Fundamental Antenna Parameters

1. Radiation PatternAn antenna radiation pattern is defined as “a graphicalrepresentation of the radiation properties of the antennaas a function of space coordinates. In most cases, theradiation pattern is determined in the far-field region.Radiation properties include radiation intensity, fieldstrength, phase or polarization.

Coordinate System

IdealizedPoint Radiator Vertical Dipole Radar Dish

Isotropic Omnidirectional Directional

Types of Radiation Patterns

Full Null BeamwidthBetween1st NULLS

Radiation Pattern Lobes

Main lobe

Side lobes

Back lobes

Radiation Pattern Lobes

Field Regions

DR1

R2

Reactive near-field region

3

1 62.0 DR

Radiating near-field (Fresnel) region

2

2 2D

R

Far-field (Fraunhofer) region

Radiation Intensity

Aside on Solid Angles

lengtharcrad0.1

r

sr0.1

2rareasurface

radianscecircumfrantotal 2224 rrSareasurfacetotal o

srr

So2

ddrds )sin(2infinitesimal areaof surface of sphere

ddr

dsd )sin(

2

Radiation Intensity

4

dUPsrW

d

dPU tot

rad

totrad

dsPPm

Wds

dPP rad

totrad

totrad

rad 2

radPrU 2

),,( rPrad decays as 1/r2 in the far fieldsince

),( U will be independent of r

Radiation Intensity

max

222

222*

),(),(

2),(

2

1~

2

1~~

2

1),,(

U

UU

EEr

U

EEEHErPrad

Radiation IntensityExamples

0.1),(

),(

4),,(),(

4),,(

max

2

2

U

UU

constP

rPrU

r

PrP

totrad

rad

totrad

rad

1. Isotropic radiator

2. Hertzian Dipole

)(sin),(

),(

)(sin42

)sin(42

1

2

1),(

0),,(

)sin(4

),,(

2

max

2

2

0

2

02222

0

U

UU

Il

r

eIlrEErU

rE

r

eIljrE

rj

rj

Directive Gain

)(14

),(4

4

),(),(),(

maxmax ydirectivit

P

UDD

P

U

P

U

U

UD

totrad

o

totrad

totradave

DirectivityExamples

0.1

0.1),(

4),(

4),(

o

totrad

totrad

o

D

P

UD

PUU

1. Isotropic radiator

2. Hertzian Dipole

2

3

)(sin2

3),(4),(

3

8

42)sin()(sin

42),(

)(sin422

1),(

0),,(),sin(4

),,(

2

2

02

0 0

2

2

0

4

2

2

0222

o

totrad

totrad

rj

D

P

UD

Ildd

lIdUP

IlEErU

rEr

eljrE

Antenna Gain

inputP

UG

),(4),(

POWER DENSITY IN A CERTAIN DIRECTION

DIVIDED BY THE TOTAL POWER RADIATED

POWER DENSITY IN A CERTAIN DIRECTION

DIVIDED BY THE TOTAL INPUT POWER

TO THE ANTENNA TERMINALS (FEED POINTS)

IF ANTENNA HAS OHMIC LOSS…THEN, GAIN < DIRECTIVITY

DIRECTIVITY

GAIN

Antenna Gain

Sources of Antenna System Loss

1. losses due to impedance mismatches

2. losses due to the transmission line

3. conductive and dielectric losses in the antenna

4. losses due to polarization mismatches

According to IEEE standards the antenna gain does not include losses due toimpedance or polarization mismatches. Therefore the antenna gain only accounts for dielectric and conductive losses found in the antenna itself. HoweverBalanis and others have included impedance mismatch as part of the antenna gain.

The antenna gain relates to the directivity through a coefficient called theradiation efficiency (et)

),(),(),( DeeeDeG dcrt

conduction losses dielectric losses

1te

impedance mismatch

Overall Antenna Efficiency

The overall antenna efficiency is a coefficient that accounts for all the differentlosses present in an antenna system.

lossesdielectricconductore

lossesdielectrice

lossesconductione

mismatchimpedanceefficiencyreflectione

mismatchesonpolarizatie

eeeeeeee

cd

d

c

r

p

cdrp

e

dcrp

t

&

)(

Reflection Efficiency

The reflection efficiency through a reflection coefficient () at the input (or feed)to the antenna.

)(

)(

12

impedanceoutputgeneratorR

impedanceinputantennaR

RR

RR

e

output

input

generatorinput

generatorinput

r

Radiation Resistance

The radiation resistance is one of the few parameters that is relativelystraight forward to calculate.

24

2

),(22

oo

totalrad

radI

dU

I

PR

Example: Hertzian Dipole

22

2

22

0 0

2

2

4

3

2

3

8

4

38

422

3

8

42)sin()(sin

42),(

2

ll

I

Il

R

Ildd

IldUP

o

o

rad

oototrad

Radiation Resistance

Example: Hertzian Dipole (continued)

0063.09.750

9.7501

079.010000

1

3

2377

377100

1

3

2

3

8

4

38

422

2

22

2

2

r

rad

o

o

rad

e

R

andl

let

ll

I

Il

R

Antenna Radiation Efficiency

radcd

radcd RR

Re

Conduction and dielectric losses of an antenna are very difficult to separate andare usually lumped together to form the ecd efficiency. Let Rcd represent the actuallosses due to conduction and dielectric heating. Then the efficiency is given as

For wire antennas (without insulation) there is no dielectric losses only conductorlosses from the metal antenna. For those cases we can approximate Rcd by:

22o

cd b

lR

where b is the radius of the wire, is the angular frequency, is the conductivityof the metal and l is the antenna length

Example Problem:

A half-wavelength dipole antenna, with an input impedance of 73 is to beconnected to a generator and transmission line with an output impedance of50. Assume the antenna is made of copper wire 2.0 mm in diameter and theoperating frequency is 10.0 GHz. Assume the radiation pattern of the antenna is

Find the overall gain of this antenna

SOLUTIONFirst determine the directivity of the antenna.

)(sin),( 3 oBU

totradP

UD

),(4),(

697.13

16

)(sin3

16

43

)(sin4),(

max0

3

2

0

3

DD

B

BD o

Example Problem: Continued

SOLUTIONNext step is to determine the efficiencies

965.0)5073

50731()1(

22

r

cdrt

e

eee

radcd

radcd RR

Re

964.09991.0965.0

9991.00628.073

73

0628.0107.52

10410102

)001.0(2

015.0

22 7

79

cdrt

cd

ocd

eee

e

b

lR

Example Problem: Continued

SOLUTIONNext step is to determine the gain

dBdBG

GG

G

DeeG cdr

14.2)636.1(log10)(

636.13

16964.0

)(sin3

16964.0),(

),(),(

100

max0

3

Antenna Type Gain (dBi) Gain over Isotropic

Power Levels

Half Wavelength Dipole

1.76 1.5x

Cell Phone Antenna(PIFA)

3.0 2.0x 0.6 Watts

Standard Gain Horn

15 31x

Cell phone tower antenna

6 4x

Large Reflecting Dish

50 100,000x

Small Reflecting Dish

40 10,000x

Effective Aperture

plane waveincident

AphysicalPload

incphysicalload WAP?

Question:

Answer: Usually NOTinc

loadeffinceffload W

PAWAP

Directivity and Maximum Effective Aperture (no losses)

Antenna #2

transmit receiver

R

Direction of wave propagation

Antenna #1

Atm, DtArm, Dr

oem DA4

2

Directivity and Maximum Effective Aperture (include losses)

Antenna #2

transmit receiver

R

Direction of wave propagation

Antenna #1

Atm, DtArm, Dr

2*2

2 ˆˆ4

)1( awocdem DeA

conductor and dielectric losses reflection losses

(impedance mismatch)polarization mismatch

Friis Transmission Equation (no loss)

Antenna #2

Antenna #1

R

transmit

Atm , D

treceiver

Arm, D

r

The transmitted power density supplied by Antenna #1at a distance R and direction rr)is given by:

24

),(

R

DPW ttgtt

t

tt)

rr)

The power collected (received) by Antenna #2 is given by:

),(),(4

4

),(

4

),(

4

),(

2

2

22

rrgrttgtt

r

rrgrttgttr

ttgttrtr

DDRP

P

D

R

DPA

R

DPAWP

Friis Transmission Equation (no loss)

Antenna #2

Antenna #1

R

transmit

Atm , D

treceiver

Arm, D

r

tt)

rr)

),(),(4

2

rrgrttgtt

r DDRP

P

If both antennas are pointing in the direction of their maximum radiation pattern:

rotot

r DDRP

P2

4

Friis Transmission Equation ( loss)

Antenna #2

Antenna #1

R

transmit

Atm , D

treceiver

Arm, D

r

tt)

rr)

2*

222 ˆˆ),(),(

4)1)(1( awrrgrttgttrcdrcdt

t

r DDR

eeP

P

conductor and dielectric lossestransmitting antenna

conductor and dielectric lossesreceiving antenna

reflection losses in transmitter(impedance mismatch)

reflection losses in receiving(impedance mismatch)

polarization mismatch

free space loss factor

Friis Transmission Equation: Example #1

A typical analog cell phone antenna has a directivity of 3 dBi at its operating frequency of 800.0 MHz. The cell tower is 1 mile away and has an antenna with a directivity of 6 dBi. Assuming that the power at the input terminals of the transmitting antenna is 0.6 W, and the antennas are aligned for maximum radiation between them and the polarizations are matched, find the power delivered to the receiver. Assume the two antennas are well matched with a negligible amount of loss.

nWwattsPr 65.142 609.344 14

375.06.0

2

2*maxmax

222 ˆˆ

4)1)(1( awrttrcdrcdt

t

r DDR

eeP

P

= 0 = 0= 1= 1 = 1

0.410

0.210

375.06800

83

10/6max

10/3max

r

t

D

D

me

e

f

c

Friis Transmission Equation: Example #2

A half wavelength dipole antenna (max gain = 2.14 dBi) is used to communicate from an old satellite phone to a low orbiting Iridium communication satellite in the L band (~ 1.6 GHz). Assume the communication satellite has antenna that has a maximum directivity of 24 dBi and is orbiting at a distance of 781 km above the earth. Assuming that the power at the input terminals of the transmitting antenna is 1.0 W, and the antennas are aligned for maximum radiation between them and the polarizations are matched, find the power delivered to the receiver. Assume the two antennas are well matched with a negligible amount of loss.

pWwattsPr 15.025164.1 781,0004

1875.00.1

2

2*maxmax

222 ˆˆ

4)1)(1( awrttrcdrcdt

t

r DDR

eeP

P

= 0 = 0= 1= 1 = 1

0.25110

64.110

1875.06800

83

10/24max

10/14.2max

r

t

D

D

me

e

f

c

Friis Transmission Equation: Example #2

A roof-top dish antenna (max gain = 40.0 dBi) is used to communicate from an old satellite phone to a low orbiting Iridium communication satellite in the Ku band (~ 12 GHz). Assume the communication satellite has antenna that has a maximum directivity of 30 dBi and is orbiting at a distance of 36,000 km above the earth. How much transmitter power is required to receive 100 pW of power at your home. Assume the antennas are aligned for maximum radiation between them and the polarizations are matched, find the power delivered to the receiver. Assume the two antennas are well matched with a negligible amount of loss.

Wwatts

Pt 82

1000000,10 36,000,0004

025.0

101002

12

2*maxmax

222 ˆˆ

4)1)(1( awrttrcdrcdt

t

r DDR

eeP

P

= 0 = 0= 1= 1 = 1

0.100010

000,1010

025.06800

83

10/30max

10/40max

t

r

D

D

me

e

f

c

Radar Range Equation

Definition: Radar cross section or echo area of a target is defined as the area when interceptingthe same amount of power which, when scattered isotropically, produces at the receiver the samepower density as the actual target.

222

4lim4

lim mW

WR

R

WW

inc

s

R

inc

Rs

(radar cross section) m2

R (distance from target) mWs (scattered power density) W/m2

Winc (incident power density) W/m2

Radar Range Equation (no losses)

Power density incident on the target is a functionof the transmitting antenna and the distance between the target and transmitter:

24

),(

t

ttgttinc

R

DPW

The amount of power density scattered by the target at the location of the receiver is then given by:

22 )4(

),(

4 rt

ttgtt

r

incs RR

DP

RWW

The amount of power delivered by the receiver is then given by:

4

),()4(

),( 2

2 rrgrrt

ttgttrsr D

RR

DPAWP

4

),(),(

)4( 2

2rrgrttgt

rtt

rDD

RRP

P ),,,( rrtt

Note that in general:

Radar Range Equation (losses)

2*

222 ˆˆ

44

),(),()1)(1( aw

rt

rrgrttgttrcdrcdt

t

r

RR

DDee

P

P

Radar Cross Section (RCS)

Definition: Radar cross section or echo area of a target is defined as the area when interceptingthe same amount of power which, when scattered isotropically, produces at the receiver the samepower density as the actual target.

222

4lim4

lim mW

WR

R

WW

inc

s

R

inc

Rs

22

2

222

2

2 4lim4lim mE

ERm

E

ER

inc

scat

Rinc

scat

R

),,,( rrtt rtrt ,

Transmitter and receiver not in the same location (bistatic RCS)

rtrt , Transmitter and receiver in the same location (usually the same antenna) called mono-static RCS

Radar Cross Section (RCS)

RCS Customary Notation: Typical RCS values can span 10-5m2 (insect) to 106 m2 (ship). Due to thelarge dynamic range a logarithmic power scale is most often used.

1log10log10

22

2 1010m

ref

mdBmdBsm

100 m2 20 dBsm

10 m2 10 dBsm

1 m2 0 dBsm

0.1 m2 -10 dBsm

0.01 m2 -20 dBsm