The concept of Virtual Synchronous Machines (VSMs) is emerging as a flexible approach for controlling power electronic converters in grid-connected as well as in stand-alone or microgrid applications. Several VSM implementations have been proposed, with the emulation of inertia and damping of a traditional Synchronous Machine (SM) as their common feature. This paper investigates a VSM implementation based on a Voltage Source Converter (VSC), where a virtual swing equation provides the phase orientation of cascaded voltage and current controllers in a synchronous reference frame. The control system also includes a virtual impedance and an outer loop frequency droop controller which is functionally equivalent to the governor of a traditional SM. The inherent capability of the investigated VSM implementation to operate in both grid-connected and islanded mode is demonstrated by numerical simulations. Then, a linearized small-signal model of the VSM operated in islanded mode while feeding a local load is developed and verified by comparing its dynamic response to the time-domain simulation of a nonlinear system model. Finally, this small-signal model is applied to identify the dominant modes of the system and to investigate their parametric sensitivity.