Radar Equation Derivation:
The radar equation relates the range of the radar to the characteristics of transmitter, receiver, target and environment. It is useful for determining the maximum range at which radar can detect a target. If the transmitting antenna used is isotropic in nature, the power density is given by,
Power density at a range, R from an isotropic antenna is
Pis = Pt/4πR2 ----------------- (1)
If a directive antenna of gain, G is used, the power density is given by,
Power density at a range, R from a directive antenna is
Pdic = PtG/4πR2 ----------------- (2)
The radiated back power density is given by,
Prerad = PtG/4πR2.σ/4πR2
Where σ – radar cross section
The received signal power, Pr = radiated poor density x effected area
ie, Pr = PtGσAe/(4π)2R4.
The maximum range of radar, Rmax is the distance beyond which the target cannot be detected. It occur when the received signal power, Pr = minimum detectable signal, Smin.
Therefore, Smin = PtGσAe/(4π)2Rmax4.
Rmax = [PtGσAe/(4π)2 Smin]1/4
This is the fundamental form of radar range equation.
If the same antenna is used for transmitting and receiving the relation between gain and effective area is
G = 4πAe/λ2
Ae=ρaA
Where, ρa – aperture
Therefore, Rmax = [Pt4πAeσAe/ λ2(4π)2 Smin]1/4
Rmax = [PtσAe2/ 4π λ2 Smin]1/4 |