dipole antenna. the dipole antenna has many practical applications can be considered as a sum...
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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)
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