Radar Range Equation
Received echo power for a monostatic radar, combining transmit power, antenna gain, target size and range.
Formula
Variables
Example
Doubling range cuts received power by 2⁴ = 16× (−12 dB)
How it works
This equation gives the power that a monostatic radar receives back from a target after the transmitted wave spreads out, reflects, and spreads out again. It combines two-way free-space spreading loss, antenna directivity, wavelength-dependent capture efficiency, and the target’s radar cross-section, which represents how strongly it reflects energy back toward the radar. The R^4 dependence comes from the signal losing strength on the outbound path and then again on the return path.
When to use it
Use this when estimating echo strength for a single-target monostatic radar in free space, especially for detection-range calculations and link-budget style checks. It is also useful for comparing how changes in antenna gain, wavelength, or target reflectivity affect received power.
Worked example, step by step
Given: P_t = 2,000 W, G = 35, λ = 0.03 m, σ = 10 m^2, R = 40,000 m Step 1: Square the gain and wavelength G^2 = 35^2 = 1,225 λ^2 = 0.03^2 = 0.0009 Step 2: Compute the numerator P_t G^2 λ^2 σ = 2,000 × 1,225 × 0.0009 × 10 = 22,050 Step 3: Compute the denominator (4π)^3 R^4 ≈ 1984.4 × (40,000)^4 (40,000)^4 = 2.56 × 10^18 Denominator ≈ 1984.4 × 2.56 × 10^18 = 5.08 × 10^21 Step 4: Divide P_r = 22,050 / (5.08 × 10^21) ≈ 4.34 × 10^-18 W Final answer: The received echo power is about 4.3 × 10^-18 W (extremely weak).
Common mistakes to avoid
- Using the formula for bistatic radar without modification; this form is for monostatic radar only
- Forgetting that range is to the target and the R^4 term makes small distance errors matter a lot
- Mixing up wavelength and frequency, or using σ as a physical area instead of radar cross-section
Did You Know?
The brutal R⁴ dependence means detecting a target at twice the range needs sixteen times the power — and is exactly why stealth aircraft chase tiny radar cross-sections instead.
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