Quarterwave
(λ/4wave) coaxial resonators are constructed by shorting the center conductor of a coaxial cable to the shield
at the far end of the circuit. The length of the cable is exactly λ/4 at the desired resonant frequency. A short
circuit is transformed to an open circuit a quarter wavelength away, so when the λ/4wave coaxial resonator is part
of an oscillator circuit, it electrically is not even present (Z~∞); however, whenever the frequency of the oscillator
attempts to go above or below the resonator's center frequency (due to load changes, temperature changes, etc.),
the λ/4wave section looks like a low impedance that works to attenuate other frequency components. It acts like
a parallel tuned L/C tank circuit. The advantage of a λ/4wave coaxial resonator over a tuned L/C tank circuit is
the much higher quality factor, "Q."
Coaxial resonators are often made from sections of normal coaxial cable (both flexible and semirigid), but many
form factors are available that are manufactured for specific frequencies and mount on printed circuit boards, or
have coaxial connectors installed on both ends. Amateur radio hobbyists often use λ/4wave resonators to filter
(trap) interference outside the working band of the radio operation. The nice thing about λ/4wave trap it that
is can be connected inline with the antenna feed cable using a simple "T" adapter (remember, it is effectively not
even there at the center frequency).
Quality factor of a λ/4wave coaxial resonator is determined by the ratio of the center frequency to the 3 dB
power bandwidth, as shown in the figure below. A set of equations is then presented for calculating the necessary
λ/4wave coaxial resonator parameters.
Quality Factor (Q) of a λ/4Wave Coaxial Resonator
Dimension Reference for λ/4Wave Coaxial Resonator Equations

Qc
Qcc
Qd Q
a b f μ σ ε_{r} n Z0
Zopt

= conductor contribution to
unloaded Q = conductor contribution to unloaded Q for copper conductors with optimum b/a
ratio of 3.59112 = dielectric contribution to unloaded Q = unloaded quality factor of a λ/4wave resonator,
including conductor and dielectric losses = outside radius of inner conductor (m) = inside radius of outer
shield (m) = resonant frequency (Hz) = permeability of the conductor (H/m) = conductivity of the conductor
(mho/m) = relative dielectric constant between a and b = resonator mode (# of 1/4 wavelengths at resonance)
= characteristic impedance of coaxial structure (Ω) = characteristic impedance that yields the highest
theoretical value of Q (b/a = 3.59112) λ

Note: These are commonly available equations; my source was Vizmuller
