1

Radiation resistance

Antennas are designed for effective radiation of electromagneticenergy.

– Equivalent circuit of an antenna

– input radiation resistance Rr

• Represents radiated energy

– input loss resistance RL

• Represents conduction and dielectric losses of the antenna

– input reactance XA

• represents the energy stored in the field near the antenna

rR

inI

2

Radiation resistance

The power radiated is equal to:

The power losses is 2

2rin

radRIW =

If Win is the input power, the radiation efficiency is:

Lr

r

in

radr RR

RWW

+==η

2

2Lin

lossRIW =

rR

inI

3

Directive gain, directivity and gainStronger in some directions

Isotropic Antenna (the reference antenna)

Same intensity for all directions

4

Let Pavg be the average Poynting vector which is the power flow density per unit area,

( )*Re21 HEPavg ×=

The total power radiated Wrad is then

φθθφθ

φθθ

ddddUW

ddRddW

Srad

Srad

sin),(

sin2

=ΩΩ=

=⋅=

∫

∫ SSPavg

where U(θ,φ) is the power flow through a unit solid angle, and is called the radiation intensity (W/sr).

avgPrU 2),( =φθ

Directive gain, directivity and gain

5

Directive gain, directivity and gain

6

DirectivityMaximum value of the directive gain in a certain direction.

Power GainRatio of the radiation intensity in a given direction to the radiation intensity of a lossless isotropic radiatorthat has the same input power.

πφθφθ4/

),(),(in

p WUG =

Directive gain GD(θ,φ)Ratio of the radiation intensity in a particular direction(θ,φ) to the average radiation intensity.

πφθφθφθ4/

),(),(),(radavg

D WU

UUG ==

Directive gain, directivity and gain

),( φθdGMaxD =

7

Example

Find the directive gain of a Hertzian dipole.

( ) φθ HE21*Re

21

=×= HEPavg

I

θ

θE

φH

( ) θβηπ

2222

2

sin32 or

Idl=avgP

( ) θβηπ

222

22 sin

32 oIdlrU == avgP

8

Example

and then

( )

θ

πφθθθ

θ

φθφθ

π π

2

2

0 0

2

2

sin23

4/sinsin

sin

),(),(

=

=

=

∫ ∫ dd

UUG

avgD

9

Example

10

Example

Directive gain and the directivity of the Hertzian dipole

Suppose the radiation efficiency is 46%,

I

θφθ 2sin23),( =DG θ

θ

dB 76.123),2/( ===∴ φπdGD

dB 16.069.046.0 −===∴ DGp

)46.0/( =inrad WW

11

Example

Find the radiation resistance of a Hertzian dipole

Suppose, Poor radiator !!

( )

( )

=

=

=

=

∫ ∫

∫ ∫

22

2

22

22

2

0 0

322

22

2

0 0

802

12

sin32

sin

λπ

βηπ

φθθβηπ

φθθ

π π

π π

dlI

dlI

dddlI

ddPP

o

o

avgr

2280

=∴λ

π dlRr

Ω=⇒= 08.001.0 rRdlλ

12

Linear dipole antenna

In dipole antennas, the current magnitude along the dipole can be represented like in a transmission line, where 2h is the length of the dipole and z=0 at the center feed-point of the dipole. ( )[ ]zhIzI m −= βsin)(

I 2h

13

Linear dipole antenna

Knowing the current distribution I(z), we can sum up the fields due to the infinitesimal segments on the antenna using the results of the Hertzian dipole.

I 2h ( )

θβθβθ

θπ

φ

θπ

ηθ

φ

sincoscoscos)(

)(2

ˆ

)(2

ˆ

hhF

FR

ejI

FReIj

jkRm

jkRmo

−=

=

=

−

−

aH

E

F(θ) is called the pattern function

14

Linear dipole antenna

Antenna pattern– E-plane pattern (pattern function versus θ for a

constant φ)

– H-plane pattern (pattern function versus φ for a constant θ=π/2)

15

Linear dipole antenna

At certain dipole lengths (≅ λ/2, λ…) called resonant lengths, the input impedance is purely resistive. For half-wavelength dipole,

The pattern pattern for a half-wavelength dipole is

Ω=≈ 73rin RZ

( )

θ

θπθ

βθβθ

sin

cos2

cos

sincoscoscos)(

=

−=

hhF

64.1=D

16

Example -- Monopole

A thin quarter-wavelength vertical antenna over a conducting ground is excited by a sinusoidal source at its base. Find the (a) radiation pattern, (b) resistance, and (c) directivity.

17

Example -- Monopole

(a) The electromagnetic field in the upper half-space due to the quarter-wave vertical antenna is the same as that of the half-wave antenna.

(b) The magnitude of the time-average Poynting vector holds for but the quarter-wave antenna radiates only into the half-space, its total radiated power is only half of a half-wave dipole. Therefore, the radiation resistance is

(c) Same as half-wave dipole

2/0 πθ ≤≤

Ω==−= 5.362/73dipole)/2 wavehalf(rr RR

64.1=D

18

Effective Area and Friis Equation

Effective Area

The effective area Ae of a receiving antenna is the ratio of the time-average power received to the time-average power density of the incident wave at the antenna.

It may be shown that is Ae related to the directive gain as:

avgLe PPA /=

),(4

2

φθπλ

De GA =

19

Effective Area and Friis Equation

Friis Equation

The time-average power density at the receiving antenna is

Consider two antennae separated by a distance r. The transmitting antenna transmits a total power Pt.

Ae1, GD1, PtAe2, GD2, PL

r

124 Dt

avg Gr

PPπ

=

20

Effective Area and Friis Equation

The power received to the load is

(Friis Equation)

( ) tDD

avgD

eavgL

PGGr

PG

APP

212

2

2

2

2

4

4

πλπλ

=

=

( ) 212

2

4 DDt

L GGrP

Pπλ

=∴