The mitigation of bridge cable vibration has long been a research hotspot with substantial developments contributed to the field. In addition to vibration control, a recent development trend incorporates auxiliary sensors, local computing modules, and data transmission devices to establish a comprehensive, integrated cable control and health monitoring system, which requires external energy input. However, an external power supply for these devices is not considered an attractive option, considering that the kinetic energy embodied in cable vibrations can be potentially harvested to cover such power demand, leading to the establishment of an independent self-powered control and health monitoring system. In this regard, we proposed an unprecedented solution for cable vibration mitigation by developing a novel H-bridge-based electromagnetic inerter damper (HB-EMID) in this work, in which HB-EMID can emulate the control behavior of an inerter damper and possess an energyharvesting function. Meanwhile, the newly proposed HB-EMID is granted with great flexibility that can alter its equivalent mechanical properties by merely adjusting the corresponding coding. Following the introduction of the system topology and working mechanism, this study applies an HB-EMID to a cable structure and systematically investigates the balanced control and energyharvesting performances, as well as its feasibility to full-scale application. Both satisfactory control performance and sufficient harvested power are confirmed through numerical validation.