antennas - introduction

2009-10 CRL 715 – Radiating Systems for RF Communications

RADIATION FROM INFINITESIMAL DIPOLERADIATION FROM INFINITESIMAL DIPOLE

Infinitesimal dipole: very short current element

Constant current distribution:Top-hat antenna


Magnetic vector potential of a current element

approximation valid for the infinitesimal dipole

The field radiated by any complex antenna in a linear medium can be represented as a superposition of the fields due to the current elements on the antenna surface.


In spherical coordinates

• cylindrical symmetry;

• angular (Θ) dependence is separable from dependence on r


Field vectors of a current element


features of the field vectors of a current element

• Longitudinal (r-components) components decrease with distance as 1/r2 or faster. Neglected in the far zone.

• Transverse component have a 1/r term – dominant at large distances

• Transverse E and H field components are orthogonal to each other (EΘ and Hφ)

• In the far zone |EΘ| = η |Hφ|


Power density of a current element

Total power of a current element calculated over a sphere:


Radiated power of a current element

Radiation resistance of a current element (ideal dipole)


DUALITY IN ELECTROMAGNETICSDUALITY IN ELECTROMAGNETICSSubstituting the quantities from one set of EM equations with the respective quantities from the dual set produces a valid equation


Dual quantities


RADIATION PATTERNRADIATION PATTERN

• Representation of the radiation properties of the antenna as a function of angular position.

Power pattern: the trace of the angular variation of the received/radiated power at a constant radius from the antenna

Amplitude field pattern: the trace of the spatialvariation of the magnitude of electric (magnetic) field at a constant radius from the antenna,


Normalized patterns

3-D and 2-D patterns


• Normalized Patterns:

Distance from origin represents magnitude

Angular position with respect to origin represents position with respect to antenna.

elevation plane


Pattern terminology

• Isotropic pattern• Directional antenna• Omnidirectional antenna


Principle patterns


Lobes


Beamwidth


RADIATION INTENSITYRADIATION INTENSITYPower per unit solid angle radiated in a given direction

Solid angle:

Elementary solid angle:


Far-zone Poynting vector (radiation power density P) and the radiation intensity are related as:

Power per unit solid angle radiated in a given direction

• Radiation intensity does not depend on distance

• The power pattern is |U(θ,ϕ)|

• The normalized power pattern is


In the far-field zone, the radial field components vanish, and the remaining transverse components of the electric and the magnetic far fields are in phase and have magnitudes related by

That is why the far-field Poynting vector has only a radial component and it is a real number corresponding to the radiation density:

Then, for the radiation intensity, we obtain in terms of the electric field


This leads to a useful relation between the power pattern and theamplitude field pattern:


DIRECTIVITYDIRECTIVITY

The ratio of the radiation intensity in a given direction and the radiation intensity averaged over all directions

Maximum directivity

Partial directivity: Directivity for a specific polarization of the field

Total and partial directivities


DIRECTIVITY AND RADIATION INTENSITYDIRECTIVITY AND RADIATION INTENSITY

Let the radiation intensity be of the form:

max max max

,

, | ,

o

o o

U B F

U B F B F

The total radiated power:

2

0 0

P , 0 , sinrad U d B F d d

The maximum directivity:

max2 2

max0 0 0 0

, | 40 4

, sin , sin , |

FD

F d d F d d F

04

A

D


A is the beam solid angle

2

max 0 0

2

max0 0

1, sin

, |

,, sin ,

, |

A

n n

F d dF

FF d d F

F

The beam solid angle is defined as the solid angle through which all the power of the antenna would flow if its radiation intensity is constant (and equal to maximum value of U) for all angles within A


APPROXIMATE EXPRESSIONS FOR DIRECTIVITYAPPROXIMATE EXPRESSIONS FOR DIRECTIVITY

Kraus’ formula — for highly directive antennas

formula of Tai and Pereira


ANTENNA GAINANTENNA GAIN

The ratio of the radiation intensity U in a given direction and the radiation intensity that would be obtained, if the power fed to the antenna were radiated isotropically.

The gain does not include losses due to impedance mismatch

Gain takes into account the efficiency of the antenna as well as its directional capabilities.

Gain vs. Directivity: antenna efficiency


ANTENNA EFFICIENCYANTENNA EFFICIENCY

Total efficiency


BEAM EFFICIENCYBEAM EFFICIENCY

Defined for each beam of the pattern: usually, the main beam is considered the ratio of the power radiated in a cone of angle Θ1 and the total radiated power (Θ1 is the first-null beam width)


FREQUENCY BANDWIDTHFREQUENCY BANDWIDTH

The range of frequencies, within which the antenna characteristics (input impedance, pattern) conform to specifications

Broadband

Narrowband


INPUT IMPEDANCEINPUT IMPEDANCE

• The impedance presented by an antenna at its terminals• The ratio of the voltage to current at a pair of terminals• The ratio of the appropriate components of the electric to

magnetic field at a point.

Generator(Zg)

Antenna

a

b

Radiatedwave

Vg

Rg

Xg

RL

Rr

XA

a

b

Ig

Antenna in Transmitting mode Thevenin equivalent


The ratio of the voltage to current at a pair of terminals, with no load attached defines the impedance of the antennas as:

The internal impedance of the generator:

g g gZ R jX

The current developed within the loop is:

1 22 2

g g gg

t A g r L g A g

gg

r L g A g

V V VI

Z Z Z R R R j X X

VI

R R R j X X

A A A A r LZ R jX R R R


The maximum power delivered to the antenna:

22

2

1

2 8

g rr g r

r L

V RP I R

R R

Under conjugate matching

The power dissipated as heat:

22

2

1

2 8

g LL g L

r L

V RP I R

R R

r L g A gR R R X X

The power dissipated as heat on the internal resistance of the generator:

2 2 22

2

1 1

2 8 8 8

g g ggg g g

r L gr L

V V VRP I R

R R RR R

rPg LP P


Power supplied by the generator during conjugate matching:

21

4

gs

r L

VP

R R

Of the total power supplied by the generator, half is dissipated as heat in the internal resistance of the generator and the other half is delivered to the antenna.

Of the power delivered to the antenna, part is radiated through the mechanism provided by the radiation resistance and the rest is dissipated as heat which influences overall efficiency of the antenna.

For a lossless antenna: All the power delivered to the antenna is radiated.

The input impedance is a function of frequency and depends on factors like geometry, method of excitation and proximity to surrounding objects.


ANTENNA EQUIVALENT AREASANTENNA EQUIVALENT AREAS

Effective Area (Effective Aperture)

The ratio of the available power at the terminals of a receiving antenna to the power flux density of a plane wave incident on the antenna (assuming wave polarization being matched to the antenna).

2

2

2 2

2

2

T TTe

i i

T T

i r L T A T

I RPA

W W

V R

W R R R X X

Effective area (m2)

Power delivered to the load (W)

Power density of the incident wave (W/m2)


Under conjugate matching conditions, the effective aperture reduces to maximum effective aperture given by:

21

8T

emi r L

VA

W R R

The ratio of maximum effective area to the physical area of an antenna is known as “Aperture Efficiency”:

emap

p

A

A

For aperture antennas:

For wire antennas:

em pA A

em pA A


Scattering Area

The equivalent area when multiplied by the incident power density is equal to the scattered or reradiated power.

Under conjugate matching:

2

28T r

si r L

V RA

W R R

Loss Area

The equivalent area when multiplied by the incident power density leads to the power dissipated as heat through RL.


2

28T L

si r L

V RA

W R R


Capture Area

The equivalent area when multiplied by the incident power density leads to the total power captured, collected or intercepted by the antenna.


2

28T r T L

si r L

V R R RA

W R R

Capture area = Effective area + Scattering area + Loss Area


MAXIMUM DIRECTIVITY & MAXIMUM DIRECTIVITY & MAXIMUM EFFECTIVE AREAMAXIMUM EFFECTIVE AREA

A1 transmits A2 receives

Radiated power density at a distance R would be:

Power received by the antenna and transferred to the load would be:

A2 transmits A1 receives


Reciprocity in antenna theory states that if antenna #1 is a transmitting antenna and antenna #2 is a receiving antenna, Ptra /Prec will not change if antenna roles are reversed

If antenna 1 is isotropic: D1=1 and its maximum effective area can be expressed as:

21

2

ee

AA

D

In general for any antenna:

2

04emA D


General expression with losses:

22

0

222

0

ˆ ˆ.4

ˆ ˆ1 .4

em t w a

cd w a

A e D

e D


POLARIZATION

The polarization of the EM field describes the time variations of the time harmonic field vectors at a given point. In other words, it describes the way the direction and magnitude of the field vectors (usually ) change in time. Polarization is a time-harmonic field characteristic.

The polarization is the figure traced by the extremity of the time-varying field vector at a given observation point.

EGGGGGGGGGGGGGG


(a) linear (b) circular (c) elliptical

Any type of polarization can be represented by two orthogonal linear polarizations, (Ex, Ey) or (EH, EV), whose fields are out of phase by an angle of δL


The polarization of any field can be represented by a set of two orthogonal linearly polarized fields.

Assume that locally a far field wave propagates along the z-axis. The far-zone field vectors have only transverse components. Then, the set of two orthogonal linearly polarized fields along the x-axis and along the y-axis, is sufficient to represent any TEMz field.

The instantaneous field of a plane wave traveling in –z direction:

0

0

ˆ ˆ; ; ;

; cos

; cos

x x y y

x x x

y y y

e z t a e z t a e z t

e z t E t kz

e z t E t kz


Case 1 - Linear polarization:

, 0,1,2,3,....y x n n

Case 2 - Circular polarization:

0 0

12 , 0,1,2...

2

12 , 0,1,2...

2

x y x y

y x

y x

e e E E

n n CW

n n CCW

If the direction of wave propagation is reversed (i.e. +z direction), the phases for CW and CCW must be reversed


Case 3 - Elliptical polarization:

0 0

12 , 0,1,2...

2

12 , 0,1,2...

2

02

02

x y x y

y x

y x

y x

y x

e e E E

n n CW

n n CCW

OR

nCW

nCCW


Polarization in terms of two linearly polarized components.

The set of two orthogonal linearly polarized fields along the x-axis and along the y-axis, is sufficient to represent any TEMz field.

The field (time-dependent or phasor vector) is decomposed into two orthogonal components:

ˆcos

ˆcos

x yx y

x x

y y L

e e e E E E

e E t z x

e E t z y

GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGG

At a fixed position (assume z =0 ):

ˆ ˆcos cos

ˆ ˆ L

x y L

jx y

e t E t x E t y

E x E y E e

GGGGGGGGGGGGGG


Linear polarization

, 0,1,2,...

ˆ ˆcos cos

ˆ ˆ

L

x y

x y

n n

e t E t x E t n y

E x E y E

GGGGGGGGGGGGGG


Circular polarization

; 0,1,2,...2

ˆ ˆ ˆ ˆcos cos 2

x y m L

x y m

E E E n n

e t E t x E t n y E E x y j

GGGGGGGGGGGGG G

clockwise (CW) orright-hand polarization

If (-z) is the direction of propagation

Counter clockwise (CCW) or Left-hand polarization


Elliptic polarization


Polarization in terms of two circularly polarized components

The total field phasor is represented as the superposition of two circularly polarized waves, one left-handed and the other right-handed.

Assuming a relative phase difference of

where eR and eL are real numbers.

The relation between the linear-component and the circular-component representations of the field polarization is easily found as


The polarization vector is the normalized phasor of the electric field vector. It is a complex-number vector of unit magnitude, i.e., *ˆ ˆ 1L L

The polarization vector takes the following specific forms:

• Linear polarization

• Circular polarization


ANTENNA POLARIZATION

• The polarization of a radiated wave (polarization of a radiating antenna) at a specific point in the far zone is the polarization of the locally plane wave.

• The polarization of a received wave (polarization of a receiving antenna) is the polarization of a plane wave, incident from a given direction, and having given power flux density, which results in maximum available power at the antenna terminals.

• The antenna polarization is defined by the polarization vector of the radiated (transmitted) wave.


• In the coordinate system of the transmitting antenna, the polarization vector of the transmitted wave is the complex conjugate of the polarization vector of the received wave in the coordinate system of the receiving antenna.

*ˆ ˆr tw w


Polarization loss factor and polarization efficiency

• Generally, the polarization of the receiving antenna is not the same as the polarization of the incident wave. This is called polarization mismatch.

• The polarization loss factor (PLF) characterizes the loss of EM power because of polarization mismatch:

2ˆ ˆi aPLF


• If the antenna is polarization matched, then PLF = 1, and there is no polarization power loss.

• If PLF = 0, then the antenna is incapable of receiving the signal.

0 1PLF

The polarization efficiency has the same physical meaning as the PLF.


FRIIS EQUATION

Relates the transmitted and the received power for a wireless link through obstacle-free space.

The transmitting and receiving antennas are in each other’sfar zone

Far-zone transmitted power:


Received power:


MAXIMUM RANGE OF WIRELESS LINK


RADAR CROSS-SECTION (RCS)

Equivalent area intercepting that amount of power which,when scattered isotropically, produces at the receiver powerdensity Ws equal to that scattered by the target itself


RADAR RANGE EQUATION

Gives the ratio of the transmitted power (fed to the transmitting antenna) to the received power after it has been scattered (re-radiated) by a target of cross-section σ


Power density of the transmitted wave at the target:

Captured power:

Re-radiated (scattered) power at the receiving antenna:

Power transferred to the receiver:


Radar range equation:

Maximum radar range:


Radiation zones

The space surrounding the antenna is divided into three regions according to the predominant field behavior. The field behavior changes very gradually as these boundaries are crossed.


• Reactive near-field region: 1r

The region immediately surrounding the antenna, where the reactive field predominates

Features of fields:

• E and H are in phase quadrature — field is reactive. No time average power flow.

• Hφ is the magnetostatic field of a current element

• Er and EΘ represent the electrostatic field of a dipole


• Radiating near-field (Fresnel) region: 1r

The radial component Er is not negligible, but the transverse components (Eθ and Hϕ ) are dominant


• Far-field (Fraunhofer) region: 1r

The angular field distribution does not depend on the distance from the source any more, i.e., the far-field pattern is already well established. The field is a transverse EM wave.

Features of fields:• no radial components; the angular field distribution is independent

of r;

• The Fields:


RADIATION REGION SEPARATION

A closed form solution of the radiation integral (vector potential) is impossible and standard approximations are applied, from which the boundaries of these regions are derived.

Consider the vector potential for a linear current source:

'

2 2 2

' '4

' ' '

j R

L

eA I l dl

R

R x x y y z z

Neglect the antenna dimensions along the x- and y-axes (infinitesimally thin wire). Then,

2 2 2 2 2 2

2 2 2 2 2 2

' ' ' '

2 ' ' 2 'cos '

R x x y y z z x y z z

x y z zz z r rz z


Using the binomial expansion

1 2 2 3 31 1 2....

2! 3!n n n n nn n n n n

a b a na b a b a b

1 22 2

1 2 1 2 3 2 22 2 2 2 2

2 2 23 2 3

2

2 'cos '

1 11 2 2

2 'cos ' 2 'cos ' ....2 2

' ' cos 1'cos ' cos sin

2 2 2

r rz z

r r rz z r rz z

z zr z z O

r r r

2 2 3 22

1 1'cos ' sin ' cos sin

2 2R r z z z

r r

This expansion is used to mathematically define the separation of the space surrounding an antenna viz. reactive near-field, the radiating near-field and the far-field region.


Far-field approximation:

Only the first two terms in the expansion are taken into account

'cosR r z

The most significant error term in R is

2 21' sin

2z

r Maximum value 2max

1' @

2 2z

r


It has been shown by many researchers through examples that for most practical antennas, with overall lengths greater than a wavelength, a maximum total phase error of /8 rad (22.5) is acceptable.

Using this criteria, the maximum phase error should always be:

2 2'2

2 8

z Dr

r

;r D r with

Far-field approximations:

'cosR r z

R r

for phase terms

for amplitude terms


Radiating near-field (Fresnel) region approximation:

If the observation point is chosen to be smaller than the far-field limit, then to limit the maximum phase error to /8, the third term in the binomial expansion of R must be retained.

2 21'cos ' sin

2R r z z

r

Maximum phase error introduced by omission of the fourth term:

3 32 2 2

2 2

1 ' 'cos sin sin sin 2cos 0

2 2

z z

r r

Maximum phase error occurs when:

1

2 2 11sin 2cos 0 tan 2


1

32

2' 2tan 2

1 'cos sin

2 8z D

z

r

30.62r l

Reactive near-field approximation:

30.62 0l r

antennas - introduction

Documents

current elementtotal

field zone

radiation density

distancethe power pattern

current elementfeatures

current elements

receivedradiated power

radiation properties