  Dynamic stability is imperative for the operation of the electric power
system. This article provides analytical results and effective stability
criteria focusing on the interplay of network structures and the local dynamics
of synchronous machines. The results are based on an extensive linear stability
analysis of the third-order model for synchronous machines, comprising the
classical power-swing equations and the voltage dynamics. The article
explicitly covers the impact of Ohmic losses in a linear approximation in power
grids, which are often neglected in analytical studies. Necessary and
sufficient stability conditions are formulated, and different routes to
instability are analysed, yielding concrete mathematical criteria applicable to
all scales of power grids, from transmission to distribution grids, as well as
microgrids. A subsequent numerical study of the criteria is presented, without
and with resistive terms, to test how tight the derived analytical results are.
