The Radar Equation
Created  Updated Mar 27, 2009
The History of Radar  Radar History: Isle of Wight Radar During The Second World War  Radar: The Basic Principle
Radar Technology: Main Components  Radar Technology: Side Lobe Suppression 
Radar Technology: Airborne Collision Avoidance
Radar Technology: Antennas  Radar Technology: Antenna Beam Shapes  Radar Technology: Monopulse Antennas  Radar Technology: Phased Array Antennas  Radar Technology: Continuous Wave Radar  Theoretical Basics: The Radar Equation
Theoretical Basics: Ambiguous Measurements  Theoretical Basics: Signals and Range Resolution
Theoretical Basics: Ambiguity And The Influence of PRFs  Theoretical Basics: Signal Processing  Civilian Radars: Police Radar  Civilian Radars: Automotive Radar  Civilian Radars: Primary and Secondary Radar
Civilian Radars: Synthetic Aperture Radar (SAR)  Military Applications: Overview  Military Radars: Over The Horizon (OTH) Radar
How a Bat's Sensor Works  Low Probability of Intercept (LPI) Radar  Electronic Combat: Overview  Electronic Combat in Wildlife
Radar Countermeasures: Range Gate PullOff  Radar Countermeasures: Inverse Gain Jamming  Advanced Electronic Countermeasures
The term 'radar equation' is a misnomer because it covers only a minuscule part of what a radar is. 'Radar range equation' would be more appropriate but, well, everybody uses the shorter term.
The equation yields an estimate of the distance at which a radar can be expected to detect a given target. Please note the careful expressions: it is an estimate and noone can guarantee that it yields accurate results because there are many influences that aren't taken care of. OK, so much for the disclaimer.
There is one fixed point in the calculation  the receiver's minimum detectable signal. After having travelled out to a target and back from it, an echo must stand out above the receiver's internal noise level so that target detection can be declared.
The bottom line is: the more you give, the more you get. Range performance gets better the stronger the outgoing radar signal is, the stronger the reflection from the target is and the more capable the receiver is at collecting the reflection and sorting it out from the noise. These three parts can be expressed in more detail:
The strength of the outgoing signal is determined by the transmitter's power and duty cycle (that is, the percentage of time during which it actually is transmitting) and the antenna's capability to concentrate it into a given direction.
The target's reflection back towards the radar is covered in a single figure of merit, 'sigma'. For some reason, sigma takes on the dimensions of square metres and is also called RCS (Radar Cross Section) but this is just another misnomer. Sigma somewhat depends on a target's size, but RCS calculation theories can fill whole volumes with chapter headings such as aspect angle, target size vs wavelength used, builtin corner reflectors, resonant features of a target's structure, statistical models of target behaviour, stealth coatings and whatnot. Large sigma values denote strong reflection. In the microwave frequency range, typical values are around 50m^{2} for a large aircraft, somewhere around 3m^{2} for a fighter and 0.0025m^{2} for a stealth aircraft.
The strength of the signal presented to the receiver is again determined by the antenna (in case it is shared between transmitter and receiver) and the minimum detectable signal level.
The antenna beam is a conical shape (and so, with increasing distance, the transmitted signal gets spread over a larger area). If the power measured at some distance is p, then at twice the distance the power is distributed over an area four times as large (which means a receiver can only detect p/4 of the power if moved from d to 2*d). This is known as the square law: increasing distance by a factor x decreases power by the factor x^{2}. The target echo strength undergoes the same square law again and therefore the received power varies with the fourth power of the distance.
Of all the influences listed above, only the antenna appears twice. And so, if your radar doesn't perform as desired then you might think of increasing transmitter power, using more time for transmission or increasing receiver sensitivity, but these measures only provide a straight 'value for money' relation. Doubling the antenna's relevant figure of merit  gain  yields an echo in the receiver that is four times as strong. Striving for more antenna gain either means asking for a bigger dish (which a customer might or might not accept for their intended carrier platform) or asking for higher frequencies, which is subject to the availability of a free spot in the electromagnetic spectrum and the existence of components for that frequency range (unless you're willing and able to develop them from scratch).
Putting all this together, the equation reads:
received power = 

Solving for range yields:
MaxRange^{4} = 

Where (constant) = (4*pi)^{3} and MDS is the minimum detectable signal power.
At first glance these equations appear to be dependant on lambda, the wavelength. However, antenna gain is inversely proportional to lambda and thus lambda gets cancelled out.
Important points to note are that:
The equation above is valid for primary radars where all the power involved is created in a single place. This is in contrast to secondary radars, where the signal is amplified within the transponder of the 'target.'
It is only valid for a monostatic arrangement  that is, with the transmitter and the receiver located on the same platform. Bistatic arrangements such as a groundbased target illuminator and an airborne seeker head need to be treated with more complicated formulae.
Generating more transmitter power is always expensive and should be avoided wherever possible. This is one of the reasons why civilian air traffic control uses secondary radar where the 'targets' are equipped with transponders that create 'answer' signals. The powersaving is tremendous because applying a 1/r^{2} law two times (for the outgoing 'question' and the incoming 'answer') is far more economical than being subjected to the 1/r^{4} law as described above.
In an electronic combat scenario, radar warning receivers enjoy greater detection ranges than radars because their trigger signals only need to travel the distance once (the 1/r^{2} law applies), whereas the radar's receiver signal is subject to the 1/r^{4} equation above. For a radar warning receiver, the equation reads:
MaxRange^{2} = 

Other Entries in This Project
History
Technology
 Basic Principle
 Main Components
 Signal Processing
 Antennae
 Side Lobe Suppression
 Phased Array Antennae
 Antenna Beam Shapes
 Monopulse Antennae
 Continuous Wave Radar
Theoretical Basics
Civilian Applications
 Police Radar
 Automotive Radar
 Primary and Secondary Radar
 Airborne Collision Avoidance
 Synthetic Aperture Radar
Military Applications
Electronic Combat
 Overview
 Electronic Combat in Wildlife
 Range Gate PullOff
 Inverse Gain Jamming
 Advanced ECM
 How Stealth Works
 Stealth Aircraft