Numerous models and formulations have been used to study synchronous machines in different applications. Herein, a unified derivation of the various model formulations, which support direct interface to external circuitry in a variety of scenarios, is presented. A synchronous machine model with magnetizing path saturation including cross saturation and an arbitrary rotor network representation is considered. This model has been extensively experimentally validated and includes most existing machine models as special cases. Derivations of the standard voltage-in, current-out formulation as well as formulations in which the stator and/or the field windings are represented in a voltage-behind-reactance form are presented in a unified manner, including the derivation of a field-only voltage-behind-reactance formulation. The formulations are compared in a variety of simulation scenarios to show the relative advantages in terms of time steps, run time, and accuracy. It has been demonstrated that selection of the formulation with the most suitable interface for the simulation scenario has better accuracy, fewer time steps, and less run time.