The present paper describes a relativistic non-linear theory of waveguide gyrotrons with the irregular electrodynamic system and inhomogeneous magnetostatic field, taking into account space-charge quasi-static and dynamic forces effect and an electron velocity angle spread. Efficiency- and amplification band optimized gyrotron variants of various designs with a circular waveguide operating H01 mode and an H11 mode are presented. Optimum waveguide systems and magnetostatic field distribution profiles are determined. The main effects due to the space-charge forces, electron velocity angle spread and dynamic deformation of the amplification band have been investigated. A comparison is drawn between gyrotron amplifiers operatingwith theH01 mode and H11 mode. The technique of synthetizing a gyro-TWT output device by frequency dependence of the operating-mode reflection coefficient hasbeen reported.