Radar Tool

Radar range equation

How far can a radar actually see? This tool predicts the maximum detection range from the radar's power, antenna, frequency and the size of the target — and shows why doubling the range is so much harder than it sounds.

The equation in plain language

R_max = ⁴√P_t · G² · λ² · σ(4π)³ · P_min · L

And yes — that outer fourth root is real. The echo's power falls with R⁴ on the way out and back, so undoing it to find the range means taking the fourth root of everything else.

Each symbol is one link in the chain between the transmitter and the faint echo:

P_t is the transmit power — how loudly the radar shouts. G is antenna gain: how well the antenna focuses that shout into a beam (it appears squared because the same antenna focuses the transmission and collects the echo). λ is the wavelength, set by the operating frequency. σ (sigma) is the target's radar cross-section — effectively how big and reflective the target looks to radio waves, measured in square metres. P_min is the weakest echo the receiver can still detect, and L collects real-world losses in cables, the atmosphere and processing.

The whole thing is raised to the power of a quarter. That's the R⁴ law biting: to double detection range you need sixteen times more transmit power, all else being equal. It's why radar engineering is a game of decibels scraped together from every part of the system, not just a bigger transmitter.

Inputs

Transmit power P_t (kW) Peak pulse power. Airport surveillance radars are typically tens of kW to low MW.

Antenna gain G (dBi) A large rotating dish is typically 30–40 dBi.

Frequency (GHz) S-band ≈ 3 GHz, X-band ≈ 10 GHz.

Target radar cross-section σ

Minimum detectable signal P_min (dBm) Set by receiver noise. −110 dBm is a reasonable classroom value.

System losses L (dB) Cables, radome, atmosphere, processing. 4–8 dB is typical.

Result

Maximum detection range—

Wavelength λ—

Echo power at that range—

Echo power at half that range—

The cyan curve is the echo power arriving back at the receiver as the target moves further away — a straight line on this log–log style plot because of the R⁴ law (−12 dB every time range doubles). Where it crosses the amber P_min line, the echo disappears into the noise: that crossing is the maximum range.

This is the simplest form of the range equation — single pulse, no integration gain, no clutter, no fluctuation statistics. Real detection range analysis adds all of those, but the shape of the problem stays exactly as shown here.