From echo power to detection
The simplified radar equation from Inverse square law tells you how much power comes back. But a radar does not detect on power — it detects on the difference between echo and noise. Every receiver generates noise of its own, and an echo is only useful once it rises far enough above it. The decisive quantity is therefore the signal-to-noise ratio (SNR).
Receiver noise follows from the physics of heat: electrons move randomly, producing a noise power proportional to temperature and bandwidth:
Set echo against noise and solve for distance, and you get the radar equation in the form designers and analysts actually use — the maximum detection range:
The new terms in the denominator matter just as much as the transmitter power in the numerator:
- Bandwidth B: a wider receiver lets in more noise. A radar chooses B as narrow as its pulse allows (B ≈ 1/PW); with pulse compression (Pulse compression and pulse integration) it regains the bandwidth without the noise penalty.
- Noise figure F: a good low-noise amplifier (LNA) directly behind the antenna keeps F low. Every dB lost here costs as much as a dB of transmitter power.
- SNRmin: the higher the required probability of detection (Pd) and the lower the permitted probability of false alarm (Pfa, Threshold, Constant False Alarm Rate and Automatic Gain Control), the more SNR is needed. For a single pulse with Pd = 90% and Pfa = 10⁻⁶ it is around 13 dB.
This is exactly where jamming attacks: a jammer does not raise the echo but the noise in the denominator. As soon as the noise in the receiver rises, Rmax shrinks with the fourth root. The J/S reasoning of Burn-through and the J/S ratio is nothing more than this equation with an extra noise source added.
Losses and fluctuation
In practice no radar reaches its theoretical range. All imperfections are lumped into a single loss factor L, which in dB is simply added up. The main items:
| Loss | Cause | Order of magnitude |
|---|---|---|
| Transmission loss | waveguides, rotary joint, duplexer — out and back | 1 – 3 dB |
| Atmospheric attenuation | absorption by oxygen and water vapour; increases with frequency and range | from negligible (L-band) to many dB (Ka-band, rain) |
| Beam shape loss | the target is not always in the centre of the beam | ~1.5 – 2 dB |
| Processing loss | non-ideal filters, CFAR (Threshold, Constant False Alarm Rate and Automatic Gain Control), digitisation | 1 – 3 dB |
| Fluctuation loss | the target's RCS jumps from scan to scan | at high Pd easily 5 – 10 dB |
That last loss needs explaining. The RCS (Radar Cross Section) of a real aircraft is not a fixed number: small changes in aspect make the reflections from fuselage, wings and engine inlets alternately reinforce and cancel each other. The echo fluctuates. The classic description of this are the Swerling models: from a steady target (Swerling 0) to targets that show a different RCS per scan (Swerling I/III) or per pulse (Swerling II/IV). A fluctuating target needs more average SNR for the same probability of detection, because the radar may catch it in a "trough".
One solution is frequency diversity: the radar works on two (or more) frequencies at once, with two transmitters sharing the same antenna through a diplexer (Cassegrain, monopulse feed and duplexer). On a different frequency the reflection pattern comes out differently, so the chance that the target sits in a trough on both frequencies at the same time is small. Pulse-to-pulse frequency agility gives the same effect (PRI modulation).
Frequency diversity is also an EPM (Introduction): a spot jammer that disrupts one frequency leaves the second channel untouched. An ESM operator who sees two simultaneous emissions with identical PRF and scan is probably looking at one radar with two channels — not at two radars.
Dwell time, hits per scan and integration
A rotating surveillance radar illuminates a target only briefly: just as long as the beam sweeps across it. That dwell time and the number of pulses that hit the target during it — the hits per scan — follow directly from beamwidth, rotation rate and PRF:
A surveillance radar with θB = 1.5°, 12 rpm (72° per second) and PRF = 1,000 Hz. Dwell time = 1.5 / 72 ≈ 20.8 ms; hits per scan ≈ 20.8 × 1 ≈ 21 pulses.
Those pulses are added up: pulse integration (Pulse compression and pulse integration). With coherent integration (phase is preserved, The transmitter: from magnetron to solid-state) the SNR grows by the factor n — 21 pulses give well over 13 dB of gain. With non-coherent integration (amplitudes only) the gain is smaller, roughly n0.8. That is why the single-pulse radar equation is too pessimistic: a real radar detects on the sum of many hits.
The flip side is a hard trade-off: rotating more slowly gives more hits and therefore more range, but refreshes the air picture less often. An early warning radar (Characteristics of the Early Warning (EW) radar) therefore turns slowly and looks far; an approach radar turns fast and looks close.
The dwell time determines how long an RWR or ESM receiver (Radar Warning Receiver (RWR)) sees the main beam: the short "blip" per revolution. From the time between two blips the operator derives the scan rate, from the length of the blip the beamwidth — together with PRF and frequency, characteristics that give away the radar type. And a jammer that only works in the main beam has just that one dwell time per scan to have an effect.
Calculate it yourself: detection range
The calculator below shows how every design choice works through. Try, for example, reducing the RCS from 1 m² (fighter from the front) to 0.01 m² (stealth): the range drops by a factor ⁴√100 ≈ 3.2 — not by a factor 100.
With the default values (100 kW, 35 dB, S-band, 1 m², 1 MHz, 3 dB, 6 dB, 13 dB) the radar reaches about 53 km — for a single pulse. Add the integration gain from Dwell time, hits per scan and integration (lower the required SNR by that gain) and the range moves out considerably. The noise floor of this receiver is around −141 dBW: the echo the radar has to find is smaller than a billionth of a billionth of a watt.
Radartutorial.eu (Christian Wolff) works through the derivation of the radar equation, the losses and model calculations step by step: radar equation and hits per scan.