A simultaneous change of voltage magnitude and grid impedance may occur during grid faults. To reproduce such impedance variation by converter-based grid emulators, the virtual impedance (VI) control and the virtual admittance (VA) control are two typical approaches. Their dynamics when used with the L-filtered grid emulator are evaluated in this paper. Two major issues, i.e., the inaccuracy of emulated low-frequency $(< 50\ \text{Hz})$ impedance by the VI control, and the adverse interactions between VA control and its inner current control. Simulation and experimental results of an L-filtered voltage source inverter confirm the correctness of analysis.