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 fields， mode hopping， which present the variation of phase difference between the two-beam-wave interactions when changing L_g. To demonstrate the effectiveness of the proposed circuit system in producing the 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 TM_n1 mode with different n. Simulation results show that 1） the oscillator can deliver two times of the output power obtained from one single oscillator at 220 GHz， 2） the two EIOs can still deliver output signals with phase difference of 0 and π when the currents of the two beams are different or the fabrication errors of the two EIO cavities are taken into account. The proposed scheme is promising in extending to phase locking between multiple EIOs， and generating higher power at millimeter-wave and higher frequencies.