Demo — Quarter-Wave RAM

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 demo opens on the X-band preset. The 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. The readouts confirm it: fundamental null 10.0 GHz, protected sub-band 6.3–13.7, band coverage 47%.

  3. Double the spacer with the Double it preset (\(d = 15\) mm). The null does not split — it moves. The fundamental drops to 5 GHz, and its third harmonic at 15 GHz slides into the window from above, so the protected sub-bands read roughly 3.1–6.9 and 13.1–16.9 GHz. The verdict box turns red: \(X\)-band has been handed back.

  4. Compare the covered fraction. Both named designs protect about 47% of 2–18 GHz and leave the other 53% wide open, and doubling \(d\) buys the low band while silently giving back \(X\)-band — exactly where fire-control threats live. Now drag the Spacer depth slider slowly between them and watch the coverage readout, because it is not constant: it sags to 33% near 10.5 mm, where one notch has drifted off the top of the sweep and the next has not yet arrived, and over the whole slider it peaks at 59% at 5.5 mm — a single wide notch covering 8.5–18 GHz that leaves everything below 8.5 GHz bare. The percentage is an accident of where the harmonic comb happens to fall inside 2–18 GHz, not a property you can design for. What never changes is the shape of the bargain: one resonant layer buys one comb of narrow windows, and whole bands get nothing.

  5. Mistune the sheet. The Sheet \(R_s\) slider carries the other half of the design. The null is bottomless only when the sheet is matched to free space at \(377\ \Omega/\text{sq}\); pull it to 100 or 800 and the notch never reaches \(-10\) dB anywhere, so coverage falls to 0%. The resistive sheet and the quarter-wave spacer have to be right together.

  6. 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 from 7.5 to 15 mm moves the protection rather than adding it — the fraction lands at 47% both times, each of the two new windows is half the width of the one it replaced, 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.

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