Since the large fault inrush current threatens the safety of power electronic devices, the synchronverter should have the fault ride-through (FRT) capability and should be equipped with a reliable protection system. Therefore, it is important to study the transient fault characteristics of the synchronverter and propose its fault current estimation method. In this paper, by establishing the equivalent model of synchronverter, an asymmetric fault current analysis method using the instantaneous symmetric component method is proposed. In addition, the current components, peak value, peak time, time required for reaching two times of the rated current, and main influencing factors of the asymmetric fault current are also analyzed. It is found that the influencing factors mainly include the control loops of synchronverter, asymmetric fault type, line impedances, and fault occurring moment. Moreover, the time required to reach two times of the rated current is approximately within the range of one hundredth of a microsecond to five milliseconds. These fault current characteristics impose strict requirements on the response speed of FRT control strategies. Finally, the simulation and experiment results verified the correctness of the proposed analysis method.