  Virtual synchronous machines are inverters with a control algorithm that
causes them to behave towards the power grid like synchronous generators. A
popular way to realize such inverters are synchronverters. Their control
algorithm has evolved over time, but all the different formulations in the
literature share the same "basic control algorithm". We investigate the
equilibrium points and the stability of a synchronverter described by this
basic algorithm, when connected to an infinite bus. We formulate a fifth order
model for a grid-connected synchronverter and derive a necessary and sufficient
condition for the existence of equilibrium points. We show that the set of
equilibrium points with positive field current is a two-dimensional manifold
that can be parametrized by the corresponding pair $(P,Q)$, where $P$ is the
active power and $Q$ is the reactive power. This parametrization has several
surprizing geometric properties, for instance, the prime mover torque, the
power angle and the field current can be seen directly as distances or angles
in the $(P,Q)$ plane. In addition, the stable equilibrium points correspond to
a subset of a certain angular sector in the $(P,Q)$ plane. Thus, we can predict
the stable operating range of a synchronverter from its parameters and from the
grid voltage and frequency. Our stability result is based on the intrinsic two
time scales property of the system, using tools from singular perturbation
theory. We illustrate our theoretical results with two numerical examples.
