Cross-field cavity amplifier with 90° phase shift

Cross-field effect

The cross-field effect is the motion of electrons under a DC electric field E and a DC magnetic field B that are perpendicular to each other. The Lorentz force F = −e(E + v × B) bends the electrons into cycloidal paths that drift at speed v = E/B, at right angles to both fields. An RF wave travelling at this drift speed can take potential energy from the electrons. This is the basis of M-type tubes (magnetron, CFA, amplitron).

Cross-field cavity power amplifier with 90° between adjacent cavities

Design choice of cavities and mode

The phase shift between adjacent cavities in mode n of an N-cavity structure is φ = 2πn/N. For φ = 90°:

90° = 360°·n / N  ->  N = 4n
choose N = 8 cavities, n = 2  -> 2 spokes
(N = 4, n = 1 also gives 90° with 1 spoke)

An 8-cavity anode is chosen. The RF pattern repeats every 4 cavities (0°, 90°, 180°, 270°), and two electron spokes are formed.

   8-cavity anode, phases (n = 2)
              0°
       270° _/ \_ 90°
          /       \
    180° | (cathode) | 180°   B0 axial
          \ 2 spokes/        E radial
       90°  \_ _/  270°
              0°
  RF in -> cavity 1        RF out <- cavity 7
  (drift section between out and in)

Design steps (outline)

1. Choose the anode voltage V₀ and anode/cathode radii b, a.
2. Set B₀ above the Hull cut-off: B₀c = √(8mV₀/e) / [b(1 − a²/b²)].
3. Choose V₀, B₀ to satisfy the Hartree condition: the electron drift angular velocity equals the RF wave's angular velocity ω/n = ω/2.
4. Size the cavities to resonate (or form a slow-wave band) at the working frequency, and cut out the coupling between output and input with a drift section so the tube cannot oscillate.

Working principle as a power amplifier

1. The heated cathode emits electrons. They move in the cross fields and drift around the cathode.
2. The input RF is coupled into the first cavity. The cavities, coupled through the interaction space, carry it as a slow wave, with 90° phase step per cavity.
3. Since the drift speed matches the wave speed, electrons stay in a fixed phase relative to the wave.
4. Phase focusing: electrons in the retarding RF phase give potential energy to the wave and move to the anode. Those in the accelerating phase gain energy and return to the cathode. Two spokes form and rotate with the wave.
5. The wave grows from cavity to cavity as the spokes give it energy. The amplified signal is coupled out at the output cavity. The drift section stops feedback, so the device amplifies instead of oscillating.

Performance: efficiency of about 40–70%, gain of about 10–20 dB, about 10% bandwidth, high peak power. Used as radar transmitter output stages.