Large transient inrush fault current in a synchronverter during a short-circuit fault can potentially damage the inverter and cause grid instability. It is therefore important to study the characteristics of and the way to restrain the inrush fault current. This paper investigates the characteristics of the synchronverter inrush fault current and proposes a new control method based on mode switching to protect synchronverters. First, an ideal voltage source inverter instantaneous inrush current calculation method for a grid symmetrical short circuit is discussed. Then, based on the inertia of synchronverter, the approximate calculation of the synchronverter instantaneous inrush current is described. The results show that the synchronverter's inrush current mainly consists of a gradually attenuated periodic component and a dc component when the phased voltage symmetry is lost. In our proposed new control approach, the synchronverter switches to the hysteresis controller to limit the output current quickly while staying connected to the grid, and supporting the grid voltage simultaneously. Simulation and experiment results verify the validity of the theoretical analysis.