To achieve broadband energy harvesting under weak excitations, this paper comprehensively investigates a novel tri-magnet levitating bistable electromagnetic energy harvester with variable potential wells (BEEH-VP). The bistable nonlinear mechanism is realized using the single-sided bipolarity of the ring magnets. Auxiliary springs are integrated into the ring magnets to dynamically self-adjust the spacing of the ring magnets, causing variable potential wells. The analytical model of BEEH-VP is derived and verified experimentally. The performance of the proposed BEEH-VP is compared with that of the bistable electromagnetic energy harvester with featuring fixed potential wells (BEEH-FP). The comparative results clearly demonstrate that the BEEH-VP exhibits superior capabilities in terms of energy output under weak excitation and possesses a broader frequency bandwidth. Bifurcation analysis for the excitation conditions reveals that the proposed structure features complex dynamic behaviors, such as intra-well, chaotic, harmonic inter-well and subharmonic inter-well vibrations. In comparison to BEEH-FP, BEEH-VP requires lower acceleration to perform the snap-through and has a larger vibration amplitude, facilitating improved output performance.