 The Radar Range Equation relates the range of radar to the characteristics of the Tx (Transmitter), Rx (Receiver), Ae(Antenna), target, and the environment.
 It is useful to determine the maximum range at which a particular radar can detect a target and it makes to understand the factor affecting radar performance.
 The radar range equation is an important tool for the following aspects:
 Assessing the performance of radar
 Designing of new radar systems
 Assessing the technical requirement for new radar procurement.
Derivation of Radar Range Equation
 If the power of radar Tx is denoted by P_t and if an isotropic antenna (which will radiate energy uniformly in all directions) is used then the power density at the distance R is given by,
 Power density at range R from an isotropic antenna,
P_{di}=\frac{P_t}{4\pi R^2}\:\:watt/m^2\:\:\:........(1)
 However, We always try to radiate the energy in one particular direction. So the directive antenna is used to concentrate the radiated power P_t in a particular direction.
 The gain of an antenna is a measure of the increased power density radiated in some direction as compared to the power density that could appear in the direction of an isotropic antenna.
 The maximum gain G of an antenna may be defined as,
G=\frac{Maximum\,power\,density\,radiated\,by\,directive\,antenna}{Power\,density\,radiated\,by\,isotropic\,antenna}
 so, Power density P_{dd} at a distance R from directive antenna of power gain G,
P_{dd}=\frac{P_t\;G}{4\pi R^2}\:\:watt/m^2\:\:\:........(2)
 Now, the target intercepts a portion of the incident energy and reradiates it in various directions. But we are only interested in the direction at which the target reradiates energy in the direction of the radar.
 The radar crosssection of the target determines the power density returned to the radar for a particular power density incident on the target.
 It is denoted by \sigma and is often called a target crosssection.
 Radiated power density back at the radar is given by P_{de},
P_{de}=P_{dd}\left (\frac{\sigma}{4\pi R^2}\right )\:\:\:\:\:........(3)
 now if we put Eq.2 into Eq.3 then we will get,
P_{de}=\left (\frac{P_tG}{4\pi R^2}\right )\left (\frac{\sigma}{4\pi R^2}\right )\:\:\:\:\:........(4)
 The radar Antenna received a portion of the echo power. If the effective area of receiving antenna is denoted A_e, the power P_r received by the radar is,
P_r=\frac{P_t\;G\;\sigma\;A_e}{(4\pi R^2).(4\pi R^2)}\:\:watts\:\:\:........(5)
 The maximum range of radar R_{max} is the distance beyond which the target can not detect. It occurs when the received signal power P_r just equals the minimum detectable signal S_{min}.
 Substituting,
P_r=S_{min} in Eq. 5
S_{min}=\frac{P_t\;G\;\sigma\;A_e}{(4\pi R^2).(4\pi R^2)}\:\:watts\:\:\:........(6)
R^4_{max}=\frac{P_t\;G\;\sigma\;A_e}{(4\pi ).(4\pi )(S_{min})}\:\:\:\:\:........(7)
 so, the final Radar range equation can be rewritten by,
\Rightarrow \boxed{ R_{max}=\left [\frac{P_tG\sigma A_e}{\left (4\pi\right )^2 S_{min}}\right ]^{1/4}}\:\:\:\:\:........(8)
where,
 P_t = Transmitter Power
 G = maximum Gain of Antenna
 A_e = Aperture area of receiving Antenna
 \sigma = Cross section area of target
 S_{min} = Minimum detectable Signal
 R_{max} = Maximum range of radar
 Equation 8 is the Best form of the Radar Range Equation.
Modified Form of Radar Range Equation
 The relation between the Antenna aperture area and the gain of the antenna can be given by,
A_e=\frac{\lambda^2}{4\pi}. G\:\:\:\:\:........(9)
 where \lambda = wavelength ( \lambda = \frac{c}{f} where c= velocity of propagation and f=fequency) can be substituted in the eq.(8) and we will get the two results,
R_{max}=\left [\frac{P_tG\sigma}{\left (4\pi\right )^2 S_{min}}(\frac{G\lambda^2}{4\pi})\right ]^{1/4}
\Rightarrow \boxed{ R_{max}=\left [\frac{P_tG^2 \lambda^2 \sigma}{\left (4\pi\right )^2 S_{min}}\right ]^{1/4}}\:\:\:\:\:........(10)
\Rightarrow \boxed {R_{max}=\left [\frac{P_tG\sigma {A_e}^2}{4\pi \lambda^2 S_{min}}\right ]^{1/4}}\:\:\:\:\:........(11)
 Eq. 10 and 11 are the same and are also called the modified form of the Radar Range Equation.
Radar Range Equation in the Form of Noise and Noise Figure
 “Noise is unwanted Electromagnetic energy which interferes with the ability of the receiver to detect the wanted signal”.
 Noise may be generated within the receiver it itself or it may enter through the receiving antenna along with the reflected signal.
 Thermal Noise or Johnson Noise is the noise, which is generated by the thermal motion of the conducted electrons in receiver input stages.
 The magnitude of the noise is directly proportional to the bandwidth and the absolute temperature of the resistive input circuit.
 the thermal noise power of a radar receiver may be written as,
N_i = k.T.B_n
where,K=Boltzman’s Constant = 1.38 x 10^{23} joules/degree ,B_n=receiver bandwidth
 Noise Figure F_n can be given by,
F_n=\frac{S_i/N_i}{S_o/N_o}
\Rightarrow F_n=(\frac{S_i}{S_o}) (\frac{N_o}{N_i})
 So, from the above figure, we can say that S_o = S_i.G and \, N_o = N_i.G + \triangle N ,
 so we can write the above equation as,
F_n=(\frac{S_i}{S_i.G}) (\frac{N_i.G + \triangle N}{N_i})
\Rightarrow F_n=1 + (\frac{\triangle N}{N_i.G})
\Rightarrow \triangle N=N_i.G.(F1)\:\:\:\:\:
\Rightarrow \boxed{\triangle N= k.T.B_n.G.(F1)}\:\:\:\:\:........(12)
 so, The minimum detectable signalS_{min} can be the noise generated from the target which is detected by the Radar, so we can write S_{min}= \triangle N,
 By substituting S_{min}= \triangle N in the eq. (12)into the eq.(11) we get,
\Rightarrow \boxed {R_{max}=\left [\frac{P_t\sigma {A_e}^2}{4\pi \lambda^2 k.T.B_n.(F1)}\right ]^{1/4}}\:\:\:\:\:........(13)
where, F = Noise Figure in ratio, K = Standard temperature to be taken 290^{0} in Kelvin
Frequently Asked Questions On Radar Range

What does radar range mean?
Answer: The radar range equation represents the physical dependences of the transmit power, which is the wave propagation up to the receiving of the echo signals.

How is the radar range of resolution calculated?
Answer: Range resolution as a distance can be calculated as follows: R_{res} = c * τ /2 Where τ is the transmitted pulse width and c is the velocity of light in the free space. Narrower beamwidth is very useful to distinguish two adjacent targets.

What are the range and bearing resolution?
Answer: Bearing, or azimuth, the resolution is the ability of a radar system to separate objects at the same range but at different bearings. The degree of bearing resolution depends on the radar beam width and target range. The range is a factor in bearing resolution because the radar beam spreads out as the range increases.

What is SNR in radar?
Answer: The signaltonoise ratio, SNR, or S/N ratio is one of the most straightforward methods of measuring radio receiver sensitivity. Signal to noise ratio defines the difference in level between the signal and the noise for a given signal level.

How far can radar detect?
Answer: This permits target detection at distances from about 500 to 2,000 nautical miles (900 to 3,700 km). Thus, an HF overthehorizon (OTH) radar can detect aircraft at distances up to 10 times that of a groundbased microwave air surveillance radar, whose range is limited by the curvature of the Earth.

Can radar detect humans?
Answer: Doppler radar cannot detect humans who are stationary or walking across the radar’s field of view. The radar can only detect the motion components that are directed towards or away from the radar.
Also Read: Radar basics Types of Radar, Principle, and Application of Radar