Radar Range Equation

Rmax = (Pt×Gt×Gr×λ²×σ / ((4π)³×Smin))^0.25

Quick answer

This calculator solves R_max = (P_t × G_t × G_r × λ² × σ / ((4π)³ × S_min))^(1/4) for Rmax. With Transmit Power 10 W, Tx Antenna Gain 0.5, Rx Antenna Gain 1.1, Wavelength 1 m, Radar Cross Section 100 cm², Min Detectable Signal 22 W, the result is 0.0335.

Calculator

Result
—

Formula

R_max = (P_t × G_t × G_r × λ² × σ / ((4π)³ × S_min))^(1/4)

Description

The radar range equation determines the maximum detection range of a radar system. The transmitted power spreads over a sphere (1/R²), reflects off the target (with radar cross section σ), spreads again on the return path (another 1/R²), and must exceed the receiver minimum detectable signal. The fourth-root dependence means that doubling the range requires 16× the transmit power. This equation is fundamental to radar system design for military, weather, aviation, automotive, and maritime applications.

Variables

  • P_t — Transmitter peak power (W)
  • G_t — Transmit antenna gain (linear, not dB)
  • G_r — Receive antenna gain (linear, not dB)
  • λ — Wavelength (m)
  • σ — Target radar cross section (m²)
  • S_min — Minimum detectable signal power (W)

Practical Notes

Typical radar cross sections: bird 0.01 m², person 1 m², car 10-200 m², aircraft 1-100 m², ship 1000-100000 m². Antenna gains must be in linear (not dB): G_linear = 10^(G_dB/10). The equation assumes free-space propagation; atmospheric attenuation, multipath, and clutter reduce actual range. To convert S_min from dBm: S_min_W = 10^((dBm-30)/10).

Symbols and units

Symbol Meaning Default unit Accepted units
Pt Transmit Power W µW, mW, W, kW
Gt Tx Antenna Gain — —
Gr Rx Antenna Gain — —
lambda Wavelength m mm, cm, m, km
sigma Radar Cross Section m² mm², cm², m²
Smin Min Detectable Signal W µW, mW, W, kW

Step by step

  1. 1. What you know: Transmit Power 10 W, Tx Antenna Gain 0.5, Rx Antenna Gain 1.1, Wavelength 1 m, Radar Cross Section 100 cm², Min Detectable Signal 22 W
  2. 2. The relation: R_max = (P_t × G_t × G_r × λ² × σ / ((4π)³ × S_min))^(1/4)
  3. 3. Substitute: Pt = 10 W, Gt = 0.5, Gr = 1.1, lambda = 1 m, sigma = 100 cm², Smin = 22 W
  4. 4. Result: Rmax = 0.0335

Worked examples

Minimum range

Given: Transmit Power 10 mW, Tx Antenna Gain 0.025, Rx Antenna Gain 0.055, Wavelength 1 mm, Radar Cross Section 10 mm², Min Detectable Signal 22 mW

Result: Rmax = 4.21 × 10⁻⁵

Low range

Given: Transmit Power 100 mW, Tx Antenna Gain 0.25, Rx Antenna Gain 0.55, Wavelength 10 mm, Radar Cross Section 1 cm², Min Detectable Signal 0.22 W

Result: Rmax = 7.49 × 10⁻⁴

Typical

Given: Transmit Power 10 W, Tx Antenna Gain 0.5, Rx Antenna Gain 1.1, Wavelength 1 m, Radar Cross Section 100 cm², Min Detectable Signal 22 W

Result: Rmax = 0.0335

High range

Given: Transmit Power 1 kW, Tx Antenna Gain 0.8, Rx Antenna Gain 1.76, Wavelength 100 m, Radar Cross Section 1 m², Min Detectable Signal 2.2 kW

Result: Rmax = 1.34

Maximum range

Given: Transmit Power 10 kW, Tx Antenna Gain 8, Rx Antenna Gain 17.6, Wavelength 1 km, Radar Cross Section 10 m², Min Detectable Signal 22 kW

Result: Rmax = 23.83

How Rmax changes with Transmit Power

Transmit Power (input) Rmax (result)
1 W 0.01884
2.5 W 0.02369
5 W 0.02817
7.5 W 0.03118
10 W 0.0335
15 W 0.03708
20 W 0.03984
40 W 0.04738
70 W 0.05449
100 W 0.05958

Embed this calculator

Paste this into any page. It keeps working when this page updates, and credits the source.

<iframe src="https://vbatpower.com/embed/radar-range/" width="100%" height="460" style="border:1px solid #1f2a37;border-radius:12px" loading="lazy" title="Radar Range Equation"></iframe>
<p><a href="https://vbatpower.com/formulas/radar-range/">Radar Range Equation</a> Calculator by vbatpower.com</p>

Frequently asked questions

How do you calculate Radar Range Equation?

Use R_max = (P_t × G_t × G_r × λ² × σ / ((4π)³ × S_min))^(1/4). With Transmit Power 10 W, Tx Antenna Gain 0.5, Rx Antenna Gain 1.1, Wavelength 1 m, Radar Cross Section 100 cm², Min Detectable Signal 22 W, Rmax works out to 0.0335.

How do you calculate radar range?

Rearrange for R_max: with Transmit Power 100 mW, Tx Antenna Gain 0.25, Rx Antenna Gain 0.55, Wavelength 10 mm, Radar Cross Section 1 cm², Min Detectable Signal 0.22 W, R_max is 7.49 × 10⁻⁴.

What is the formula for Radar Range Equation?

The formula is R_max = (P_t × G_t × G_r × λ² × σ / ((4π)³ × S_min))^(1/4). The radar range equation determines the maximum detection range of a radar system.

Related Concepts

All Signal & RF formulas →
Wavelength Free Space Path Loss Signal-to-Noise Ratio Nyquist Sampling Rate Link Budget Antenna Gain Skin Depth Radar Range Equation

Need more features?

Save calculations, import telemetry data, simulate battery discharge, and collaborate with your team.

Try the App