Simple derivation of the expression for the synchronous field in circular cylindrical cavity gyrotrons

It is shown that the expression for the synchronous field at any point in a circular cylindrical cavity can be easily obtained using Graf´s formula. The result is applied to the calculation of cavity-beam coupling impedances. Results for the ratio of coupling impedances in tubes using magnetron injection guns and tubes in which the guiding centre of the electron beam is on the cavity axis are given. For operation at half the cyclotron magnetic field, the coupling impedance is much higher for tubes of the second type.