Impedance calculations for traveling-wave gyrotrons operating at harmonics of the cyclotron frequency in circular and rectangular overmoded waveguides

The interaction impedance of traveling-wave gyrotrons has recently been defined for harmonics of the cyclotron frequency. The interaction impedance is a function of waveguide geometry, mode numbers, orbit radius, and orbit location within the guide. This paper treats all of the electromagnetic harmonics in all circular and rectangular waveguide modes. Analytic expressions for the magnitude of the interaction impedance are developed, and are evaluated by both numerical and exact analytic methods. Results show that the impedance of any single harmonic over the waveguide cross section may be expressed as a linear combination of a small number of simple entire-domain functions, the corresponding coefficients of which are directly calculated by numerical integration. Since the coefficients of a particular harmonic are expressible as a function of only two variables, a three-dimensional plot clearly illustrates the optimum combination of parameters. Contours of interaction impedance as a function of position in the waveguide are shown and optimum geometrical arrangements to achieve highest impedance values for particular harmonics are demonstrated.