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Antenna & Propagation Antenna Parameters

1

Chapter 2 – Antenna

Parameters

1. Antenna Radiation

2. Figure of Merit

3. Efficiency

4. Radiation Pattern

5. Antenna Impedance

6. Friss formula

Antenna & Propagation Antenna Parameters

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Objective

To revise various antenna parameters from a system point of view:

• what they are

• how they are calculated

• where they can be used

Antenna & Propagation Antenna Parameters

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Prerequisites

• Units for an angle: radian vs degree

• Definition of solid angle W:

– steradians (sr)

– degrees2

• Power density and radiation intensity

• Radiated power from a Poynting vector

• Radiation from a Hertzian dipole

• Radiation from a magnetic current element

o180=π

2

2

deg41253180.44 =

=π

ππ

Antenna & Propagation Antenna Parameters

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Solid Angle: Steradian

r

Surface area dA

Solid angle dΩRadius r

Subtended angles dθ and dφ

Sphere

2r

dAd =Ω

2

2 sin

r

ddr φθθ=

φθθ ddsin=

x

z

πφθθπ π

4sin2

0 0==Ω ∫ ∫ ddSolid angle of a sphere, steradians

Antenna & Propagation Antenna Parameters

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Co-ordinate System

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Isotropic Antenna

• A theoretical point source radiating power equally in all

directions, 100% efficiency.

• Power Density: (W/m2)

• Directivity and Gain: D = G = 1 = 0 dBi

24 r

PS T

π=

Antenna & Propagation Antenna Parameters

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ANTENNA RADIATION

1. Radiation pattern

2. Radiation power density

3. Radiation intensity

4. Half-power beamwidth

Antenna & Propagation Antenna Parameters

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Cutting Plane

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Radiation Pattern

• Plot of |Eq| as a function of (q, f) at a fixed r, giving rise to a 3-

dimensional radiation pattern.

• E-plane pattern:

– plot of |Eq| vs q (f = constant, e.g. f = 0)

• H-plane pattern:

– plot of |Eq| vs f (q = constant, e.g. q = p/2)

• Distinguish between:

– polar and rectangular (Cartesian) plots

– linear (ratio) and logarithmic (dB) scales

– main lobe (main beam), side lobes (minor lobes), nulls

– HPBW (qHP, fHP) and FNBW (qFN, fFN): qHP ~ qFN/2

Antenna & Propagation Antenna Parameters

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Polar Pattern

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Rectangular Pattern

Beam Efficiency (BE):

>90% for radiometry,

astronomy, radar, etc. rx_or_tx_power_total

FNBW_within_power_rx_or_txBE =

Lobes, nulls, and beamwidths (HPBW and FNBW)

Antenna & Propagation Antenna Parameters

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Radiation Properties

Main lobe

Half-power

Beamwidth

HPBW

Side Lobe

Back

lobe

Minor

lobes

First-nulls

Beamwidth

FNBW

Nulls

Antenna & Propagation Antenna Parameters

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E-Plane and H-Plane

r

eIdzjE

rjβ

θθβ −

=)(sin30

θθβ

θ sinsin30

Kr

IdzE ==

where K is a constant,

usually normalize K=1.

E-plane H-plane

z

x

θ

x

y

φ

Hertzian Dipole

Antenna & Propagation Antenna Parameters

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FIGURE OF MERIT

1. Directivity

2. Gain

3. Antenna reciprocity

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Gains: D, G

• Directive Gain: D(q,f) or GD(q,f)

• Directivity:D

• Power Gain: G or GP

• Relationship between D and G:

• Relationship between D and HPBW:

DG η=

Antenna D (ratio) D (dB)

isotropic 1 0

Hertzian dipole 1.5 1.76

λ/2 dipole 1.64 2.15η = antenna efficiency

D and G are power ratios.

HPHPHP

Dφθππ.

44=

Ω=

o

HP

o

HPHP

Dφθ

ππ

.

41253

180.4

2deg

2

=Ω

=

Antenna & Propagation Antenna Parameters

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EIRP and ERP

• Effective Isotropic Radiated Power (EIRP):

• Effective Radiated Power (ERP):

TT GPEIRP .=

dipoleG

EIRPERP

2/λ

=

dBdBmdBm EIRPERP 15.2−=

dBdBmdBm ERPEIRP 15.2+=

EIRP is preferred in theoretical treatment for its simplicity.

ERP is sometimes used in practice as λ/2 dipole is readily available.

64.1

..

2/

TT

dipole

TT GP

G

GPERP ==

λ

Antenna & Propagation Antenna Parameters

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Effective Area

• The effective area of an antenna can be used to calculate the received

power

• Q1. A thin (diameter = 5 mm) half-wave dipole at 900 MHz (mobile phone)

has Gain = 1.64.

– How long is the antenna? l = = m = mm

– Calculate its effective area. Ae= = m2 = mm2

– Assume a rectangle with the area Ae. If one side of the rectangle equals

the antenna length, what is the dimension of the other side? Compare

this with the antenna diameter!

• Q2. What is the effective area of an isotropic antenna?

17

SAtyPowerDensiAP eeR .. == GAe πλ4

2

=where

Antenna & Propagation Antenna Parameters

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Effective Length or

Effective Height• For an antenna receiving a far-field wave, the effective length

le is defined by:

– where

• Voc = open-circuit voltage at antenna terminals

• Ei = incident electric field

• le = effective length

• For an electrically short dipole, le = lphysical/2

18

e

i

OC l.EV =

Antenna & Propagation Antenna Parameters

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Reciprocity Theorem

• The Lorentz Reciprocity Theorem (which states that the E

and H fields generated by two different sources at the same

frequency must satisfy certain conditions) can be applied to

antennas to show that:

the receiving and transmitting

properties of an antenna can be

interchanged.• For example, the radiation pattern, feed-point impedance,

and gain are the same when a wire dipole antenna is used for

either transmitting or receiving.

Antenna & Propagation Antenna Parameters

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EFFICIENCY

1. Antenna efficiency

2. Radiation efficiency

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Radiation Resistance: Rr

• Rr is a fictitious resistance to account for the power radiated

by an antenna.

• Power radiated, (E,H = rms value)

• (1)

•

• But, according to (I = rms value) (2)

• Therefore, equating (1) & (2)

21

∫∫= dSExHPr .

∫∫=ππ

φθθπ

0

222

0

sin 120

ddrE

Pr

∫∫=ππ

φθθπ

0

222

0

2 sin

120

1ddr

E

IRr

rr RIP 2=

Antenna & Propagation Antenna Parameters

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Radiation Resistance of an

Electrically Short Dipole

• Electrically short means dl < l/10

• Hertzian dipole is an electrically short dipole

• For example:

– If dl/l = 0.01, then Rr = 0.08 W

– If dl/l = 0.1, then Rr = 7.90 W

22

22

2 79080

=

=λλ

πdldl

Rr(Ω)

Antenna & Propagation Antenna Parameters

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Feed-Point Impedance: Za

• Za = antenna impedance at its feed-point.

• Za is complex generally.

• Za can be determined by numerical methods, such as

Moments Method, FDTD, etc.

• For a dipole with total l<l/2 (or monopole with h<l/4), Xa is

negative (i.e. capacitive).

23

aaa jXRZ +=Za

Antenna & Propagation Antenna Parameters

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Antenna Efficiency: η

• Definition:

• In a lossless antenna,

• In an antenna with loss,

• Losses are due to:

– conduction (ohmic) loss

– dielectric loss

– reflection (mismatch)

• Total efficiency:

24

lossr

r

RR

R

losspowerradiatedpower

radiatedpower

+=

+=

η

ra RR =

lossra RRR +=

)1(2

Γ−== dcrdc ηηηηηη

Antenna & Propagation Antenna Parameters

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ANTENNA IMPEDANCE

1. Input Impedance

2. Bandwidth

Antenna & Propagation Antenna Parameters

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Feed-Point Impedance: Za

• Za = antenna impedance at its feed-point.

• Za is complex generally.

• Za can be determined by numerical methods, such as

Moments Method, FDTD, etc.

• For a dipole with total l<l/2 (or monopole with h<l/4), Xa is

negative (i.e. capacitive).

26

aaa jXRZ +=Za

Antenna & Propagation Antenna Parameters

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Dipole Antenna

Antenna & Propagation Antenna Parameters

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Feed-Point Impedance of a

Dipole Antenna• Dipole Za represented by a 4-element equivalent circuit.

• L, C, and R elements are determined from the physical dimensions of dipole, independent of frequency.

• <10% error in Ra and Xa for dipole length (2h) up to 0.6λ.

• Equivalent circuit can be used in electronic circuit simulation software such as PSPICE.

h = dipole half-length

a = dipole radius

Antenna & Propagation Antenna Parameters

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Equivalent Circuit

[Tang, Tieng & Gunn, IEEE Trans.AP, Jan. 1993, pp.100-103]

pFah

hc

7245.0)/2log(

0674.1231 −

=

[ ]pF

ahhc

−−

= 02541.0861.0)/2log(

89075.02

8006.032

[ ] 02389.02

31 )/2(40754.7)/2log(41288.0 −+= ahahR

[ ] HahhL µ6188.0)/2log(4813.12.0012.1

31 −=

Ω− k27408.7

Antenna & Propagation Antenna Parameters

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ANTENNA BANDWIDTH

1. Operating Frequency

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Antenna Bandwidth:

Frequency Range

• (There are various definitions of antenna bandwidth.)

• The bandwidth of an antenna is the band of frequencies, over which it is considered to perform acceptably.

• The wider the range of frequencies a band encompasses, the wider the bandwidth of the antenna.

• Antennas are ordered pre-tuned by the manufacturer, for use in a specified band segment.

• The trade-off in designing an antenna for a wider bandwidth is that it would generally not have as good of performance in comparison to a similar antenna that is optimized for a narrower bandwidth.

Antenna & Propagation Antenna Parameters

32Rick Graziani graziani@cabrillo.edu 32

Antenna Bandwidth:

Frequency Range

Antenna & Propagation Antenna Parameters

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FRIIS FORMULA

1. Definition

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Definition

The Friis Transmission Equation is used to calculate the power

received from one antenna (with gain G1), when transmitted

from another antenna (with gain G2), separated by a distance R,

and operating at frequency f or wavelength lambda.

Antenna & Propagation Antenna Parameters

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Frii’s Transmission Equation

• Example:

– For a transmitter-receiver distance of 20 m at 10.2 GHz,

with PT = 100 W, PR = 3 mW, and GR = 15 dB, (a) what is

the required GT? (b) what is the maximum power density

at a point 20 m from the transmitter?

– Ans. (a) 18.4 dB, (b) 1.44 W/m2 or 0.144 mW/cm2

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24 r

GPS TT

π=

πλ

π 4.

4.

2

2

RTTeR

G

r

GPASP == RT

T

R GGrP

P2

4

=πλ

S = power density

Ae = effective area

Antenna & Propagation Antenna Parameters

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Simple Questions• What is the difference between POWER GAIN, DIRECTIVE

GAIN, and DIRECTIVITY of an antenna?

• What is the difference between EIRP and ERP?

• What is the significance of the effective area of a receiving

antenna?

• What is the significance of the radiation resistance of a

transmitting antenna?

• How would the feed-point impedance (sometimes called the

antenna impedance) affect the performance of an antenna?

• Derive the Frii’s Transmission Equation and/or the Radar

Equation.

• What is the significance of the G/T ratio of a satellite

receiving station?36