Radar Range Equation
Maximum detection range for a monostatic radar from transmit power, gain, RCS, and receiver sensitivity — Skolnik's classic point-target form.
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The engineering
The radar range equation is the link budget of the radar world: energy spreads out over 4πR² on the way to the target, the target re-radiates σ worth of it, and it spreads over 4πR² again coming home. Set the received power equal to the receiver's minimum detectable signal and solve for R — that fourth root is the whole personality of radar design. This card uses the deterministic point-target monostatic form (same antenna transmits and receives, so G appears squared).
The R⁴ dependence is brutal and forgiving at the same time: doubling detection range costs 12 dB somewhere in the budget, but a 1 dB error anywhere only moves the range about 6%. Fold detection statistics, integration gain, and pattern/propagation factors into S_min and the loss term — this form knows nothing about probability of detection or Swerling fluctuation. If your answer looks too good, check that S_min already includes the required SNR, not just receiver noise floor.
Where this math comes from
Christian Hülsmeyer patented a ship-collision echo detector in 1904, but nobody wrote down how far it could see. The range equation as an engineering budget emerged from the frantic pre-war radar programs — Robert Watson-Watt's 1935 Daventry demonstration bounced a BBC transmitter off a Heyford bomber, and within months British and American teams were trading power for range on paper before they bent any metal.
The wartime MIT Radiation Laboratory turned the equation into doctrine, and Norton and Omberg's 1947 paper 'The Maximum Range of a Radar Set' formalized every term. Merrill Skolnik's 1962 Introduction to Radar Systems — chapter 2 of which this card implements — made it the first equation every radar engineer learns, with Marcum and Swerling's 1950s detection statistics supplying the honest value of S_min.
- 1904Christian HülsmeyerPatents the telemobiloscope — first radio echo detection of ships.
- 1935Robert Watson-WattDaventry experiment proves aircraft detection by radio; British radar program launches.
- 1947K. A. Norton & A. C. Omberg'The Maximum Range of a Radar Set' formalizes the range equation term by term.
- 1954J. I. Marcum & P. SwerlingDetection statistics for fluctuating targets — the rigorous meaning of S_min.
- 1962Merrill SkolnikIntroduction to Radar Systems makes the range equation chapter 2 of every radar education.
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