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An analysis of an electron beam which interacts with a chain of coupled resonators is presented. Several important characteristics of traveling-wave tubes which employ periodic slow-wave circuits are described. It is found that, even for a lossless circuit, the gain does not become large near either pass band edge although the interaction impedance does become very large. Furthermore, useful amplification is found to occur outside the normal circuit pass band, particularly when the frequency is below the low-frequency cutoff where the circuit presents an inductive reactance to the beam. The problem of matching uniform transmission lines to the periodic circuit is discussed from the equivalent circuit point of view and it is shown that the terminating impedance which produces no reflection from the output end of the circuit when the beam is present may be appreciably different from that required when the beam is absent. The method of analysis applies to spatial harmonic operation, including backward spatial harmonics, as well as to synchronously tuned multicavity klystrons.