Loading....
Coupon Accepted Successfully!

 

Solved Problems-2

Problem-2
Determine the radiation resistance of the following antennas:
  1. Infinitesimal dipole or Hertzian dipole antenna of lengths Description: Description: 30209.png 
  2. Half-wave dipole antenna
  3. Quarter-wave monopole antenna
Solution
(a) For a Hertzian dipole antenna, the radiation resistance is given as,
 
Description: Description: 30202.png
 
For a length of Description: Description: 30195.png Description: Description: 30188.png
 
For a length of Description: Description: 30180.png Description: Description: 30171.png
 
For a length of Description: Description: 30162.png Description: Description: 30155.png

(b) The electric and magnetic fields of a half-wave dipole antenna in the far-field region are given as,
 
Description: Description: 30144.png
 
The time-average power density in the far field of a half-wave dipole antenna in free space is obtained as,
 
Description: Description: 30135.png
 
The total radiated power by the half-wave dipole antenna is,
Description: Description: 51270.png
 
Due to symmetry in the integrand, we can write,
 
Description: Description: 30116.png
 
so that the total radiated power is written as,
 
Description: Description: 30109.png
 
Let us evaluate this integration.
 
Let Description: Description: 30102.png ∴ Description: Description: 30095.png
 
Description: Description: 30088.png
 
Substituting (1 + x) = y in the first integrand and (1 - x) = y in the second integrand, we have,
 
 
Description: Description: 30080.png
 
Replacing the variable, πy = p, we get,
 
 
Description: Description: 30071.png
 
(Since Description: Description: 30062.png)
 
Description: Description: 30052.png
 
Therefore, the radiation resistance is given as,
 
Description: Description: 30043.png
 
Description: Description: 30035.png
 
where, Description: Description: 30024.png
 
Description: Description: 30017.png
 
In free space, replacing η = 120π, the radiation resistance is given as,
 
Description: Description: 30009.png
 
Description: Description: 30002.png
 
(c) As obtained earlier, the time-average power density in the far field of a half-wave dipole antenna in free space is,
 
Description: Description: 29995.png
 
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,
 
Description: Description: 51868.png
 
Let us evaluate this integration.
 
Let Description: Description: 29976.png ∴ Description: Description: 29968.png
 
Description: Description: 29959.png
 
Substituting (1 + x) = y in the first integrand and (1 - x) = y in the second integrand, we have,
 
 
Description: Description: 29951.png
 
Replacing the variable, πy = p, we get,
 
Description: Description: 29943.png
 
(Since Description: Description: 29932.png)
 
Description: Description: 29925.png
 
For free space, replacing η = 120π, we have,
 
Description: Description: 29918.png
 
Therefore, the radiation resistance is given as,
 
Description: Description: 29911.png
 
Description: Description: 29904.png
 





Test Your Skills Now!
Take a Quiz now
Reviewer Name