Magnetron with 45° phase shift as power amplifier

Cross-section of a magnetron with 45° between adjacent cavities

In mode n, the phase difference between adjacent cavities of an N-cavity anode is φ = 2πn/N. For φ = 45°, N = 8n, so an 8-cavity anode in the n = 1 mode gives 45°. The pattern repeats every 8 cavities, so there is one rotating spoke. (A 16-cavity anode with n = 2 gives two spokes.)

   8 cavities, phase of each cavity's RF field
              0°
       315° _/ \_ 45°
          /       \
    270° | (cathode) |  90°     B0 axial
          \  spoke  /          E radial
       225° \_ _/ 135°
             180°
   RF in -> cavity 0°      RF out <- cavity 315°

Functioning as a power amplifier

A magnetron is normally an oscillator. It works as a power amplifier (as in a cross-field amplifier, amplitron type) when an RF input is injected into the cavity chain and the drive conditions are set so that it does not oscillate by itself.

1. Fields: a heated cathode at the centre and the anode block give a radial DC field E (anode voltage V₀). A magnet gives an axial field B₀, slightly above the Hull cut-off B₀c = √(8mV₀/e) / [b(1 − a²/b²)]. Electrons drift around the cathode at the E × B speed.
2. Input: the weak RF signal is coupled into one cavity. The cavities are coupled through the interaction space, so the signal moves from cavity to cavity with a 45° phase step. This forms a slow wave that rotates around the cathode at angular velocity ω/n.
3. Synchronism: V₀ and B₀ are adjusted (Hartree condition) so that the electron drift velocity equals the wave's phase velocity.
4. Phase focusing and energy transfer: electrons in the retarding phase of the RF field give potential energy to the wave and move toward the anode. Electrons in the accelerating phase gain energy and return to the cathode. The useful electrons form a spoke that stays locked to the wave and keeps giving it energy.
5. Growth and output: the wave grows as it travels from the input cavity to the output cavity. The amplified signal is coupled out with a loop. A drift section (or a non-resonant gap) between output and input stops the amplified signal from reaching the input again, so no self-oscillation occurs.

Performance: efficiency of about 40–70%, gain of about 10–20 dB, bandwidth of about 10%, high peak power. Used in radar transmitters.

Bunching effect

Bunching is the grouping of electrons into dense packets at regular intervals. In O-type tubes (klystron, TWT) it comes from velocity modulation followed by drift: fast electrons catch up with slow ones. In magnetrons the same idea appears as phase focusing. Electrons in favourable (retarding) RF phase stay in the interaction, while unfavourable ones are removed, so electrons gather into spokes. The bunched electrons give energy to the RF field in step with it.