Prof. D. R. Wilton

Notes 22Notes 22

Antennas and Radiation Antennas and Radiation

ECE 3317

[Chapter 7]

Antenna Radiation

Antenna Radiation

We consider here the radiation from an arbitrary antenna.

The far-field radiation acts like a plane wave going in the radial direction.

+-

x

y

z

r

, ,r r

r "far field"

S

Antenna Radiation (cont.)

The far-field has the following form: ˆ ˆE E E

ˆ ˆH H H

0

E

H

0

E

H

x

y

z

E

HS

x

y

z

HE

S

Antenna Radiation (cont.)

The far-field Poynting vector is now calculated:

*

* *

* *

0 0

22

0 0

1S E H

21 ˆ ˆ ˆ ˆE E H H21

ˆ E H E H2

EE1ˆ E + E

2

EE1ˆ

2

r

r

r

0

E

H

0

E

H

Antenna Radiation (cont.)

Hence we have

22

0

1ˆS E E

2r

2

0

EˆS

2r

or

Note: in the far field, the Poynting vector is purely real (no reactive power flow).

Radiation Pattern

The far field always has the following form:

0

E , , E ,jk r

Fer

r

E ,F Normalized ar electric fieldf

In dB:

10

E ,dB , 20log

E ,

F

Fm m

,m m direction of maximum radiation

Radiation Pattern (cont.)The far-field pattern is usually shown vs. the angle (for a fixed angle ) in polar coordinates.

10

E ,dB , 20log

E ,

F

Fm m

0 dB

30°30°

60°

120°

150° 150°

120°

60°

m

0

-10 dB

-20 dB

-30 dB

Radiated Power

The Poynting vector in the far field is

2

20

E , 1ˆS , ,

2

F

r rr

The total power radiated is then given by

2

2 22

00 0 0 0

E ,ˆS sin sin

2

F

radP r r d d d d

Hence we have

2

2

0 0 0

1E , sin

2F

radP d d

Directivity

In dB,

2

S ,,

/ 4r

rad

D rP r

The directivity in a particular direction is the ratio of the power density radiated in that direction to the power density that would be radiated in that direction if the antenna were an isotropic radiator (radiates equally in all directions).

10, 10 log ,dBD D

The directivity of the antenna in the directions (, ) is defined as

Note: The directivity is sometimes referred to as the “directivity with respect to an isotropic radiator.”

Directivity (cont.)

The directivity is now expressed in terms of the far field pattern.

2

S ,,

/ 4r

rad

D rP r

2

20

22

2

0 0 0

E , 12

, 41

E , sin2

F

F

rD r r

d d

Therefore,

2

22

0 0

4 E ,,

E , sin

F

F

D

d d

Hence we have

Directivity (cont.)

Two Common Cases

Short dipole wire antenna (l << 0): D = 1.5

Resonant half-wavelength dipole wire antenna (l = 0 / 2): D = 1.643

y

+h

z

x-h

feed

2l h

Beamwidth

The beamwidth measures how narrow the beam is. (The narrower the beamwidth, the higher the directivity).

HPBW = half-power beamwidth

Gain and Efficiency

The radiation efficiency of an antenna is defined as

radr

in

Pe

P Prad = power radiated by the antenna

Pin = power input to the antenna

The gain of an antenna in the directions (, ) is defined as

, ,rG e D

The gain tells us how strong the radiated power density is in a certain direction, for a given amount of input power.

10, 10 log ,dBG G In dB, we have

Infinitesimal Dipole

The infinitesimal dipole current element is shown below.

x

y

z

Il

The dipole moment (amplitude) is defined as I l.

From Maxwell’s equations we can calculate the fields radiated by this source (see chapter 7 of the textbook).

The infinitesimal dipole is the foundation for many practical wire antennas.

Infinitesimal Dipole (cont.)

0

0

0

0 20

0 20 0

00

I 1 11 cos

2

I 1 1 1( ) 1 sin

4 ( )

I 1 1( ) 1 sin

4

jk rr

jk r

jk r

lE e

r jk r

lE j e

r jk r jk r

lH jk e

r jk r

The exact fields of the infinitesimal dipole in spherical coordinates are

Infinitesimal Dipole (cont.)

0

0

0

0

I( ) sin

4

I( ) sin

4

jk r

jk r

l eE j

r

l eH jk

r

In the far field we have:

0

0

I( )sin

4I

( )sin4

F

F

lE j

lH jk

Hence, we can identify

Infinitesimal Dipole (cont.)

The radiation pattern is shown below.

0

I( )sin

4F l

E j

-9 -3 -6

0 dB

30°30°

60°

120°

150° 150°

120°

60°

Infinitesimal Dipole (cont.)

The directivity of the infinitesimal dipole is now calculated

0

I( )sin

4F l

E j

2

22

0 0

4 E ,,

E , sin

F

F

D

d d

2

22

0 0

4 sin,

sin sin

D

d d

Hence

Infinitesimal Dipole (cont.)

Evaluating the integrals, we have

2

22

0 0

2

2

0

2

3

0

2

4 sin,

sin sin

4 sin

2 sin sin

2sin

sin

2sin

4 / 3

D

d d

d

d

Hence, we have

23, sin

2D

Infinitesimal Dipole (cont.)

23, sin

2D

-9 -3 -6

0 dB

30°30°

60°

120°

150° 150°

120°

60°

, 1.5D

, 1.0D o54.7

0

I( )sin

4F l

E j

The far-field pattern is shown, with the directivity labeled at various points.

Wire Antenna

00

0

II( ) sin

sinz k h z

k h

A good approximation to the current is:

A center-fed wire antenna is shown below.

y

+h

z

I (z)

x-h

feed

2l h

I 0

Wire Antenna (cont.)

00

0

II( ) sin

sinz k h z

k h

A sketch of the current is shown below.

resonant dipole (l = 0 / 2, k0h = / 2)

+h

-h

l

short dipole (l <<0 / 2)

0I( ) I 1z

zh

+h

-h

l

0I( ) I cos2

zz

h

Short Dipole

short dipole (l <<0 / 2)

0I( ) I 1z

zh

+h

-h

l

Wire Antenna (cont.)

The average value of the current is I0 / 2.

0

1I I

2effl l

0

0

I( )sin

4I

( )sin4

effF

effF

lE j

lH jk

Infinitesimal dipole:

0

0

I( )sin

4I

( )sin4

F

F

lE j

lH jk

Short dipole:

Wire Antenna (cont.)For an arbitrary length dipole wire antenna, we need to consider the phase radiated by each differential piece of the current.

Far-field observation point

, ,x y z

00

I 1( ) sin

4jk rl

E j er

0

0

1( )sin I

4

h jk R

h

eE j z dz

R

Wire antenna:

Infinitesimal dipole:

1

22 2 2'R x y z z y

+h

z

x-h

feed

r

R

dz' z'

I (z')

Wire Antenna (cont.)

Far-field observation point

z

y

+h

x-h

feed

r

R

dz'

22 2

2 2 2 2

2 2

2

2

2

'

' 2 '

' 2 '

'1 2

'1 2

R x y z z

x y z z zz

r z zz

z zzr

r r

z z zr

r r r

, ,x y z

Wire Antenna (cont.)

Far-field observation point

z

y

+h

x-h

feed

r

R

dz'

2

'1 2 cos 1 2 cos 1 cos

cos

z z z zR r r r

r r r r

r z

, ,x y z

1 12

xx Note: 1x

cos /z r

Wire Antenna (cont.)

Far-field observation point

z

y

+h

x-h

feed

r

R

dz'

, ,x y z

0

0 0

0

0

cos

0

cos

0

cos0

1( )sin I

4 cos

1( )sin I

4 1 / cos

1( )sin I

4

h jk r z

h

hjk r jk z

h

hjk rjk z

h

eE j z dz

r z

e ej z dz

r z r

ej e z dz

r

Wire Antenna (cont.)

We define the array factor of the wire antenna:

0 cosIh

jk z

h

AF z e dz

We then have the following result for the far-field pattern of the wire antenna:

0

0

1( )sin

4

jk reE j AF

r

Wire Antenna (cont.)

Using our assumed approximate current function we have

0 cos00

0

Isin

sin

hjk z

h

AF k h z e dzk h

The result is (derivation omitted)

0 0 0

20 0

I cos( cos ) cos( )2

sin sin

k h k hAF

k h k

Wire Antenna (cont.)

In summary, we have

0 0 0

20 0

I cos( cos ) cos( )2

sin sin

k h k hAF

k h k

0

0

1( )sin

4

jk reE j AF

r

0

0 0 00

0 0

I cos( cos ) cos( )1( )

2 sin sin

jk r k h k heE j h

r k h k h

Thus, we have

Wire Antenna (cont.)

For a resonant half-wave dipole antenna

0

00

cos cosI1 2

( )2 / 2 sin

jk reE j h

r

0

0

/ 4

/ 2

h

k h

/ 2, 1.643D The directivity is

Wire Antenna (cont.)

Results

2l h

Wire Antenna (cont.)

Radiated Power:

22

0 0 0

22

0 0 00

0 0 00 0

1E , sin

2

I cos( cos ) cos( )1 1( ) sin

2 2 sin sin

FradP d d

k h k hj h d d

k h k h

0 0 00

0 00 0k

Simplify using

22

0 0 00

0 00 0

I cos( cos ) cos( )1 1( ) sin

2 2 sin sinrad

k h k hP j d d

k h

Wire Antenna (cont.)

Performing the integral gives us

2

0 0 00

0 00

I cos( cos ) cos( )1 12 ( ) sin

2 2 sin sinrad

k h k hP j d

k h

2 2

0 0 0 0

0 0

I cos( cos ) cos( )sin

4 sin sinrad

k h k hP d

k h

The result is then

Wire Antenna (cont.)

The radiation resistance is defined from 2

0

1P I

2rad radR

Z0 Zin

in in inZ R jX

in radR R

Circuit Model

For a resonant antenna (l 0/2), Xin = 0.

y

+h

z

I (z)

x-h

feed

2l h

2

0

2P

Irad

radR

Wire Antenna (cont.)

The radiation resistance is now evaluated.

2

0

2P

Irad

radR

This yields the result

2 2

0 00

0 0

cos( cos ) cos( )1

2 sin( ) sinrad

k h k hR d

k h

2 Dipole: 00,

4 2h k h

73radR

Wire Antenna (cont.)

The result can be extended to the case of a monopole antenna

h

Feeding coax

0 / 4h

1

2monopole dipolein inZ Z

36.5radR

Wire Antenna (cont.)

This can be justified as shown below.

+

-

dipole

Vdipole I0

Virtual ground

+

Vmonopole

I0

-

1

2monopole dipoleV V

Receive Antenna

The Thévenin equivalent circuit of a wire antenna being used as a receive antenna is shown below.

+-VTh

ZTh

Th inZ Z

ˆV EincTh effl l

0

0

/ 2, ( )

2cos cos , / 2 ( )

2eff

l l

ll l

electrically small

resonant

+

-VThEinc

l = 2h l̂ direction of wire axis