
R  0.62 D 3 
(1.9)
Outer boundary
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E field components lose their inphase condition and approach
time-phase quadrature.
jI m L sin e
H 
4r
Er 
I m L cos( ) e  jr
2r 2
E 
j I m L sin( ) e  jr
4r
(1.10a)
(1.10b)
H and E components approach
time phase, which is an indication of
the formation of time-average power
flow in the outward (radial) direction.
 jr
(1.10c)
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Chap 1: Antenna Basic
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Far field region (r>>1)
“That region of the field of an antenna where the angular field
distribution is essentially independent of the distance from the
antenna”
Inner boundary
R  2D 2 / 
(1.11)
As r increases, Er will be smaller than E because Er is inversely
proportional to r2. Thus fields can be further simplified to,
Er  E  H r  H   0
 jr
I m Le
E  j
4r
sin 
I m Le
H  j
4r
(1.12a)
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 jr
sin 
(1.12b)
Chap 1: Antenna Basic
20
Antenna Terminology
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Chap 1: Antenna Basic
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Reciprocity
Reciprocal devices – devices that exhibit the
same radiation pattern for transmission as for
reception.
Reciprocity may not hold for solid state
antennas composed of non-linear
semiconductors or ferrite materials and active
antennas.
Allows measurement to be made in either
transmission mode or receiver mode –
depending on convenience.
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Chap 1: Antenna Basic
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Radiation intensity
• Radiation intensity:- In a given direction, the
power radiated from an antenna per unit solid
angle
• It is a far field parameter, and can be obtained
by multiplying radiation density (magnitude of
Poynting vector) with the square of distance
• Its denoted by U
• Unit is Watts per steradian (W/sr)
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Chap 1: Antenna Basic
23
U  ,    r 2 Pave
Pave 

1

Re E  H
2

(1.13)
Total radiated power can be expressed in radiation intensity as,
Prad   Pave dS   Pave r 2 sin  d d
S
S
  U  ,  sin  d d
S

2


 0  0
U ave 
Prad
4
U  ,   d
(1.14)
Average radiation intensity is total radiated
power over total solid angle
(1.15)
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Chap 1: Antenna Basic
24
Lets find the radiation intensity of the short dipole


1

Pave  Re E  H
Poynting vector gives the
2
average power density
1
radiated
 Re E H  aˆ r
2
I m2 L2  2
2
(1.16)
ˆ

sin

a
r
32 2 r 2


2 2 2
I
2
2
mL 
U  ,    r Pave  
sin

2
32
Radiation intensity can be simply referred to as
(1.17)
U  U max sin 2 
(1.18)
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Chap 1: Antenna Basic
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Beam solid angle (beam area)
What is solid angle?
Its like the angle in 3D, one
sphere has 4 solid angle
Beam solid angle - The solid angle through which all the power would be
radiated if the power per unit solid angle (radiation intensity) equals the
maximum value over the beam area A.
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Chap 1: Antenna Basic
26
Given by the integral of the normalized radiation intensity or
power pattern over a sphere
2

U  ,  sin  d d   U d

 
A  
0 0
(1.19)
Lets calculate the beam solid angle of the short dipole
 A   U N d
Normalized
radiation
intensity
2
 

2
sin
  sin  d d 
 0  0
(1.20)
8
 
3
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Antenna pattern
• Also called as radiation pattern
• Defined as “the spatial distribution of a quantity that
characterises the electromagnetic field generated by an
antenna”.
• The quantities that are most often used are power flux
density, radiation intensity, directivity, phase, polarisation
and field strength.
• Radiation pattern can be pictured in a single 3D or three
2D diagrams (three orthogonal planes – XY, YZ and ZX or
)
  00 ,   900 and   900
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Chap 1: Antenna Basic
28
Example of radiation pattern of a horn antenna
3D pattern
This is the spatial
distribution of
power radiated –
also called power
pattern
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Chap 1: Antenna Basic
29
Lets plot the short dipole radiation pattern
Radiation intensity
of a short dipole is,
3D pattern
U  U max sin 2 
Y
Z
X
  00 ,   900
  90
YZ and XZ plane pattern cut
XY plane pattern cut
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Chap 1: Antenna Basic
0
30
Analysis of the short
dipole radiation pattern
â
Direction
of short
dipole
2D pattern cut
Length represent the relative
field strength, maximum at
=900, minimum at =00,
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Direction of
Electric field
radiation,
always
tangent to
the pattern
â
Chap 1: Antenna Basic
31
E-plane and H-plane
• Is defined only for single linear polarised antenna
• The radiation pattern that contains the electric field is
called the E-plane cut or pattern
• Automatically the other plane, which contains the
Magnetic field is called the H-plane
For the short dipole, the left
pattern cut is the E-plane
and the right is H-plane
E- and H-plane does not mean anything for a dual
polarised or circularly polarised antenna!
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Chap 1: Antenna Basic
32
Isotropic and Omni-directional radiator
• Isotropic:- A hypothetical, lossless antenna having equal
radiation intensity in all direction.
Pr  4r 2
Pt
(1.21)
For an isotropic radiator, the power
density is given by dividing the total
radiated power equally over the
surface of the sphere
• Omni-directional:- An antenna having an essentially nondirectional pattern in a given plane of the antenna and a
directional pattern in any orthogonal plane.
A typical example is the wire dipole (short dipole)
– non directional in XY plane
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Chap 1: Antenna Basic
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Directivity
• The ratio of the radiation intensity in a given
direction from the antenna to the radiation
intensity averaged over all directions
• Tells us how well the antenna is radiating
towards a particular direction
• For an isotropic antenna, the directivity is
equal to unity
• Does not take into account the efficiency of
the antenna
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Chap 1: Antenna Basic
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Directivity in terms of radiation intensity and beam solid angle
D( ,  ) 
U ( , )
U ave
D( ,  )  1
D( ,  ) 
4U  , 
A
4
U ( , )
U  , d
(1.22)
Lets find the directivity of short dipole
4 sin 2 
D ,   
 1.5 sin 2 
8 
3
Maximum directivity of a short dipole is 1.5
and is along the XY plane with z=0
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Chap 1: Antenna Basic
35
Effective Area
In a situation where the incoming EM wave is normal to the entire
surface of a receiving antenna, the power received is:
Pr   Pave  dS  Pave S
(1.23)
However, in reality that is not the case. Thus, the concept of effective
area (or effective aperture) is usually employed in the analysis of
receiving antennas.
Effective area is defined as: Ae 
Relationship between
directivity and effective area
Trim2410 EMG6343:Part I
Pr
Pave
D2
Ae 
4
Chap 1: Antenna Basic
Also the ability of an
antenna to extract
energy from a passing
EM wave
(1.24)
36
Derivation of Effective Area for a Hertzian
Dipole
Z in  Rrad  jX in
Voc  EL
Equivalent circuit of a
receiving Hertzian
Dipole
Open circuit
voltage
induced on the
antenna
Trim2410 EMG6343:Part I
Antenna equivalent
impedance
Z L  RL  jX L
External circuitry –
equivalent impedance
Chap 1: Antenna Basic
37
For maximum power transfer – impedance of antenna and
external circuitry should be congugate matched!
Z L  Z in* RL  Rin X L   X in
Thus, power delivered to
external circuitry is,
2
For Hertzian dipole,
L
Rrad  80  
 
Voc
1  Voc 
Pr  
 Rrad 
2  2 Rrad 
8Rrad
2
2
2
(1.25)
E 2 2
Solving both, Pr 
640 2
E2
E2
Also, Pave 

2 o 240
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Chap 1: Antenna Basic
Pr
2
2
Ae 
 1.5
D
Pave
4
4
38
Effective length
• Represents the effectiveness of an antenna as radiator or
as collector of electromagnetic wave energy.
• Same concept as effective aperture, but for wire
antenna.
(1.26)
Voc  EL
Open circuit
voltage induced
at the antenna
terminal
Trim2410 EMG6343:Part I
Effective
length of the
antenna
Electric field at
the antenna
Chap 1: Antenna Basic
39
Gain and efficiency
• Gain is the ratio of the radiation intensity, in a
given direction, to the radiation intensity that
would be obtained if the power accepted by
the antenna were radiated isotropically
• Gain does not include losses arising from
impedance and polarization mismatches
• Gain and directivity is related by the efficiency
of the radiating element.
• If there is no loss in the antenna, gain equals
directivity
G  kD
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Chap 1: Antenna Basic
40
Major lobe & Minor lobe
• Major lobe is also called main lobe
• Defined as “the radiation lobe containing the direction of
maximum radiation”
• In certain antennas, such as multi-lobed or split beam
antennas, there may exist more than one major lobe
• Minor lobe - A radiation lobe in any direction other than that
of the major lobe
• When its adjacent to the main lobe its called side lobe
• Side lobe level – maximum relative directivity of the highest
side lobe with respect to the maximum directivity of the
antenna
• Back lobe – refers to a minor lobe that occupies the
hemispheres in a direction opposite to that of the major lobe.
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Chap 1: Antenna Basic
41
Radiation pattern lobes
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Chap 1: Antenna Basic
42
Beamwidth - half-power beamwidth
and first null beamwidth
• The width of the main beam or major lobe in terms of angles
or radians.
• Half-power beamwidth (also known as 3dB beamwidth) and
first null beamwidth is of interest
• 3dB beamwidth - In a radiation pattern cut containing the
direction of the maximum of a lobe, the angle between the
two directions in which the radiation intensity is one-half the
maximum value
• Normally related to the resolution
• Narrow beam requires large antenna dimensions
• First-null beamwidth – In a radiation pattern cut, the angle
between the two nulls adjacent to the main beam
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Chap 1: Antenna Basic
43
Half power beamwidth and
first null beamwidth can be
calculated or graphically
deduced, if the radiation
pattern is known.
If the half power
beamwidth are known, it
can be used to
approximate the
directivity,
4
41000
Dapprox 
 o o
 HP  HP  HP  Hp
Appx beam area A
(1.27)
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Chap 1: Antenna Basic
44
Radiation resistance
Radiation resistance Rr of an antenna is the hypothetical resistance that
would dissipate the same amount of power as the radiated power Rr.
Lets find the radiation resistance for a short dipole
Prad   Pave  dS
2 
I m2 L2  2 2
 
sin 
2 2
32 r
0 0
r 2 sin  d d
I m2 L2  2  3

2  sin  d
2
32
0
First find the total
power radiated
I m2 L2  2


2
12
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Chap 1: Antenna Basic
45
2
Replace  and ,
2L
Prad  40   I m2
 
The power is equivalent to the power dissipated in a fictitious
resistance Rr by a current Im
2
1 2
2L
Prad  I m Rr  40   I m2
2
 
Thus, the radiation resistance is given by,
L
Rr  80  
 
2
2
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Chap 1: Antenna Basic
46
Bandwidth
• The range of frequencies within which the
performance of the antenna, with respect to
some characteristic, conforms to a specified
standard
• Normally expressed as a fraction of centre
frequency
• Normally used standards - Impedance
bandwidth; Gain bandwidth; Radiation
pattern bandwidth; side lobe level;
beamwidth; polarisation; beam direction
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Chap 1: Antenna Basic
47
Polarisation (of an antenna)
• In a given direction from the antenna, the
polarisation of the wave transmitted by the
antenna
• Polarisation of a wave describes the shape and
locus of the tip of the E vector at a given point
in space as a function of time.
• General locus is ellipse – elliptically polarised
• Under certain conditions – ellipse becomes a
circle – circular polarisation, or straight line –
linear polarisation.
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Chap 1: Antenna Basic
48
Polarisation of a wave
• When E field is traced in clockwise direction –
right-hand polarisation, otherwise left-hand
polarisation
• Note that polarisation rotation is opposite the
direction of rotation of E field as a function of
distance at a fixed point in time
• Common usage is with linear polarisation,
vertical and horizontal
• Both antenna and wave polarisation must
match for maximum power transfer.
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Chap 1: Antenna Basic
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Chap 1: Antenna Basic
50
Half Wave Dipole
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Chap 1: Antenna Basic
51
Half wave Dipole
Consist of a thin wire fed at midpoint via a
transmission line
/2
From Hertzian dipole we know that the
magnetic vector potential (phasor) at
arbitrary point P, due to a single dipole of
length L and carrying current Im is
Consider as
combination of
Hertzian Dipoles
Trim2410 EMG6343:Part I
LI m  jr
Az 
e
4r
Chap 1: Antenna Basic
(1.28)
52
In the Hertzian dipole, it was assumed that the
current distribution along the length is constant.
The actual current distribution is not precisely
known.
It is determined by solving Maxwell’s equations
subject to the boundary conditions on the
antenna, but the procedure is mathematically
complex.
However, sinusoidal
current distribution
approximates the
distribution obtained by
solving the boundaryvalue problem and is
commonly used in
antenna theory.
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A triangular
current
distribution is
also possible,
but would give
less accurate
results.
Chap 1: Antenna Basic
53
Consider the half wave dipole as a
chain of Hertzian dipoles
z
dz
The phasor current
along the dipole is:
I  I m cos z
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Chap 1: Antenna Basic
54
Magnetic vector potential at P due to a differential length dl is

dz

P
r
dl  dz
I m cos z
dAz 
dz e  jr 
4r 
r
l
/2
r  r
Error in magnitude
with this
approximation is
negligible
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Chap 1: Antenna Basic
r  r  z cos
Error in phase is
significant, thus
replace with this
approximation
55
Thus,
I m  jr  / 4 jz cos
Az 
e  e
cos z dz


/
4
4r
From the integral table, utilize
e az a cos bz  b sin bz 
 e cos bz 
a 2  b2
az
We obtain,


I m e cos cos  
2

Az 
2r sin 2 
 j r
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Chap 1: Antenna Basic
(1.29)
56
Follow the steps as in the derivation of Hertzian dipole:
1. Express Az into Ar, A and A,
2. Get Magnetic and Electric field by using the curl of A
Ar  Az cos  ,
A   Az sin  ,


jI m e cos cos  
1
2

 aˆ
H    A 


2r sin 
 jr
(1.30a)


jI m e cos cos  
1
2
 aˆ
E
 H 

j
2r sin 
 jr
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Chap 1: Antenna Basic
A  0
For both
electric and
magnetic field,
discard the
1/r3 and 1/r2
terms
(1.30b)
57
Characteristic of half wave dipole
Probably the
most popular
type of
antenna
Input is balanced – thus require a
balanced transmission line
Input
impedance =
73 + j42.5
A good
candidate for
array
application
If L=0.485
Input impedance = 73
Half of this antenna is called
monopole antenna
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58
Friis Transmission Formula
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Chap 1: Antenna Basic
59
Relationship between transmission and
reception of electromagnetic waves
Dt 2
At 
4
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Dr 2
Ar 
4
Chap 1: Antenna Basic
60
For an directional antenna, power density at receiver is,
Pt Dt
Pt 4At
Pt At
PDr 

 2 2
2
2
2
4R
4R 
R
Power received by the receiver,
Pt At Ar
Pr  PDr Ar  2 2
R 
Thus, Friis transmission formula is,
Pr At Ar
 2 2
Pt R 
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Chap 1: Antenna Basic
(1.31)
61