**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

P

_{is }= P_{t}/4πR^{2}----------------- (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

P

_{dic}= P_{t}G/4πR^{2}----------------- (2)
The radiated back power density is given by,

P

_{rerad}= P_{t}G/4πR^{2}.σ/4πR^{2}
Where σ – radar cross section

The received signal power, Pr = radiated poor density x effected area

ie, Pr = P

_{t}GσA_{e}/(4π)^{2}R^{4}.
The maximum range of radar, R

_{max}is the distance beyond which the target cannot be detected. It occur when the received signal power, Pr = minimum detectable signal, S_{min}.
Therefore, S

_{min}= P_{t}GσA_{e}/(4π)^{2}R_{max}^{4}.
R

_{max}= [P_{t}GσA_{e}/(4π)^{2}S_{min}]^{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 = 4πA

_{e}/λ^{2}
A

_{e}=ρ_{a}A
Where, ρ

_{a}– aperture
Therefore, R

_{max}= [P_{t}4πA_{e}σA_{e}/ λ^{2}(4π)^{2}S_{min}]^{1/4}^{ }

R_{max} = [P_{t}σA_{e}^{2}/ 4π λ^{2} S_{min}]^{1/4} |

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