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A linear and non-linear analysis of a crossed-field gyrotron operating in the TE[sub m11] whispering gallery mode is presented. The starting oscillation condition, the effect of the internal conductor in a coaxial cavity on the cut-off frequency of the mode and the spacing between adjacent modes are considered. The effect of the applied voltage on the cyclotron frequency of the electron, the starting oscillation beam power and the electron efficiency are also considered. It is shown that 41% efficiency can be attained for m=2. These results are compareci with those in a cylindrical cavity. [ABSTRACT FROM AUTHOR]