Misconceptions about equivalent circuits for periodic microwave structures

This paper seeks to prove that an analysis of a slow wave structure coupled to an electron beam as based on a likely "equivalent" circuit may not agree with a field analysis, in which a solution of Maxwell\´s equations leads to a different equivalent circuit. We first describe a typical slow wave structure by an "equivalent" circuit for the beam-circuit interaction. Then we solve Maxwell\´s equations for one of the cavities with its fields expanded in the one resonant -mode of the "cold" structure. We obtain an equivalent circuit for the electric and magnetic field amplitudes, as excited by the beam, which differs from the first "equivalent" circuit. We then solve the beam equations, excited by cavity electric field and subject to the periodicity condition. The result is a determinantal equation for the allowed phase shift per period, which is checked by a solution of Maxwell\´s equations with the fields expanded in the one resonant 0-mode of the "cold" structure. The growing waves near cutoff by our field analysis differ significantly from these waves in the likely "equivalent" circuit. We conclude that only the field analysis and its unambiguous equivalent circuit is reliable. Our analysis corresponds very generally to using a variational expression for , in which the trial fields are expanded in sets of resonant cavity modes.