Free Coaxial Cavity Resonator & Filter Tool

1. Principles of Quarter-Wave Coaxial Resonators

In high-power VHF/UHF repeater duplexers, transmitter notch filters, and receiver front-end preselectors, ordinary lumped LC filters fail due to coil resistive losses (low $Q < 200$) and dielectric breakdown.

A quarter-wave ($lambda/4$) coaxial cavity resonator consists of a hollow conductive tube of inside diameter $D$ enclosing a concentric center rod of diameter $d$. One end of the rod is firmly grounded (short circuit), while the other end is open-circuited. At the open end, the standing wave reaches maximum RF voltage, while current peaks at the grounded base.

2. The 76.7-Ohm Optimal Diameter Ratio for Maximum $Q$

The unloaded quality factor ($Q_u$) of a coaxial cavity is governed entirely by skin-effect RF surface resistance in the inner and outer conductors: $$alpha_c propto rac{R_s}{2 Z_0} left( rac{1}{d} + rac{1}{D} ight)$$ Minimizing conductor attenuation yields the classic transcendental ratio: $$rac{D}{d} = 3.5911 implies Z_0 = 60 ln(3.5911) approx 76.7 ext{ }Omega$$ At this specific geometry, the unloaded $Q_u$ is maximized and scales directly with cavity diameter $D$: $$Q_u approx k cdot D cdot sqrt{f_0}$$ A silver-plated $100 ext{ mm}$ ($4 ext{-inch}$) cavity at $146 ext{ MHz}$ achieves an astounding $Q_u > 3,000$!

3. Capacitive Foreshortening & Tuning Screw

A pure quarter-wave rod with a free-space open end would have length $ell = c / (4 f_0)$. In practice, an adjustable tuning screw or capacitive disk creates an end capacitance $C_t approx 1 - 3 ext{ pF}$ to ground. This capacitive loading shortens the resonant rod length according to: $$Z_0 cot(eta ell) = rac{1}{omega C_t}$$ Foreshortening the rod by $10% - 25%$ allows fine screw tuning across the frequency band while keeping the cavity physically compact.