5. Cavity Resonators // Microwave Engineering // bhaswanth

5.1 Closing the ends

Take a rectangular waveguide section of length $d$ and short both ends with conductive walls. The waves bouncing back and forth between the shorts set up standing waves. At specific frequencies, the standing-wave pattern fits perfectly: the cavity resonates.

The resonant frequencies for a rectangular cavity of dimensions $a \times b \times d$ supporting mode TE $_{mnp}$ or TM $_{mnp}$ are

$\boxed{f_{mnp} = \frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2 + \left(\frac{p}{d}\right)^2}}$

The $p$ index counts half-wavelengths along the cavity length. Compared to the open waveguide formula, an extra discrete dimension has been quantized.

Bell analogy. A bell rings at a discrete set of frequencies determined by its size, shape, and material. Strike it, and only those frequencies sustain; everything else dies out within a few oscillations. A microwave cavity is exactly the same idea: hit it with broadband noise, only the resonant modes ring, and they ring for a long time at fixed frequencies that depend purely on the cavity's dimensions and the speed of light.

5.2 The Q factor

The "ringing time" is captured by the quality factor Q,

$Q = 2\pi \frac{\text{energy stored per cycle}}{\text{energy dissipated per cycle}}$

For an LC tank with a real inductor, Q rarely exceeds 200 because of ESR in the inductor. A microwave cavity made from polished copper at room temperature can hit $Q = 5{,}000$ to $20{,}000$ . Cooled to liquid-nitrogen temperatures, $Q > 100{,}000$ is achievable. Superconducting cavities for particle accelerators reach Q over $10^{10}$ .

The reason: the only loss mechanism is wall conductor loss. There is no ESR in a wire (because there is no wire), no leakage in a capacitor (because the cavity volume is air or vacuum), and no radiation (because the cavity is closed). The wall current dissipates a tiny amount each pass, and the rest of the energy keeps bouncing.

A high-Q cavity has a narrow resonance: bandwidth $\Delta f = f_0 / Q$ . For $Q = 10{,}000$ at 10 GHz, $\Delta f = 1$ MHz. This sharpness is exploited in:

Frequency standards. A cavity tuned to a known frequency provides a stable reference. Old microwave instrumentation used cavity wavemeters as built-in frequency references.
Narrowband filters. Coupled-cavity filters for radar receivers deliver superb selectivity.
Slow-wave structures. Multi-cavity klystrons and magnetrons use cavity Q to support sustained oscillation.
Atomic clocks. A microwave cavity holds the interrogation field that probes cesium or rubidium hyperfine transitions.

5.3 Excitation: probe, loop, aperture

How do you get energy into a closed metal box?

Probe coupling (E-field): Insert a small wire through a small hole in the wall, oriented along the local $\vec{E}$ direction at the spot. Maximum coupling at $\vec{E}$ -field maxima inside the cavity; zero coupling at $\vec{E}$ -field nulls. The probe is the inner conductor of a coaxial line; the cavity wall is the outer.
Loop coupling (H-field): A small wire loop with its plane perpendicular to the local $\vec{H}$ field. Couples to $H$ -field maxima.
Aperture coupling: A hole between the cavity and an adjacent waveguide. Field components leak through the hole; size and shape determine coupling strength.

Each method couples in (and out, by reciprocity). Practical cavities have two ports: one to feed in, one to extract the resonant signal. The cavity's loaded Q (with both ports) is lower than the unloaded Q (no ports) because the coupling itself dissipates energy.

5.4 Cavity inside a klystron

Cavities are not just filters. Inside a klystron vacuum tube, a re-entrant cavity (where the metal walls poke inward to leave a small gap) provides the high-impedance gap where the electron beam interacts with the RF field. The cavity is doing double duty: it stores energy at the operating frequency, and it provides a localized strong electric field where the beam is modulated. Section 7 covers klystron operation in detail.