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Mode stability analysis in the beam—wave interaction process for a three-gap Hughes-type coupled cavity chain

Based on space-charge wave theory, the formulae of the beam—wave coupling coefficient and the beam-loaded conductance are given for the beam—wave interaction in an N-gap Hughes-type coupled cavity chain. The ratio of the non-beam-loaded quality factor of the coupled cavity chain to the beam quality factor is used to determine the stability of the beam—wave interaction. As an example, the stabilities of the beam—wave interaction in a three-gap Hughes-type coupled cavity chain are discussed with the formulae and the CST code for the operations of the 2π, π, and π/2 modes, respectively. The results show that stable operation of the 2π, π, and π/2 modes may all be realized in an extended-interaction klystron with the three-gap Hughes-type coupled cavity chain.