This paper investigates the current sensorless control of bidirectional converters operating both in continuous (CCM) and discontinuous conduction mode (DCM). Often the current control loop is the inner part of a nested control system. For a specific application of such a nested structure, we design two extended state observers (ESO) for the voltage and current control loops, respectively. Our specific application is a power filter (ripple eliminator) connected to a DC bus. For the outer voltage loop, the ESO is to estimate the DC bus disturbance; whereas for the inner current loop, the inductor current and aggregate disturbance caused by plant uncertainty are estimated by another ESO. The controllers are designed based on the active disturbance rejection control (ADRC) principle. Since no current sensors are necessary, the cost and size of the converter can be reduced without sacrificing performance. To improve the efficiency, the converter works under DCM near zero-current crossings. Some modifications to the observer are proposed to guarantee seamless transitions between the two modes. The proposed algorithm is verified by simulation of a 1kW bidirectional converter.