# Solved Problems-2

Problem-2
Determine the radiation resistance of the following antennas:
1. Infinitesimal dipole or Hertzian dipole antenna of lengths
2. Half-wave dipole antenna
3. 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 + x) = y in the first integrand and (1 - x) = y in the second integrand, we have,

Replacing the variable, Ï€y = p, we get,

(Since )

Therefore, the radiation resistance is given as,

where,

In free space, replacing Î· = 120Ï€, the radiation resistance is given as,

(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 + x) = y in the first integrand and (1 - x) = y in the second integrand, we have,

Replacing the variable, Ï€y = p, we get,

(Since )