Radar Range Equation | Radar Range Equation Derivation

• 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 factors affecting radar performance.

• The radar range equation is an important tool for the following aspects:

1. Assessing the performance of radar
2. Designing of new radar systems
3. Assessing the technical requirement for new radar procurement.

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Also Read: MTI Radar | Moving Target Indicator Radar

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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 re-radiates it in various directions. However, we are only interested in the direction in which the target re-radiates energy in the direction of the radar.

• The radar cross-section 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 cross-section.

• 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 the 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 re-written 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.

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Also Read: Types and Parameters of Radar Antenna

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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.

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Also Read: Applications of Radar

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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 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.(F-1)\:\:\:\:\:

\Rightarrow \boxed{\triangle N= k.T.B_n.G.(F-1)}\:\:\:\:\:........(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.(F-1)}\right ]^{1/4}}\:\:\:\:\:........(13)

where, F = Noise Figure in ratio, K = Standard temperature to be taken 290⁰ in Kelvin

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Also Read: Radar basics- Types of Radar, Principle, and Application of Radar

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Frequently Asked Questions On Radar Range