A small-signal analysis of the electron cyclotron backward-wave oscillator

In an earlier report on the construction and performance of the electron cyclotron backward-wave oscillator, it was shown, through physical arguments, that in an unloaded waveguide supporting the dominant mode, an electron having transverse rotation at its cyclotron frequency will interact with RF fields of approximately equal frequency. This transverse motion will deliver energy to the RF E fields and interact with the RF H fields, thus producing longitudinal bunching. A small-signal analysis is presented in this paper. With the use of the normal mode expansion analysis, the circuit equation is obtained by considering the normal mode in approximate synchronism with the beam. The RF current is computed by considering electron motion under the dc and circuit fields, but neglecting RF space-charge fields. Combining these equations leads to a sixth-order equation of propagation constants. Two waves are far from synchronism and are therefore neglected; the remaining four are two waves which originate from the "fast cyclotron waves" and two waves which originate from the forward and reflected circuit waves. The "fast cyclotron wave" so obtained has a different meaning from the usual definition and is discussed in detail. Theoretical start-oscillation current is found to depend critically on the reflection coefficient at the electron gun end. Proper adjustment of this parameter leads to excellent agreement between the theoretical and experimental start-oscillation currents.