Dipole AntennaDipole Antenna

The Dipole antennaThe Dipole antenna

• Has many practical applications• Can be considered as a sum (integration) of

Hertzian dipoles elements

l/2

-l/2

z'

dz' --

R

r

Punto de observación

DIPOLO LARGO

Prime variables refer to distance to the source from the origin.

r

zeIA

jkr

z

4

:dipole malinfinitesifor

dAz I z' dz' e jkR

4R

DipoleDipolel/2

-l/2

z'

dz' --

R

r

Punto de observación

DIPOLO LARGO

R

dzezIdA

jkR

z

4

)'(

R r2 2rz' cos z' 2 r 12z'

rcos

z' 2

r 2

Using Cosine Law:

Using the Taylor series:

(1 x)1/ 2 1 1

2x

1

2 4x2

13

2 4 6x3 . ..

Then it is:

22

sin2

'cos'

r

zzrR

• To ignore the terms of 1 / r in below impose a condition, which is that the phase error in the third term contributes in R is less than pi/ 8 (22.5 °). Evaluating the worst case, ie when sin(theta) = 1 and z '= z'max = l / 2, is obtained,

•

• where the minimum value of r (rff) is to reduce the error in the phase,

22

sin2

'cos'

r

zzrR

8

kz' 2

2r

2l / 2 2

2r

l2

4r

r ff 2l2

Substituting ...Substituting ...

Az

4I z' e jk r z' cos

r z' cos l /2

l /2

dz'

Az

4re jkr I z' e jkz' cos

l /2

l /2

dz'

In 'Far Field' can be approximated to:

Which can be written as: where:

Az

4re jkr I z' e jkz' cos

l / 2

l /2

dz'

Note I() is Fourier transformation of I(z')

If know If know AAzz……

can find can find HH and and EE..

H 1

rAz r sin

E H jAz sin

H jk

Az sin

Az

4re jkr I z' e jkz' cos

l /2

l /2

dz'

I need to know the distribution of current in the dipole I()

What is measured is:What is measured is:

Which can be approximated to:

z' z' z'

I(z') I(z') I(z')

l/2

-l/2 -l/2 -l/2

l/2 l/2

Im l /2

l /2

sin kl

2 z'

e jkz' cos dz'

'2/sin)'( zlkIzI m

To integrate:To integrate:

• It is divided into two integrals:

Im l /2

l /2

sin kl

2 z'

e jkz' cos dz'

2Im sin kl

2 z'

0

l /2

cos kz' cos dz'

2sin

2coscos

2cos2

)(k

klklI

Im

EE y y HHøø are proportional to sin are proportional to sin II((););

therefore the pattern of power (not standardized) random length dipole is;

Now we will see special cases.

2

sin

2coscos

2cos

)(

klkl

P

DipoleDipole

• Half wavelength dipole (l=/2)

• Full wavelength dipole (l=)

• 3/4 wavelength dipole (l=3/4)