GATE EC 2012 Electromagnetics Solution Q 7

GATE EC 2012 Electromagnetics Solution Q 7

The radiation pattern of an antenna in spherical co-ordinates is given by-

F\left(\theta\right)=\cos^4\theta,\;0\leq\theta\leq\pi/2

The directivity of the antenna is____.

(a) 10 dB

(b) 12.6 dB

(c) 11.5 dB

(d) 18 dB

Ans :- (a)

Explanation

Directivity is given as,

D=\frac{4\pi u_{max}}{\pi_{rad}}

where, \pi_{rad} is radiated power given by,

\pi_{rad}=\int_0^{2\pi}\int_0^{\pi/2}\cos^4\theta\;\sin\theta\;d\phi

Also written as,

\pi_{rad}=\int_\theta\int_\phi u\left(\theta,\phi\right)d\Omega

where d\Omega is solid angle and d\Omega=\sin\;\theta\;d\theta\;d\phi.

\pi_{rad}=\int_0^{2\pi}\int_0^{\pi/2}\cos^4\theta\sin\;d\theta\;d\phi \pi_{rad}=2\pi\int_0^{\pi/2}\cos^4\theta\sin\;d\theta \pi_{rad}=-2\pi\left[\frac{\cos^5\;\theta}5\right]_0^{\pi/2} \pi_{rad}=\frac{2\pi}5 \therefore D=\frac{4\pi u_{max}}{\left({\displaystyle\frac{2\pi}5}\right)}=10\;u_{max} =10\;{\left[F\left(\theta\right)\right]}_{max}=10

= 10\;\log_{10}D=10\;dB.

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