The tools provided so far give enough information for understanding the “radar range equation”, the most fundamental formula for radar operation. As its name implies, it gives the possible maximum range of a radar, as determined by the following factors:

Electrical noise. This is a function of environmental noise, which tends to be unpredictable, and the noise inherent in the electronic systems of the receiver. A radar pulse echo return must be above the noise threshold for a target to be detected.

Transmitter power. As mentioned, this is a function of pulse power and PRF, as well as antenna gain.

Receiver gain. This is a function of the receiver antenna gain and the sensitivity of the receiver electronics.

Attenuation due to range. The power of a radar beam will fall off with the square of distance. Since the radar must pick up the return echo of the transmit pulse, which also falls off by the square of distance, that means that the strength of a return pulse falls off by the fourth power of the distance to the target.

Target “radar cross section (RCS)”. The RCS of a target is effectively its reflectivity to radar. RCS varies with the material being illuminated, for example metal surfaces tend to be more reflective than plastic surfaces, and with the physical configuration of the surfaces. A smooth surface tends to be less reflective than a jagged rough surface. The RCS of a target tends to be highly variable, depending on the viewing angle of the target. An aircraft that is very bright to radar from one angle may be almost invisible from another, and its radar return may change drastically as it flies around.

Atmospheric attenuation. This is the trickiest of all the factors to estimate, since it can vary wildly given different atmospheric conditions. It is usually just given as a flat constant, since it is hard to do much better in practice.
This gives a simplified version of the radar range equation:

power * gain * RCS
———————– > noise
attenuation * range^4

There are many variations of this equation, usually providing greater detail or modified to demonstrate the capabilities of different radar configurations. The basic idea is simple: the capability of a radar to detect a target is directly proportional to its transmit power, its receiver gain, and the RCS of a target; and inversely proportional to the atmospheric attenuation and the fourth power of the range.