A radar operating at 1.5GHz uses a peak power of 2.5MW and have a range of 100 nmi for objects whose radar cross section is 1m2. The minimum receivable power of the receiver is 2×10^(-13) Watt. What is the smallest diameter of the antenna reflector could have, assuming it to be full parabolid with η=0.65.

Given

$f=1.5 GHz \\ P_t = 2.5 MW \\ R_{\max}=100nmi=100 \times 1.8518=185.18 km \\ σ=1m^2 \\ P_r=2×10^{-13} Watt \\ d=? \\ η=0.65$

we know that,

$R_{\max}=\bigg[\dfrac{pt Ae^2 σ}{4π λ^2 δ_{\min}}\bigg] \\ 185.18 \times10^3= \dfrac{(2.5 \times 10^6 \times Ae^2 \times 1)}{4π\times \bigg(\dfrac{3 \times 10^8}{1.5 \times 10^9}\bigg)^2 \times 2 \times 10^{-13}} ∴Ae^2= \dfrac{(1.85 \times 10^5 )^4 \times 4π \times (40 \times 10^{-3} ) \times 2 \times 10^{-13}}{2.5 \times 10^6} Ae=6.8765$

But Ae=η.A

$η.A=6.8765 \\ 0.65×πr^2=6.8765 \\ ∴r^2=\dfrac{6.8765}{π \times 0.65}$

$r^2=3.3674 \\ ∴r=1.8350m. \\ ∴d=2r=3.670m$