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.