Calculation of thermal initial velocity spread based on the non-parallel electrode plate model

Microwave electron vacuum devices convert the DC electrical energy carried by electrons into microwave energy through the motion of electrons in a vacuum. This principle determines that the electron beam and the high-frequency field are the two main components enabling microwave tubes to achieve oscillation or amplification. Consequently, the quality of the electron beam naturally becomes a crucial factor affecting the performance and efficiency of microwave tubes. One important metric for evaluating electron beam quality is velocity spread. For gyrotrons, research has shown that a 5% velocity spread can result in an 11% reduction in overall tube efficiency. However, previous studies on quantifying velocity spread have been conducted under relatively ideal conditions. To achieve a more accurate calculation of velocity spread, this study starts with the potential calculation of non-parallel electrode plates and establishes a cathode emission model with a certain inclination angle. Based on the Maxwell-Boltzmann distribution, the model calculates the influence of thermal initial velocity spread in thermionic electron emission. The results are compared with previous studies and electromagnetic simulation software simulations, showing that the non-parallel electrode plate model produces numerical results that are closer to the simulation results than previous models.