Demo — Range Equation Explorer

Demo — Range Equation Explorer#

The radar range equation has six knobs, and all of them sit under a fourth root. This demo turns the equation into a slider rig so you can feel the 1/4-power “efficiency”: a big change in any term produces only a quarter of that change (in dB) in detection range.

The equation#

\[ R_\text{max} = \left[\frac{P_t\, G_t\, G_r\, \lambda^2\, \sigma}{(4\pi)^3\, S_\text{min}}\right]^{1/4} = K\,\sigma^{1/4}. \]

A 12 dB change in any single term moves \(R_\text{max}\) by only 3 dB — a factor of 2 in range.

Interactive demo#

Open in full screen

Walkthrough#

  1. Start from the defaults (an S-band acquisition set) and read \(R_\text{max}\) — 474 km for a 1 m² target, the reading’s worked example.

  2. Drop \(\sigma\) by 12 dB. Watch \(R_\text{max}\) halve. The side panel reports your most recent adjustment step (not the cumulative drag), and every step obeys \(\Delta R_\text{max} \approx \Delta\sigma/4\) — the quarter-efficiency made explicit, increment by increment.

  3. Add 12 dB of transmit power \(P_t\). The range only doubles. To double range you needed a 16× power increase — the radar designer’s burden.

  4. Drag the \(\sigma\) slider and watch the marker ride the curve. The \(R_\text{max}\)-vs-\(\sigma\) plot is the L1 fourth-power law; the marker tracks your slider, and the dashed reference line marks the 474 km worked example.

  5. Toggle “Show L1 \(K\sigma^{1/4}\) overlay.” The Lesson 1 approximation lies on top of the full equation — confirming \(K\) just bundles every non-\(\sigma\) term.

  6. Try to double range using only \(S_\text{min}\). You need a 16× (12 dB) sensitivity improvement — far harder than it sounds, since the noise floor sets the limit.

Key observations#

  • Every term is fourth-rooted. Whatever you change, \(R_\text{max}\) moves by a quarter of that change in dB.

  • RCS is the target’s lever; power and gain are the radar’s — and both run out, but the radar’s run out first in practice.

  • The overlay is a definition, not a proof. \(K\) is defined as every non-\(\sigma\) term collected, so the dashed curve must lie on the solid one. That is the point: \(R_\text{max} = K\sigma^{1/4}\) loses no information — it is bookkeeping, not approximation.

Source#

MATLAB bundle · L3_RadarRangeEquation.m↓

The companion script evaluates the worked S-band example and confirms the 474 km → 84 km detection range collapse for a −30 dBsm target.

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