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The synchronous machine is one of the critical components of electric power systems. Modeling of synchronous machines is essential for power systems analyses. Electric machines are often interfaced with power electronic components. This work presents an advanced synchronous machine modeling, which emphasis on the modeling and simulation of systems that contain a mixture of synchronous machines and power electronic components. Such systems can be found in electric drive systems, dc power systems, renewable energy, and conventional synchronous machine excitation. 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. Selection of the formulation with the most suitable interface for the simulation scenario has better accuracy, fewer time steps, and less run time. Brushless excitation systems are widely used for synchronous machines. As a critical part of the system, rotating rectifiers have a significant impact on the system behavior. This work presents a numerical average-value model (AVM) for rotating rectifiers in brushless excitation systems, where the essential numerical functions are extracted from the detailed simulations and vary depending on the loading conditions. The proposed AVM can provide accurate simulations in both transient and steady states with fewer time steps and less run time compared with detailed models of such systems and that the proposed AVM can be combined with AVM models of other rectifiers in the system to reduce the overall computational cost. Furthermore, this work proposes an alternative formulation of numerical AVMs of machine-rectifier systems, which makes direct use of the natural dynamic impedance of the rectifier without introducing low-frequency approximations or algebraic loops. By using this formulation, a direct interface of the AVM is achieved with inductive circuitry on both the ac and dc sides allowing traditional voltage-in, current-out formulations of the circuitry on these sides to be used with the proposed formulation directly. This numerical AVM formulation is validated against an experimentally validated detailed model and compared with previous AVM formulations. It is demonstrated that the proposed AVM formulation accurately predicts the system's low-frequency behavior during both steady and transient states, including in cases where previous AVM formulations cannot predict accurate results. Both run times and numbers of time steps needed by the proposed AVM formulation are comparable to those of existing AVM formulations and significantly decreased compared with the detailed model.