Antennas - Educypediaeducypedia.karadimov.info/library/antenna_basic.pdfAntennas are in general...

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


Wave

Antennas


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


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


Wave

Zeq = Req + jXeq

P t R Ieq( ) = 1

2

2

Vg

Zg

Antennas


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


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


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


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


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


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


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


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


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


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


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


The generals solutions for the wave 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


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


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


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


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


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


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


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


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


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


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


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


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 2

0 0

22 3

0

2

sin ( )

Isin

2 4

I4

3 4

totP d d r P t

zr d

r

z

� �

�

�

��

�

��

�

�

� ��

� ��

� �

�

�

�

�

Antennas


1 2tot totP P�

1totP

2totP

Antennas


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


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


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


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


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


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


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


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


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


Linear Antennas

Consider a dipole with wires of length comparable to thewavelength.

z

'

'r

r

'i� i

�

'z

z

1L�

2L

Antennas


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


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


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


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


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


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


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


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


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


From the linear antenna applet

Radiation Pattern for E and H Power Radiation Pattern

Antennas


For short dipoles of length 0.0005 λ to 0.05 λRadiation Pattern for E and H Power Radiation Pattern

Antennas



Antennas


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


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


Examples of long wire antennasRadiation Pattern for E and H Power Radiation Pattern

Antennas



Antennas - Educypediaeducypedia.karadimov.info/library/antenna_basic.pdfAntennas are in general...

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