Question

The normalised electric field radiation pattern of an antenna is given as
$f\left(\theta \right)=co{s}^4\theta ;0\le \theta \le \frac{\pi }{2},0\le \varphi \le 2\pi$ and zero else where.
then the directivity of the antenna is

Answer 1

The power pattern of the antenna is,
$P\left(\theta \right)={\left|F\right(\theta \left)\right|}^2=co{s}^8\theta$
∵ The maximum value of $P\left(\theta \right)$ is 1, the directivity can be written as
$D=\frac{4\pi }{{\int }_{\varphi =0}^{2\pi }{\int }_{\theta =0}^{\pi /2}co{s}^8\theta sin\theta d\theta d\varphi }$
Substituting,
$cos\theta =t\Rightarrow -sin\theta d\theta =dt$
We get, $D=\frac{4\pi }{2\pi {\int }_0^1{t}^{8}dt}=\frac{4\pi }{2\pi \times \left(\frac{t^9}{9}\right){|}_0^9}$
or D = 18
∴ D(dB) = 10 log18 = 12.55 dB