Matching theory and PIC simulation for klystron input cavity with microwave beam

The matching theory based on an equivalent circuit model is outlined that self-consistently determines the modulation of a klystron input cavity for an arbitrary coupling of the input microwave to the cavity and arbitrary cavity parameters. The model including a mutual inductance, a beam equivalent capacitance and an equivalent resistance is established, along with two expressions for the input power and the reflected power. An expression is derived for the power required to maintain a desired cavity gap voltage for the case when the coupling is perfectly matched to the input microwave, and also for the case of arbitrary coupling. Expressions for the frequency of the input microwave and externally-loaded quality factor (Q) leading to the matching condition are also derived. When the complex coupling coefficient equals 1, expressions are made for the beam-loaded Q and externally-loaded Q. It is concluded that when the complex coupling coefficient equals 1, the operating frequency equals to the resonant frequency of the input cavity, the beam-loaded Q equals to the externally-loaded Q, and the beam equivalent capacitance is much less than 1 will lead to the approximate matching condition. This conclusion is consistent with the traditional theory. By comparing the input power, it is concluded that the input power corresponding to the matching condition is less than the input power corresponding to the approximate matching condition. When the complex coupling coefficient equals 1, comparison shows that if the beam capacitance is neglected, the input power corresponding to the new theory equals to the input power corresponding to the traditional theory. By the comparison of the 2D PIC simulation and the matching theory, we can get the approximate value of the beam impedance. We have found the frequency of the input microwave, the externally-loaded Q and the minimum input power according to the matching theory.