An analytic theory describing the conversion e-ciency of the electron beam in multi-gap coupled cavity is developed. Through calculating the electronic conversion e-ciencies in three-gap coupled cavity for modes 2…, … and …=2 respectively, the mechanism of the beam- wave interaction in output circuit is discussed in this paper. 1. INTRODUCTION Extended-interaction klystron (EIK) is invented to achieve a large power, wide frequency bandwidth and high gain in the millimeter/sub-millimeter wave length range (1{3). Multi-gap coupled cavity in it is a key technique for enhancing the gain-bandwidth product and power capability. Based on kinematics theory, by calculating the conversion e-ciency of the electron beam in single gap, the beam-wave interaction was analyzed in paper (4). On this basis, the efiects of the distance of gap-gap, R.F. gap voltage on the conversion e-ciency were discussed in paper (5). But the analysis was carried out under the assumption of narrow gaps separated by drift tubes and neglected the beam-wave interaction of other modes in the multi-gap cavity. Because of strong electromagnetic coupling between the adjacent cavities, there are multiple resonant modes which have a very great difierence of the electric fleld intensity on the gap. The mechanism of the beam- wave synchronization and coupling for these modes is the key technology investigation of multi-gap coupled cavity chain. Based on kinematics theory, this paper will present the simulation results of the conversion e-ciencies in three-gap coupled cavity and discuss the efiect of drift tube length on beam-wave interaction for modes 2…, … and …=2.