Question 8.7: Find the radiation resistance of an infinitesimal dipole.
Find the radiation resistance of an infinitesimal dipole.
The radiated power P_{rad} from the Hertzian dipole is computed using (8.23).
Substituting the terms for the Hertzian dipole into (8.23), using Z_{0} = 120π and k = 2π/ λ and employing the definition (8.38), we obtain
P_{rad} = \frac{Z_{0} (k \mathscr{L} )^{2} I_{av}^{2}}{12\pi } (8.23)
R_{rad} = \frac{2 P_{rad}}{I_{0}^{2}} (8.38)
R_{rad} = 80 \pi ^{2} \left\lgroup\frac{ \mathscr{L} }{\lambda } \right\rgroup ^{2} \left\lgroup\frac{ I_{av} }{I_{0}} \right\rgroup ^{2} = 80 \pi ^{2} \left\lgroup\frac{ \mathscr{L} }{\lambda } \right\rgroup ^{2}where a uniform current distribution is assumed, or I_{av} = I_{0}. If there were a triangular current distribution depicted in the next example, we would obtain I_{av}= I_{0} / 2. In this case, the radiation resistance is 1/4 of the previous value.
R_{rad} = 20 \pi ^{2} \left\lgroup\frac{ \mathscr{L} }{\lambda } \right\rgroup ^{2}The radiation resistance for an infinitesimal dipole for \mathscr{L}/λ < 0.15 is shown in the figure below. The small values for the radiation resistance show that this antenna is not very efficient. This antenna will be discussed further in Example 8.11.
