This study describes a phase-locking mechanism for a relativistic extended interaction oscillator (REIO) that employs a precisely engineered double-ridge waveguide. The mechanism effectively synchronizes four REIOs operating in 3π mode within an angular array through an equivalent coupling loop design. By optimizing the coupling loop structure, the cutoff frequency is reduced, and the coupling strength is properly maintained, ensuring efficient phase-locking with phase jitter confined within ±1.5°. The collective output power of the four REIOs is nearly four times that of a single free-running REIO, producing 550 MW of 5.68 GHz microwaves with an electronic efficiency of 43.5%.
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