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Sensors & detection
Radar range equation
Maximum range at which a monostatic radar can detect a target of a given radar cross-section.
- Ref
- SEN-01
- Rev
- v1
- Field
- Sensors & detection
- Result
- Rmax [km]
where λ = c / f. G and Smin enter as linear ratios; the calculator takes them in dB and converts.
Power spreads out on the way to the target, a little is reflected, and it spreads out again on the way back. Because the spreading happens twice, received power falls with the fourth power of range, so range grows only as the fourth root of everything that helps you.
Calculator
- Wavelength λ9.993 cm
- Gain, linear1,000 ×
- Smin1×10−13 W
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Range grows as the fourth root of transmit power, target RCS and 1/Smin. Doubling any one of them adds about 19%. Doubling the range needs 16× the power.
- Antenna gain acts twice. +3 dB of gain gives about 41% more range; +3 dB of transmit power gives about 19%.
- Frequency is a trap. At fixed gain a higher frequency shortens range; at fixed antenna size it lengthens it. Always say which one you are holding constant.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (km) | Local step | Change in result |
|---|---|---|---|---|---|---|
| G Antenna gain | 20 to 45 dBi | 14.98 | 266.3 | 251.4 | +1 dBi | +12.2 % |
| σ Target radar cross-section | 0.01 to 100 m² | 14.98 | 149.8 | 134.8 | +10% | +2.41 % |
| Smin Minimum detectable signal | −120 to −90 dBm | 26.63 | 149.8 | 123.1 | +1 dBm | −5.59 % |
| Pt Peak transmit power | 1 to 1,000 kW | 14.98 | 84.23 | 69.25 | +10% | +2.41 % |
| f Frequency | 1 to 18 GHz | 19.34 | 82.04 | 62.7 | +10% | −4.65 % |
Sweep of Peak transmit power
| Peak transmit power (kW) | Maximum detection range (km) |
|---|---|
| 1 | 14.98 |
| 2.371 | 18.59 |
| 5.623 | 23.06 |
| 13.34 | 28.62 |
| 31.62 | 35.52 |
| 74.99 | 44.08 |
| 177.8 | 54.69 |
| 421.7 | 67.87 |
| 1,000 | 84.23 |
Sweep one variable
Variables and how each one contributes
Maximum detection range
What the equation gives you. Everything else feeds it.
Peak transmit power
Power radiated by the transmitter during the pulse. Range scales as Pt1/4. Quadrupling power gives only 41% more range; doubling range needs 16× the power.
Antenna gain
How strongly the antenna focuses power, relative to an isotropic radiator. Gain acts on both transmit and receive, so it enters squared and range scales as G1/2. +6 dB of gain doubles the range.
Frequency
Operating frequency. Sets the wavelength λ = c / f. Enters through λ. With gain held fixed, range falls as f−1/2. With a fixed-size antenna the gain rises as f², and the net effect reverses: range rises as f1/2.
Target radar cross-section
The effective echoing area of the target, not its physical size. Range scales as σ1/4. A target with 10× smaller RCS (−10 dB) is detected at about 44% less range.
Minimum detectable signal
Weakest echo the receiver can detect at the required detection performance. Range scales as Smin−1/4. A receiver 3 dB more sensitive sees about 19% further.
Worked example with the values above
- Convert to SI. Pt = 100,000 W. G = 1030/10 = 1,000. λ = c/f = 0.09993 m (9.993 cm). Smin = 10(−100 − 30)/10 = 1×10−13 W.
- Numerator: PtG²λ²σ = 9.986×108.
- Denominator: (4π)³Smin = 1.984×10−10.
- Divide to get 5.032×1018, then take the fourth root: 47,360 m = 47.36 km.
Assumptions and limits
- Monostatic radar: one antenna transmits and receives.
- Free-space propagation: no atmospheric loss, ground reflection or ducting.
- Single pulse, no integration, no system losses (real systems divide by a loss factor L ≥ 1).
- Target at the centre of the beam with a fixed RCS.
- Smin is an input. In practice Smin = kT0BF(S/N)min, so it depends on bandwidth, noise figure and the required detection performance.
Common mistakes
- Mixing dB and linear values. The equation needs G and Smin as ratios and powers, not decibels.
- Reading dBm as dBW. They differ by 30 dB, which is a factor of 1000 in power.
- Treating σ as the physical size of the target. RCS depends on shape, material, aspect angle and frequency.
- Using G once. With separate transmit and receive antennas the product is GtGr.
Statistics and uncertainty
Detection is probabilistic. Smin follows from the signal-to-noise ratio needed for a chosen probability of detection and false-alarm rate, and real target RCS fluctuates from look to look (the Swerling models). Treat the result as the nominal range for that design probability, not a hard edge: at that range the target is detected with the stated probability, not with certainty.
Where it is used
- Range budgets for surveillance and fire-control radars.
- Estimating how much a reduction in radar cross-section cuts detection range.
- Comparing S-band and X-band designs, with gain and aperture treated consistently.
Maths you need
- Exponents and roots: the fourth root is the heart of the equation
- Decibels: converting dB to a ratio and back: ratio = 10dB/10
- Wavelength and frequency: λ = c / f, with c ≈ 3×108 m/s