# 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 $\sqrt{2}$ in range.
## Interactive demo
Open in full screen
## Walkthrough
1. **Start from the defaults** (an S-band fire-control set) and read $R_\text{max}$ — around 470 km for a 1 m² target.
2. **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.
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. **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.
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 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.