Engineering & Science
Monostatic Radar Range Equation: Maximum Detection Range (Rmax) & Fourth-Power Law
Calculate maximum monostatic radar detection range Rmax = ((Pt * G^2 * lambda^2 * sigma) / ((4*pi)^3 * Smin))^0.25 in kilometers and echo power Pr in dBm.
Why doubling radar detection range requires 16x transmitter power!
Unlike one-way radio links (Friis 1/R^2 spreading), a monostatic radar pulse suffers spherical 1/(4*pi*R^2) spreading on the outbound path to the target AND a second 1/(4*pi*R^2) spreading on the reflected echo back to the antenna! Combining antenna effective aperture Ae = G * lambda^2 / (4*pi) with target radar cross-section sigma (m^2) yields received echo power Pr = (Pt * G^2 * lambda^2 * sigma) / ((4*pi)^3 * R^4)!
Solving for the maximum range Rmax where Pr equals minimum detectable receiver signal Smin (W) gives the fourth-root radar equation: Rmax = ((Pt * G^2 * lambda^2 * sigma) / ((4*pi)^3 * Smin))^0.25! For an X-band marine/airborne radar (Pt = 100 kW = 100,000 W, antenna gain G = 1,000 [30 dBi], wavelength lambda = 0.03 m [10 GHz], target RCS sigma = 2.0 m^2, receiver threshold Smin = 1.0e-13 W [-100 dBm]), Rmax = 30.86 km—and boosting peak power 16-fold to 1.6 MW doubles Rmax to 61.72 km!
| X-Band Radar Configuration (G=1000, lam=0.03 m) | Peak Power Pt & Target RCS | Max Range Rmax (km) | Scaling Insight |
|---|---|---|---|
| Baseline 100 kW Transmitter | Pt = 100,000 W, sigma = 2.0 m^2 | 30.86 km | Echo at Rmax = -100 dBm |
| 16x Power Upgrade (1.6 MW) | Pt = 1,600,000 W, sigma = 2.0 m^2 | 61.72 km (Exact 2.0x!) | 16^0.25 = 2x detection range |
| Stealth Target (sigma = 0.0002 m^2) | 10,000x smaller RCS | 3.09 km (0.1x range!) | (10^-4)^0.25 = 10x range cut |
Maximum Radar Detection Range Rmax (km)
Evaluates the fourth root (^0.25) of (Pt * G^2 * lambda^2 * sigma) / ((4*PI())^3 * Smin) and divides meters by 1,000.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Radar Parameters", ROUND((((B2 * (B3 ^ 2) * (B4 ^ 2) * B5) / (((4 * PI()) ^ 3) * B6)) ^ 0.25) / 1000, 2))Minimum Detectable Signal Smin Converted to dBm (10 * LOG10(Smin * 1000))
Expresses the receiver sensitivity floor Smin (Watts) in decibels relative to 1 milliwatt (dBm).
=IF(B6 <= 0, "Invalid Radar Parameters", ROUND(10 * LOG10(B6 * 1000), 1))Try the example
Use the original values or change them to test your understanding.