Cavity Resonator
A closed metal box supports a discrete set of resonant modes with very high Q — the basis of cavity filters, oscillator tanks and wavemeters. This tool gives the resonant frequency of a rectangular or circular cavity for a chosen TE/TM mode, and the unloaded (conductor-limited) Q of the dominant rectangular TE₁₀ℓ mode. Copper cavities routinely reach Q in the tens of thousands.
Equations & Parameters ▸
rect: \(f_{mn\ell}=\dfrac{c}{2}\sqrt{(m/a)^2+(n/b)^2+(\ell/d)^2}\)
circ: \(f=\dfrac{c}{2\pi}\sqrt{(p/a)^2+(\ell\pi/d)^2}\) (p = J-zero for TM, J′-zero for TE)
\(Q_{c}=\dfrac{(kad)^3 b\,\eta}{2\pi^2 R_s\,(2\ell^2a^3b+2bd^3+\ell^2a^3d+ad^3)}\) (TE₁₀ℓ)
circ: \(f=\dfrac{c}{2\pi}\sqrt{(p/a)^2+(\ell\pi/d)^2}\) (p = J-zero for TM, J′-zero for TE)
\(Q_{c}=\dfrac{(kad)^3 b\,\eta}{2\pi^2 R_s\,(2\ell^2a^3b+2bd^3+\ell^2a^3d+ad^3)}\) (TE₁₀ℓ)
| a, b, d | Rectangular cavity width, height, length (mm). Circular: a = radius, d = length. |
| Mode | Mode indices m,n,ℓ (rectangular) or a named circular mode. |
| σ | Wall conductivity (S/m) for the unloaded Q. Copper 5.8×10⁷. |
| Q | Conductor-limited unloaded Q (dielectric assumed lossless). |
Reference: D. M. Pozar, Microwave Engineering, 4th ed., §6.3 (rectangular & circular cavity resonators).
Inputs
a×b×d box
mm
widthmm
rect onlymm
axialTE₁₀₁ dominant
S/m
For QResults
Resonance
Resonant frequency—
Free-space wavelength—
Mode—
Quality factor
Surface resistance Rs—
Unloaded Q (TE₁₀ℓ)—
Diagram