Cross-field effect and multicavity klystron amplifier

Cross-field effect

The cross-field effect is the motion of electrons in a DC electric field E and a DC magnetic field B that are perpendicular to each other. The force F = −e(E + v × B) bends the electron path into a cycloid that drifts at speed v = E/B, at right angles to both fields. Electrons moving in step with an RF wave give it their potential energy. This principle is used in M-type devices such as the magnetron, CFA and amplitron.

Multi-cavity klystron amplifier

A multi-cavity klystron is an O-type (linear-beam) microwave power amplifier. It has an electron beam passing through three or more re-entrant cavities separated by drift spaces.

Construction

 cathode  input    idler     idler    output
  gun    cavity   cavity    cavity   cavity  collector
 |>==== [ C1 ] == [ C2 ] == [ C3 ] == [ C4 ] ====|
          ^RF in      drift spaces       v RF out
      <------- axial focusing magnet ------->

• Electron gun (cathode, focusing electrode, anode) gives a uniform beam at v₀ = √(2eV₀/m) ≈ 0.593×10⁶ √V₀ m/s.
• Input (buncher) cavity, one or more intermediate (idler) cavities, and an output (catcher) cavity. Each is a re-entrant resonator with a narrow gap (grids) that the beam passes through.
• Drift tubes between cavities, with no field inside.
• A solenoid or PPM magnet keeps the beam focused.
• A collector to collect the spent beam.

Principle of operation

1. Velocity modulation: the input RF sets up a gap voltage V₁ sin ωt in the buncher. An electron crossing at time t₁ leaves with v = v₀[1 + (β_i V₁ / 2V₀) sin ωt₁], where β_i is the beam coupling coefficient.
2. Bunching: in the drift space, faster electrons catch up with slower ones that left earlier, so the beam becomes density-modulated. The bunching parameter is X = β_i V₁ θ₀ / (2V₀), where θ₀ = ωL/v₀ is the drift transit angle.
3. Intermediate cavities: the bunched current excites each idler cavity. Since the cavity has a high Q, the induced gap voltage is much larger than V₁. It re-modulates the beam and gives tighter bunches. Each extra cavity adds about 15–20 dB of gain.
4. Energy extraction: the catcher sits where the bunches are tightest. The fundamental RF current is i₂ = 2I₀J₁(X), which is largest at X = 1.841 (J₁ = 0.582). The bunches induce a current in the catcher, and its gap field slows them, so kinetic energy is converted to amplified RF. The output is coupled out through a loop or waveguide.
5. The spent electrons hit the collector.

Performance: total gain of 40–60 dB, peak power up to tens of MW, efficiency of 35–50%. Synchronous tuning gives the most gain; stagger tuning gives more bandwidth. Used in radar, UHF TV transmitters and accelerators.