This paper introduces an efficient time-domain methodology for the transient and periodic steady-state analysis of the synchronous machine. A state space model of the synchronous machine in phase coordinates is used to accurately represent its transient and steady-state behavior. In the proposed methodology, the transient behavior of the machine is reproduced in detail and once the fault is removed, a Newton technique based on a numerical differentiation (ND) procedure is applied for the acceleration of the convergence of state variables in the time domain to the limit cycle and, thus, for obtaining a fast periodic steady-state solution. The efficiency of the methodology is compared against a brute force (BF) procedure and, in addition, with a widely accepted machine model of an Electromagnetic Transient Program (EMTP-ATP), which is also used for validation purposes.