# Demo — Quarter-Wave RAM
A Salisbury screen makes one clean promise: at its design frequency the echo cancels. This demo models that screen as a shorted transmission-line stub, plots its reflection across 2–18 GHz, and lets you watch the $-10$ dB notch snap to a single band — then double the spacer and see it hand $X$-band right back. The lesson is not the null; it is how little of the spectrum any single null covers.
## The model
An air spacer of depth $d$ over the metal skin behaves as a shorted stub with input impedance $Z_\text{spacer} = j Z_0 \tan(\beta d)$, where $\beta = 2\pi f/c$. The matched resistive sheet $R_s$ sits in parallel with that stub, so the surface impedance seen by an incoming wave is
$$
Z_\text{in} = \frac{R_s \cdot j Z_0 \tan(\beta d)}{R_s + j Z_0 \tan(\beta d)},
\qquad
\text{reflect}_{\text{dB}}(d) = 20\log_{10}\left|\frac{Z_\text{in} - Z_0}{Z_\text{in} + Z_0}\right|.
$$
With $Z_0 = R_s = 377\ \Omega$, the reflection drops toward $-\infty$ wherever $\tan(\beta d)$ blows up — at the quarter-wave frequency and its odd harmonics. The two cases are $d_1 = c/(4\times 10\ \text{GHz}) = 7.5$ mm (quarter-wave at $X$-band) and $d_2 = 2 d_1 = 15$ mm.
## Interactive demo
Open in full screen
## Walkthrough
1. **Read the reflection curve.** The vertical axis is reflection in dB; lower is better. The horizontal dashed line sits at $-10$ dB — the 90%-absorbed threshold. Anywhere the curve dips below it, the screen is doing its job.
2. **Find the 7.5 mm notch.** The $d = 7.5$ mm curve plunges to a deep null at 10 GHz and holds $-10$ dB from about 6.3 to 13.7 GHz — one slice of the band.
3. **Double the spacer to 15 mm.** The single $X$-band null splits into two, at 5 and 15 GHz (the odd harmonics), with $-10$ dB coverage running roughly 3.1–6.9 and 13.1–16.9 GHz.
4. **Compare the covered fraction.** Either design protects about 47% of 2–18 GHz and leaves the other 53% wide open. Doubling $d$ buys the low band but silently gives back $X$-band — exactly where fire-control threats live.
5. **Open the "Intercept asymmetry" tab.** A reused L11 view: the radar pays $1/R^4$ to see you while your RWR pays $1/R^2$ to hear it — the detect-before-detected band RAM and shaping are buying into.
## Key observations
- **The null is a promise kept at one frequency.** Between the harmonics the phase geometry drifts and the echo returns; $-10$ dB coverage is only about $\pm37\%$ of $f_0$.
- **Broader is not free.** Doubling the spacer moves the protection, it does not add it — coverage stays near 47%, and the depth (and weight) doubled to get it.
- **No single resonant layer defends VHF-to-$K_u$.** That is why real LO skins go layered (Jaumann), magnetic (iron-loaded), and structural — buying bandwidth with thickness, weight, and maintenance hours.
## Source
MATLAB bundle · L27_QuarterWaveRAM.m↓
The companion script sweeps $f$ from 2 to 18 GHz, builds the parallel-stub impedance for both spacer depths, plots the two reflection curves against a $-10$ dB reference line, and prints the null frequencies and covered bands so the class can compare the 47% either design protects against the 53% it leaves open.