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The fundamental mode frequency of A6 magnetron is investigated theoretically by the equivalent circuit method, and an exact theoretical formula as a function of structure dimensions is derived. The theoretical results are compared with the simulation and experimental results, and the relative errors, which are calculated according to the theoretical and simulation or experimental frequencies, are less than 3%. So the validity of the theoretical formula is verified. In addition, this approach can also be used in other configurations but some changes need to be made in the formula according to the specific structures.

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High-current pulsed electron beam sources are the core components of high-power microwave systems. In order to meet the requirements of future applications, one needs to improve the performance of electron beam sources in terms of vacuum insulation, beam transportation, and thermal management. In this paper, we report about our recent progress in the development of high-current vacuum electron beam sources. In order to meet the vacuum maintenance requirements of high-power microwave tubes, a high-electric field ceramic vacuum interface is designed and fabricated based on the ceramic metal brazing technique. In our experiments, a stable operation of the ceramic vacuum interface is demonstrated in the 10 Hz repetition mode with withstand voltage of larger than 600 kV and pulse width of about100 ns. Besides, a cold cathode is developed using SiC nanowires, and an average beam current density of 1.2 kA/cm 2 is achieved under the electric field strength of 90 kV/cm. Compared with traditional velvet cathodes, the characteristics of the SiC nanowire cathode, such as macro-electrical stability, emission uniformity, and operating life have been significantly improved. Furthermore, a high-current electron beam collector has been developed for relativistic backward wave oscillator tubes. A spiral flume is designed in the collector to meet the requirement of both high specific energy and low flow rate. It shows that the withstand heat flow density is in the order of 10 12 W/m 2 , which is suitable for the long pulse and repetitive operation of the system. These results represent a significant step towards the practical application of long-life high-power microwave systems.INDEX TERMS High-current sources, vacuum interface, ceramic flashover, nano-cathode, collector, thermal control, high-power microwave devices.

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