In this paper, a scheme of commonly-resonated extended interaction circuit system based on high order TM_n1 mode is proposed to lock the phases of two extended interaction oscillators (EIOs) for generating high power at G-band. Two separate EIOs are coupled through a specific single-gap coupling field supported by a designed gap waveguide with length (L_g), which form the phase-locked EIOs based on the commonly-resonated system. As a whole system, the system has been focused on with mode analysis based on different single-gap coupling field, mode hopping, which presents the variation of phase difference between the two-beam-wave interaction when changing the L_g. To demonstrate the effectiveness of the proposed circuit system in producing phase locking, we conducted particle-in-cell (PIC) simulations to show that the interesting mode hopping occurs with the phase difference of 0 and π between the output signals from two output ports, corresponding to the excitation of the TMn1 mo