# Solved Problems-2

Problem-2

Determine the radiation resistance of the following antennas:

- Infinitesimal dipole or Hertzian dipole antenna of lengths
- Half-wave dipole antenna
- Quarter-wave monopole antenna

Solution

(a) For a Hertzian dipole antenna, the radiation resistance is given as,

For a length of
For a length of
For a length of

(b) The electric and magnetic fields of a half-wave dipole antenna in the far-field region are given as,
The time-average power density in the far field of a half-wave dipole antenna in free space is obtained as,
The total radiated power by the half-wave dipole antenna is,
Due to symmetry in the integrand, we can write,
so that the total radiated power is written as,
Let us evaluate this integration.
Let âˆ´
Substituting (1 +
Replacing the variable,
(Since )
Therefore, the radiation resistance is given as,
where,
In free space, replacing Î· = 120

(c) As obtained earlier, the time-average power density in the far field of a half-wave dipole antenna in free space is,
As the monopole is fed by a perfectly conducting plane at one end, it radiates only through a hemispherical surface. Therefore, the total radiated power by the monopole antenna is,
Let us evaluate this integration.
Let âˆ´
Substituting (1 +
Replacing the variable,
(Since )
For free space, replacing Î· = 120
Therefore, the radiation resistance is given as,

(b) The electric and magnetic fields of a half-wave dipole antenna in the far-field region are given as,

*x*) =*y*in the first integrand and (1 -*x*) =*y*in the second integrand, we have,*Ï€y*=*p*, we get,*Ï€*, the radiation resistance is given as,*x*) =*y*in the first integrand and (1 -*x*) =*y*in the second integrand, we have,*Ï€y*=*p*, we get,*Ï€*, we have,