  We study the global asymptotic behavior of a grid-connected constant field
current synchronous generator (SG). The grid is regarded as an "infinite bus",
i.e. a three-phase AC voltage source. The generator does not include any
controller other than the frequency droop loop. This means that the mechanical
torque applied to this generator is an affine function of its angular velocity.
The negative slope of this function is the frequency droop constant. We derive
sufficient conditions on the SG parameters under which there exist exactly two
periodic state trajectories for the SG, one stable and another unstable, and
for almost all initial states, the state trajectory of the SG converges to the
stable periodic trajectory (all the angles are measured modulo $2\pi$). Along
both periodic state trajectories, the angular velocity of the SG is equal to
the grid frequency. Our sufficient conditions are easy to check
computationally. An important tool in our analysis is an integro-differential
equation called the {\em exact swing equation}, which resembles a forced
pendulum equation and is equivalent to our fourth order model of the
grid-connected SG. Apart from our objective of providing an analytical proof
for a global asymptotic behavior observed in a classical dynamical system, a
key motivation for this work is the development of synchronverters which are
inverters that mimic the behavior of SGs. Understanding the global dynamics of
SGs can guide the choice of synchronverter parameters and operation. As an
application we find a set of stable nominal parameters for a 500 kW
synchronverter.
