Transmission-Line Resonator
A short piece of transmission line, open- or short-circuited at one end, resonates like a lumped RLC circuit — the basis of distributed filters and oscillator tanks. A half-wave line and a quarter-wave line each act as a series or parallel resonator depending on the termination. This tool gives the resonant length, the unloaded Q set by the line's loss (Q = β/2α), and the equivalent RLC near resonance.
Equations & Parameters ▸
\(\beta=\dfrac{2\pi f_0\sqrt{\varepsilon_r}}{c},\quad Q_0=\dfrac{\beta}{2\alpha}=\dfrac{\pi}{\alpha\lambda_g}\)
short λ/2 & open λ/4 → series RLC; open λ/2 & short λ/4 → parallel RLC
series: \(R=Z_0\alpha\ell\); parallel: \(R=Z_0/(\alpha\ell)\)
short λ/2 & open λ/4 → series RLC; open λ/2 & short λ/4 → parallel RLC
series: \(R=Z_0\alpha\ell\); parallel: \(R=Z_0/(\alpha\ell)\)
| Type | Half-wave or quarter-wave, open- or short-circuited. |
| Z0, f0 | Line impedance (Ω) and resonant frequency (GHz). |
| α | Line attenuation (dB/m). Sets the unloaded Q. |
| εr | Effective dielectric constant, for the guide wavelength. |
Reference: D. M. Pozar, Microwave Engineering, 4th ed., §6.1 (transmission-line resonators).
Inputs
Termination
Ω
ImpedanceGHz
ResonancedB/m
For QGuide λ
Results
Resonator
Resonant length ℓ—
Guide wavelength λg—
Behaviour—
Quality & equivalent RLC
Unloaded Q₀—
Equivalent R—
Equivalent L / C—
Diagram