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The analytical expressions of the beam-wave coupling coefficients and the beam-loaded conductance in an N-gap coupled cavity are derived based on space-charge wave theory. Through calculating the relations of the beam-wave coupling coefficient and the normalized beam-loaded conductance to the gap number, beam voltage and perveance for 2π mode, the mechanism of the beam-wave synchronization and coupling in the multi-gap coupled cavity are discussed. The results show that, with the increase of N(≥2), the beam-wave coupling efficiency and the normalized beam-loaded conductance vary with beam voltage more rapidly and there is a maximum value for the absolute squared value of the coupling coefficient |MN|2 and a maximum value and a minimum value for the normalized beam-loaded conductance gb. The magnitudes of these extrema increase with the increase of gap number N, and the corresponding voltage is close to the synchronization voltage. The increase of the perveance could make the voltage difference between two extremums of gb increase, the magnitudes of these extrema decrease, and the beam-wave coupling efficiency fall.

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.

In this paper, the analytical expressions of the beam-wave coupling coefficient and the beam-loaded conductance in the N-gap Hughes-type coupled cavity used in an extended- interaction klystron are derived based on the space-charge wave theory. The stability of the circuit is discussed through calculating the quality factor of the electron beam. The theoretical analyses show that with the increase of N, the stability of operating mode (2π) becomes more sensitive to the beam voltage, and that the parasitical oscillation may more easily occur and is difficult to suppress. In addition, the increase of the perveance and the decrease of the external loaded quality factor may both cause the instability of the system. The electric field intensities on the gap are greatly different among the modes 2π, π and π/2, which may be a new subject for improving the power capability and the bandwidth in klystron development.