Electron Trajectories Analysis in Spatial Harmonic Magnetrons Using Epitrochoidal Curve Theory

The manuscript presents the implementation of epitrochoidal curve theory for the study and analysis of electron trajectories in the boundary conditions of the designed spatial harmonic magnetron (SHM). A model is created based on the epitrochoidal curve theory to study the trajectories of a particle/point with derived equations, offering insights into the understanding of magnetrons by a fundamental theoretical framework. The constants of the theory, namely “a,” “b,” “h,” and their correlation, have been analyzed. Constant “a” controls the expansion radius and thus relates to the dc voltage, constant “b” controls the gyration radius and relates to the magnetic field, and constant “h” controls the periodicity and perturbation of the trajectories and thus relates to the RF voltage. The model provides some derived equations enabling the mapping of these constants within the boundary constraints of the designed SHM to understand certain important phenomena in magnetrons and SHMs. The boundary condition solutions from the model provide the initial guesses of constants, which are further optimized by radial expansion study and trajectory analysis. The trajectories of the generated electron hub reveal the dominance of electron backbombardment phenomena at the cathode when the rotational radial vector rtot < rm mean radius, and anode current collection domination when the stationary radial vector rstat > rm mean radius. Thus, the model highlights the significance of the mean radius and optimizes the ratio of radial expansion constants “a” and “b” and the RF constant “h,” which plays a significant role in the working of magnetron physics. Using the model, the optimum values of the parameters for 22 spokes in SHM have been obtained as amax = 1.2342, amin = 0.6996, bmax = 0.0561, bmin = 0.0318, hmax $= \pm 0.2960$ , and hmin $= =\pm 0.0080$ .