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#
A 12 dB change in any single term moves \(R_\text{max}\) by only 3 dB — a factor of \(\sqrt{2}\) in range.
Interactive demo#
Walkthrough#
Start from the defaults (an S-band fire-control set) and read \(R_\text{max}\) — around 470 km for a 1 m² target.
Drop \(\sigma\) by 12 dB. Watch \(R_\text{max}\) halve. The side panel shows \(\Delta\sigma = -12\) dB → \(\Delta R_\text{max} \approx -3\) dB ≈ \(\Delta\sigma/4\) — the quarter-efficiency made explicit.
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.
Sweep \(\sigma\) across the Plotly curve. The \(R_\text{max}\)-vs-\(\sigma\) plot is the L1 fourth-power law; the vertical marker tracks your slider.
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.
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 proves the abstraction. \(R_\text{max} = K\sigma^{1/4}\) is not a simplification that loses information; it is the full equation with constants collected.
Source#
MATLAB · code/L3_RadarRangeEquation.m↓
The in-class script evaluates the worked S-band example and confirms the 472 km → 84 km detection range collapse for a −30 dBsm target.