antenna lecture notes
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Antennas
© Amanogawa, 2001 – Digital Maestro Series 1
Antennas
Antennas are transducers that transfer electromagnetic energybetween a transmission line and free space.
TransmittingAntenna
I
I
Transmitter
Transmission LineElectromagnetic
Wave
ReceivingAntenna
I
IReceiver
Transmission LineElectromagnetic
Wave
Antennas
© Amanogawa, 2001 – Digital Maestro Series 2
Here are a few examples of common antennas:
Linear dipole fedby a two-wire line
Ground plane
Linear monopolefed by a single wireover a ground plane
Coaxial groundplane antenna
Parabolic (dish) antenna
Linear elements connectedto outer conductor of thecoaxial cable simulate theground plane
Uda-Yagi dipole array
Passive elements
Loop dipole
Log−periodic array
Loop antennaMultiple loop antenna wound
around a ferrite core
Antennas
© Amanogawa, 2001 – Digital Maestro Series 3
From a circuit point of view, a transmitting antenna behaves like anequivalent impedance that dissipates the power transmitted
The transmitter is equivalent to a generator .
I
I
Transmitter
Transmission Line
TransmittingAntenna
I
I
Transmitter
Transmission LineElectromagnetic
Wave
Zeq = Req + jXeq
P t R Ieq( ) = 1
2
2
Vg
Zg
Antennas
© Amanogawa, 2001 – Digital Maestro Series 4
A receiving antenna behaves like a generator with an internalimpedance corresponding to the antenna equivalent impedance.
The receiver represents the load impedance that dissipates the timeaverage power generated by the receiving antenna.
I
I
Transmission Line
ReceivingAntenna
I
I
Receiver
Transmission Line
P t R Iin( ) = 1
2
2
Veq
Zeq
ElectromagneticWave
ZinZR
Antennas
© Amanogawa, 2001 – Digital Maestro Series 5
Antennas are in general reciprocal devices, which can be used bothas transmitting and as receiving elements. This is how theantennas on cellular phones and walkie −talkies operate.
The basic principle of operation of an antenna is easily understoodstarting from a two −wire transmission line , terminated by an opencircuit .
Vg
Zg
| I |
| I |
Note: This is the returncurrent on the second wire,not the reflected currentalready included in thestanding wave pattern .
ZR → ∝Open circuit
Antennas
© Amanogawa, 2001 – Digital Maestro Series 6
Imagine to bend the end of the transmission line, forming a dipoleantenna . Because of the change in geometry, there is now anabrupt change in the characteristic impedance at the transitionpoint, where the current is still continuous. The dipole leakselectromagnetic energy into the surrounding space, therefore itreflects less power than the original open circuit ⇒ the standingwave pattern on the transmission line is modified
Z0Vg
Zg
| I |
| I |
| I0 |
| I0 |
Antennas
© Amanogawa, 2001 – Digital Maestro Series 7
In the space surrounding the dipole we have an electric field. Atzero frequency (d.c. bias) , fixed electrostatic field lines connect themetal elements of the antenna, with circular symmetry .
EE
Antennas
© Amanogawa, 2001 – Digital Maestro Series 8
At higher frequency , the current oscillates in the wires and the fieldemanating from the dipole changes periodically. The field linespropagate away from the dipole and form closed loops.
Antennas
© Amanogawa, 2001 – Digital Maestro Series 9
The electromagnetic field emitted by an antenna obeys Maxwell’sequations
Under the assumption of uniform isotropic medium we have thewave equation:
Note that in the regions with electrical charges ρ
j
j
E H
H J E
� �
� �
� � � �
� � � �
� �
� � �
2
2
E H J E
H J E
J H
j j
j
� � � � � ��
� �
� � �
� �� � � � � � �� �
� �� � � � � � � �
� � � �
� � � �
� � �
� �
� �2 2E E E E� �� �� � � �� � � � � � ��� � � �
Antennas
© Amanogawa, 2001 – Digital Maestro Series 10
In general, these wave equations are difficult to solve, because ofthe presence of the terms with current and charge. It is easier touse the magnetic vector potential and the electric scalar potential .
The definition of the magnetic vector potential is
Note that since the divergence of the curl of a vector is equal tozero we always satisfy the zero divergence condition
We have also
B A� � ���
� �B A 0� � � � � � � ���
� �j jj E AE H A 0�� � �� � � � � � � � � �� � �� �� ��
Antennas
© Amanogawa, 2001 – Digital Maestro Series 11
We define the scalar potential φ first noticing that
and then choosing (with sign convention as in electrostatics)
Note that the magnetic vector potential is not uniquely defined,
since for any arbitrary scalar field ψ
In order to uniquely define the magnetic vector potential, thestandard approach is to use the Lorenz gauge
� � 0�� � �� �
� � � �E A E Ajj �� � �� � � � � � �� � � �� ��� ��
� �B A A �� � � � � � � �� ��
jA 0� �� ��� � ��
Antennas
© Amanogawa, 2001 – Digital Maestro Series 12
From Maxwell’s equations
From vector calculus
� � � �
j
j
j j
1H B J E
B J E
A J A
� ��
� � � �
� � � � � �
� � � � � � �
� � � �� � � � � � � � ��
� � � �
� � �
� ��
� � � � j2 2AA A J A � � � �� � � �� � � � � � � �� � � � ���� � ��
� �j jAA � � � �� � � ���� � � � �� � � ���
Lorenz Gauge
� � � � 2� � � � � � � � � �� � �
Antennas
© Amanogawa, 2001 – Digital Maestro Series 13
Finally, the wave equation for the magnetic vector potential is
For the electric field we have
The wave equation for the electric scalar potential is
2 2 2 2A A A A J� �� � �� � � � � � �� � � � �
� �
� �2 2
D E
A
Aj
j j j� �� �
�� � �
�
�� � � �
�
�� � � �� � � � � � � �
� � � � �� � � ��
�
�
� �
2 2 2 2 �� � � � � � � �
�� � � � � � �
Antennas
© Amanogawa, 2001 – Digital Maestro Series 14
The wave equations are inhomogenoeous Helmholtz equations ,which apply to regions where currents and charges are not zero.
We use the following system of coordinates for an antenna body
z
x
y
r�
r '�
r r '�� �
Radiating antenna body
Observation pointr
r
J( ')
( ')�
� �
�
dV '
Antennas
© Amanogawa, 2001 – Digital Maestro Series 15
The general solutions for the wa ve equations are
The integrals are extended to all points over the antenna bodywhere the sources ( current density , charge ) are not zero. The effectof each volume element of the antenna is to radiate a radial wave
� �� � j r r
V
J r er dV
r r
''
A '4 '
��
�
� ��
����� �� �
� �� �
� �� � j r r
V
r er dV
r r
''1
'4 '
��
���
� ��
����� ��
�� �
j r re
r r
'
'
�� �
�
� �
� �
Antennas
© Amanogawa, 2001 – Digital Maestro Series 16
Infinitesimal Antenna
z
x
y
r r r '� �� � �
r ' 0��
Infinitesimal antenna body
Observation point
J(0)
(0)�
�
dV '
S
z
I constant phasor�
�z
Dielectric medium (ε , µ)
Antennas
© Amanogawa, 2001 – Digital Maestro Series 17
The current flowing in the infinitesimal antenna is assumed to beconstant and oriented along the z−axis
The solution of the wave equation for the magnetic vector potentialsimply becomes the evaluation of the integrand at the origin
� � � �
� � z
S r S
V r z
V
i
S zI J ' J 0 '
'J ' I
� � � � �
�
��� �
� �
�
� �
1H A
IA
4 1E H
j r
z
z ei
r
j
� ��
�
� �
��
� � � � � � � � � �
���
� �
� �
Antennas
© Amanogawa, 2001 – Digital Maestro Series 18
There is still a major mathematical step left. The curl operationsmust be expressed in terms of polar coordinates
z
x
y
r�
�
i��
i��
i�
ri�
Azimuthal angle
Elevation angle
Antennas
© Amanogawa, 2001 – Digital Maestro Series 19
In polar coordinates
� � � �
� � � �
� � � �
r
r
r
r
r
i r i r i
rr
A rA r A
A A ir
A r A ir r
r A A ir r
2
sin
1A
sin
sin
1sin
sin
1 1
sin
1
�
�
�
�
�
�
�
�
� � �� � �
� � �
� �� �� �� �� �� �� �� �
� �� �� �� �� �� �� �� �� �� �
� � �
�
�
�
�
Antennas
© Amanogawa, 2001 – Digital Maestro Series 20
We had
For the fields we have
j r
z z r
j r
z ei i i i
r
j z ei
r r
ith
j
wI
A cos sin4
I 1A 1 sin
4
�
�
�
�
�
� �
� �
�
�
� � �
� �� �� � �� �� �
�� � � � �
�� �
j rj z e
ir j r
I1 1H A 1 sin
4
�
��
� � �
� � �� � � � �� �� �
�� ��
Antennas
© Amanogawa, 2001 – Digital Maestro Series 21
The general field expressions can be simplified for observationpoint at large distance from the infinitesimal antenna
� �
� �
j r
r
j z e
j r
ij r j r
ij r j r
2
2
I1E H
4
1 12 cos
1 1sin 1
�
��
� � � �
� �
� �
�� � � �
� � �� �� � �� � ��
�� �� � � �� �
�� � �
�� �
�
�
� �r r
j r j r2
1 1 21 1
��
� ��� �� � � ��
Antennas
© Amanogawa, 2001 – Digital Maestro Series 22
At large distance we have the expressions for the Far Field
• At sufficient distance from the antenna, the radiated fields areperpendicular to each other and to the direction of propagation .
• The magnetic field and electric field are in phase and
These are also properties of uniform plane waves.
E H H�
��
� �� � �
j rj z e
ir
IH sin
4
�
��
�
��
���
j rj z e
ir
IE sin
4
�
��
� �
��
���
r2� ��
Antennas
© Amanogawa, 2001 – Digital Maestro Series 23
However, there are significant differences with respect to a uniformplane wave:
• The surfaces of constant phase are spherical instead of planar,and the wave travels in the radial direction
• The intensities of the fields are inversely proportional to thedistance, therefore the field intensities decay while they areconstant for a uniform plane wave
• The field intensities are not constant on a given surface ofconstant phase. The intensity depends on the sine of theelevation angle
The radiated power density is
� �2*
22
1 1( ) Re E H
2 2
Isin
2 4
r
r
P t i H
zi
r
��
�
��
�
� � �
� �� � �� �
�� � �
��
Antennas
© Amanogawa, 2001 – Digital Maestro Series 24
The spherical wave resembles a plane wave locally in a smallneighborhood of the point ( r, θ, ϕ ).
z( )P t�
E
H�
r�
J�
�
Antennas
© Amanogawa, 2001 – Digital Maestro Series 25
Radiation Patterns
Electric Field and Magnetic Field
z
x
or HE �
y
x
Plane containing the antenna
proportional to sinθ
Plane perpendicular to the antenna
omnidirectional or isotropic
Fixed r
Antennas
© Amanogawa, 2001 – Digital Maestro Series 26
Time−average Power Flow (Poynting Vector)
z
x
( )P t�
Plane containing the antenna
proportional to sin2θ
Plane perpendicular to the antenna
omnidirectional or isotropic
y
x
Fixed r
Antennas
© Amanogawa, 2001 – Digital Maestro Series 27
Total Radiated Power
The time−average power flow is not uniform on the spherical wavefront. In order to obtain the total power radiated by the infinitesimalantenna, it is necessary to integrate over the sphere
Note: the total radiated power is independent of distance. Althoughthe power decreases with distance, the integral of the power overconcentric spherical wave fronts remains constant.
2 20 0
22 3
0
2
sin ( )
I2 sin
2 4
I43 4
totP d d r P t
zr d
r
z
p p
p
j q q
bhp q qp
bp hp
=
Ê ˆD= Á ˜Ë ¯
Ê ˆD= Á ˜Ë ¯
Ú Ú
Ú
Antennas
© Amanogawa, 2001 – Digital Maestro Series 28
1 2tot totP P�
1totP
2totP
Antennas
© Amanogawa, 2001 – Digital Maestro Series 29
The total radiated power is also the power delivered by thetransmission line to the real part of the equivalent impedance seenat the input of the antenna
The equivalent resistance of the antenna is usually called radiationresistance . In free space
2 22 2I1 4 2 1 2
I I2 3 4 2 3
eq
tot eq
R
z zP R
�� � ��
�
� �� � � �� � � � �� � � �� � � �� �
�� �
�������
� � � �2
2820 01o
oeqo
zR �
�� �
�
�� �
� ��
� � ��
� �
Antennas
© Amanogawa, 2001 – Digital Maestro Series 30
The total radiated power is also used to define the average powerdensity emitted by the antenna. The average power densitycorresponds to the radiation of a hypothetical omnidirectional(isotropic) antenna , which is used as a reference to understand thedirective properties of any antenna.
z
x
( , )P t �
aveP
Power radiation pattern of anomnidirectional average antenna
Power radiation patternof the actual antenna
Antennas
© Amanogawa, 2001 – Digital Maestro Series 31
The time −average power density is given by
The directive gain of the infinitesimal antenna is defined as
� �
Surface of wave front
22
2
Total Radiated Powe
2
r
I1I
12 3 44 4
ave
tot
P
zPz
rr r
�� ��
� �� �
� �
� �� � � � �� �
��
12 22
2
( , , ) I Isi( , )
3sin
2
n2 4 3 4ave
P t r z z
P r rD
� �� �
��
�
�� � � � � �� � � �� � � �� �� � � �� �
�
� � �
Antennas
© Amanogawa, 2001 – Digital Maestro Series 32
The maximum value of the directive gain is called directivity of theantenna. For the infinitesimal antenna , the maximum of thedirective gain occurs when the elevation angle is 90 °
The directivity gives a measure of how the actual antenna performsin the direction of maximum radiation, with respect to the idealisotropic antenna which emits the average power in all directions.
� � 23max ( , ) sin 1Directivi .t
2 2y 5D
� �
� �� � �� �
x
z
90�
aveP maxP
Antennas
© Amanogawa, 2001 – Digital Maestro Series 33
The infinitesimal antenna is a suitable model to study the behaviorof the elementary radiating element called Hertzian dipole .
Consider two small charge reservoirs, separated by a distance ∆z,which exchange mobile charge in the form of an oscillatory curent
+
−
( )I t
t
z
+
−
+
−+
−
+
− +
−
Antennas
© Amanogawa, 2001 – Digital Maestro Series 34
The Hertzian dipole can be used as an elementary model for manynatural charge oscillation phenomena. The radiated fields can bedescribed by using the results of the infinitesimal antenna .
Assuming a sinusoidally varying charge flow between thereservoirs, the oscillating current is
�
current flowing
out of resercharge on
reference resevoir rvoir
cos( ) ( ) ( ) Iphaso
or
o od d
I t q t q t j qdt dt
� �� � �������
Radiationpattern
oq
oI
Antennas
© Amanogawa, 2001 – Digital Maestro Series 35
A short wire antenna has a triangular current distribution, since thecurrent itself has to reach a null at the end the wires. The currentcan be made approximately uniform by adding capacitor plates .
The small capacitor plate antenna is equivalent to a Hertzian dipoleand the radiated fields can also be described by using the results ofthe infinitesimal antenna . The short wire antenna can be describedby the same results, if one uses an average current value giving thesame integral of the current
max 2oI I�
Imax
Ioz
Io
Antennas
© Amanogawa, 2001 – Digital Maestro Series 36
Example − A Hertzian dipole is 1.0 meters long and it operates atthe frequency of 1.0 MHz, with feeding current I o = 1.0 Ampéres.Find the total radiated power.
For a short dipole with triangular current distribution and maximumcurrent Imax = 1.0 Ampére
�
�
� �
8 6
22 2
Hertzian dipol
3 10 10 300
1 m
4 2 1 2120 ( ) ( 1 1 )
3 4 12 300
4.39
e
mW
o o
otot
I z
c f
z
I zP
�
� � � ��
� �
� � � � � �
� �� � �� �
�
�
max 2 4.39 / 4 1.09 mWo totI I P� � � �
Antennas
© Amanogawa, 2001 – Digital Maestro Series 37
Time−dependent fields - Consider the far −field approximation
� � � �
� �
� � � �
( )
2
I sinRe H Re
4
I sinRe cos( ) sin
I sins
I sinsin(
in
( )4
Re
( )
4
4
E
)
j t rj t
j t
j ze i e
r
zi j t r j t
E t
H t
zi t r
r
t
r
e
r
r
zi
r
� ���
�
�
�
�
�
�
� � �
� � �
� �
�
�
�
�
�
�
�� � � !
� "�
� � � � !� "
�
�
� � �
�
�
�
�
��
�
�
�
��
��
�
Antennas
© Amanogawa, 2001 – Digital Maestro Series 38
Linear Antennas
Consider a dipole with wires of length comparable to thewavelength.
z
'
'r
r
'i� i
�
'z
z
1L�
2L
Antennas
© Amanogawa, 2001 – Digital Maestro Series 39
Because of its length, the current flowing in the antenna wire is afunction of the coordinate z. To evaluate the far−field at anobservation point, we divide the antenna into segments which canbe considered as elementary infinitesimal antennas .
The electric field radiated by each element , in the far−fieldapproximation, is
In far−field conditions we can use these additional approximations
'I
' sin '4 '
j rj z e
E ir
�
��
� �
� �
��
r r z
'
' 'cos
�� �
Antennas
© Amanogawa, 2001 – Digital Maestro Series 40
The lines r and r’ are nearly parallel under these assumptions.
z
' �
'r
r
'z
z2L
'cosz
This length is neglected if
' 'cosr r z � �
Antennas
© Amanogawa, 2001 – Digital Maestro Series 41
The electric field contributions due to each infinitesimal segmentbecomes
The total fields are obtained by integration of all the contributions
you cannotneglect here
'cos
4 'cos
I'
4
j r j zj z e e
E ir z
� �
��
� � �
� �
�
��
�
you can neglect here
sin
��
2
1
2
1
cos
cos
E sin I( )4
H sin I( )4
j rL j z
L
j rL j z
L
j ei z e dz
r
j ei z e dz
r
��
��
�
� �
� �
�
�
�
�
�
�
� �
� �
�
�
��
��
Antennas
© Amanogawa, 2001 – Digital Maestro Series 42
Short Dipole
Consider a short symmetric dipole comprising two wires, each oflength L << λ . Assume a triangular distribution of the phasorcurrent on the wires
The integral in the field expressions becomes
� �
� �max
max
1 0I( )
1 0
I z L zz
I z L z
� #�� � �
cosmax
since short
1
dipolefor a
cos
2max 1
1
2I( ) I( )
2
z
L Lj
L L
j
z Lz e dz z dz
z L L
I
e�
�
�� �
��
�
� � �
�
� �
�
� ������
�
Antennas
© Amanogawa, 2001 – Digital Maestro Series 43
The final expression for far−fields of the short dipole are similar tothe expressions for the Hertzian dipole where the average of thetriangular current distribution is used
�
�
average current
max
max
max
E sin 24 2
sin4
H sin4
j r
j r
j r
j e Ii L
r
j I L ei
r
j I L ei
r
z�
�
�
�
� �
� �
� �
� �
�
�
�
�
�
� � �
�
�
��
�
��
Antennas
© Amanogawa, 2001 – Digital Maestro Series 44
Half−wavelength dipole
Consider a symmetric linear antenna with total length λ/2 andassume a current phasor distribution on the wires which isapproximately sinusoidal
The integral in the field expressions is
maxI( ) cos( )z I z��
� �4
cos maxmax 2
4
2 coscos cos
2sin
j z II z e dz
�
� �
� �
� ��� ��
Antennas
© Amanogawa, 2001 – Digital Maestro Series 45
We obtain the far−field expressions
and the time −average Poynting vector
max
max
cosE cos
2 sin 2
cosH cos
2 sin 2
j r
j r
j e Ii
r
j e Ii
r
�
�
�
� �
� �
�
�
�
�
� ��� �
� ��� �
��
��
22max
2 2 2
cos( ) cos
28 sinr
IP t i
r
� �
� �
� ��� �
��
Antennas
© Amanogawa, 2001 – Digital Maestro Series 46
The total radiated power is obtained after integration of thetime −average Poynting vector
The integral above cannot be solved analytically, but the value isfound numerically or from published tables. The equivalentresistance of the half −wave dipole antenna in air is then
� �
2.4376
22max 0
2max
1 cos1 1
2 4
10.193978
2
eq
tot
R
uP I du
u
I
��
� �
�
�
�
�� �� � �� �
� �
����������
�������
( 2) 0.193978 73.07eqR�
�
� � � �
Antennas
© Amanogawa, 2001 – Digital Maestro Series 47
The direction of maximum radiation strength is obtained again forelevation angle θ =90° ande we obtain the directivity
The directivity of the half −wavelength dipole is marginally betterthan the directivity for a Hertzian dipole (D = 1.5).
The real improvement is in the much larger radiation resistance ,which is now comparable to the characteristic impedance of typicaltransmission line.
2max2 2
22max2 2
( , ,90 ) 81.641
142.4376
8
tot
I
P t r rD
P rI
r
�
� �
��
��
�� � �
�
�
Antennas
© Amanogawa, 2001 – Digital Maestro Series 48
From the linear antenna applet
Radiation Pattern for E and H Power Radiation Pattern
Antennas
© Amanogawa, 2001 – Digital Maestro Series 49
For short dipoles of length 0.0005 λ to 0.05 λRadiation Pattern for E and H Power Radiation Pattern
Antennas
© Amanogawa, 2001 – Digital Maestro Series 50
Radiation Pattern for E and H Power Radiation Pattern
Antennas
© Amanogawa, 2001 – Digital Maestro Series 51
Radiation Pattern for E and H Power Radiation Pattern
Antennas
© Amanogawa, 2001 – Digital Maestro Series 52
For general symmetric linear antennas with two wires of length L, itis convenient to express the current distribution on the wires as
The integral in the field expressions is now
� � � �� �maxI sinz I L z�� �
� �
� � � �� �
cosmax
max2
sin
2cos cos cos
sin
Lj z
L
I L z e dz
IL L
� �
� ��
�� �� �� �
� �
�
Antennas
© Amanogawa, 2001 – Digital Maestro Series 53
The field expressions become
� � � �� �
� � � �� �
2
1
2
1
cos
max
cos
max
E sin I( )4
cos cos cos2 sin
H sin I( )4
cos cos cos2 sin
j rL j z
L
j r
j rL j z
L
j r
j ei z e dz
r
j I ei L L
r
j ei z e dz
r
j I ei L L
r
��
�
��
�
�
�
� �
� �
�� �
� �
�
�
� ��
�
�
�
�
�
�
� �
� �
� �
� �
�
�
��
�
��
�
Antennas
© Amanogawa, 2001 – Digital Maestro Series 54
Examples of long wire antennasRadiation Pattern for E and H Power Radiation Pattern
Antennas
© Amanogawa, 2001 – Digital Maestro Series 55
Radiation Pattern for E and H Power Radiation Pattern
Antennas
© Amanogawa, 2001 – Digital Maestro Series 56
Radiation Pattern for E and H Power Radiation Pattern
Antennas
© Amanogawa, 2001 – Digital Maestro Series 57
Radiation Pattern for E and H Power Radiation Pattern
Antennas
© Amanogawa, 2001 – Digital Maestro Series 58
Radiation Pattern for E and H Power Radiation Pattern
Antennas
© Amanogawa, 2001 – Digital Maestro Series 59
Radiation Pattern for E and H Power Radiation Pattern
Antennas
© Amanogawa, 2001 – Digital Maestro Series 60
Radiation Pattern for E and H Power Radiation Pattern
Antennas
© Amanogawa, 2001 – Digital Maestro Series 61
Radiation Pattern for E and H Power Radiation Pattern
Antennas
© Amanogawa, 2001 – Digital Maestro Series 62
Radiation Pattern for E and H Power Radiation Pattern
Antennas
© Amanogawa, 2001 – Digital Maestro Series 63
Radiation Pattern for E and H Power Radiation Pattern
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