As a promising current sensor for high-frequency pulsed current monitoring, printed circuit board (PCB) Rogowski coils demonstrate distinct advantages, including extended bandwidth capability and inherent immunity to magnetic saturation. However, their practical implementation is constrained by an intrinsic trade-off between bandwidth optimization and sensitivity enhancement. To address this challenge, a high-sensitivity, wide-bandwidth current sensor is designed in this paper. Through analysis of the PCB Rogowski coil structure and the distribution parameter model of the coil, an optimal sampling resistance value is selected, and a new type of second-order RC integration circuit is designed. In addition, the effects of integrating resistance, integrating capacitance, the number of turns in the coil, the thickness of the PCB, and the inner and outer diameters of the coil on the operating bandwidth and sensitivity of the sensor are analyzed and investigated to determine the optimal parameters of PCB Rogowski coils and provide a basis for the selection of coil parameters. Finally, a multi-PCB Rogowski coil split-board output method is proposed to improve the sensitivity and measurement accuracy of the sensor while keeping its operating bandwidth unchanged. Simulation and experimental test results show that the multi-PCB Rogowski coil split-board output method can improve the sensitivity and measurement accuracy of the sensor while keeping its operating bandwidth constant.