Lecture 4
Hertzian
dipole
(The elemental
electrical dipole)
HERTZ ANTENNA

The Hertzian dipole (Fig.4-1) is a theoretical short
dipole (short length of straight wire smaller than the
wavelength L<<λ) with a uniform current along its
length driven by an AC current of frequency ω.
Infinite small
length of wire
(Fig.4-1)
A Hertzian dipole is a thin, linear conductor whose length L is very
short compared with the wavelength λ; L should not exceed λ/50.
Hertzian dipole consists a short conducting wire of length ℓ
terminated in two small conductive spheres or disks
(capacitive loading). Fig.(4-2)
Fig.(4-2) Hertzian dipole antenna
We assume the current in the wire to be uniform and to vary
sinusoidally with time
i(t )  I . cos t  Re[ Ie jt ]
Since the current vanishes at the ends of the wire, charge
must be deposited there .The relation between the charge
and the current is
dq (t )
i (t )  
dt
In phasor notation q(t )  Re[ Qe jt ]
Or
We have
I   jQ
I
Q 
j
The positive sing is for the charge on the upper end and the
negative sign for the charge on the lower end
Radiation pattern of a Hertzian dipole
The magnitude of the normalized field strength (with respect
to the peak value) versus θ for a constant ϕ (an E-plane
pattern) and the magnitude of the normalized field strength
versus ϕ for    2 (the H-plane pattern) is shown in
Fig.(4-3)
H-plane pattern
E-plane pattern
Fig.(4-3 B) Radiation pattern of a Hertzian dipole
Where F(θ) and F(ϕ) is the maximum radiation
(4-4)
Radiated Electric Field
Radiation Resistance and Radiated Power
for Hertzian dipole
The radiation resistance (Rrad), defined as the value of resistor
driven by the current in the dipole antenna which would lead to
the some average power dissipation as the average radiated
2  2
power
Rrad 
3
( )

The term   0  0  120  377 is called the impedance of free
space and has standard value
12 F
7 H


8
,
85

10
Permeability of air 0
 0 4 10
m Permittivity
m
of air
I 02 2  2
[
( ) ]
Radiated power is Prad 
2
3 
Where
I 0 -current in the dipole


-length of the dipole
-wave length
I 2
( I ) 2
0
Or radiated power Prad  ( )  Rrad 
12
2
2
(  )
And
Rrad   0
6
0  377 and   2  ,we get
 2
2  2
 80 ( )  790( ) 
Substituting for
Rrad
  2 

-phase constant

rad
m
The directivity D of the antenna is defined as the ratio of the
maximum power density to the average power density
( Prad ) max
D
 3  1,5
2
( Prad )average
Radiation characteristics of a Hertzian Dipole
Normalized radiation field pattern
E ( ,  )
 sin 
E ( ,  ) max
S ( ,  )
Pn( ,  ) 
 sin 2 
S ( ,  ) max
En( ,  ) 
Normalized radiation power pattern
Average radiation intensity
Directivity
Gain
D( ) 
Prad
( I ) 2
Uaverage 
 0 
....( w )
2
sr
4
48
P
U ( )
 rad  1,5 sin 2 
Uaverage 4
G  Dmax  1,5 (assuming loss less)
Effective length
e 
Iaverage
   ...(m)
I max
e 2 0
 2 0
2 0
Ae 


 0,1192 ....m 2

4 Rrad
3160
4  790( ) 2
Effective aperture

Beam Area
A 
Poynting vector
4
41000
 2  2 sr...or... A 
 27333..(deg rees ) 2
3
D
D
S ( ,  ) 
E2 ( ,  )  E2 ( ,  )
Z0
.... w
m 2