This paper presents the application of the cell mapping method to determine the domain of stability of a synchronous machine with excitation control. Excitation control and the synchronous machine dynamics are formulated mathematically and solved by step-by-step integration. At each step, the solution is converted to the form of cells. The cell mapping algorithm is used to find all the asymptotically stable equilibrium points and periodically stable equilibrium states and their domains of stability. Then a compaction scheme is used to plot all equilibrium states and their domains of stability. A description of the cell mapping method, construction of the cell map, compaction scheme, cell mapping algorithm, test system, dynamic equations and the results are presented in this paper. The results show that the domain of stability is expanded due to the introduction of excitation control. Several advantages of the cell mapping approach to find the domain of stability of a dynamic system, such as the simplicity and versatility of the cell mapping algorithm, adaptability of the computer program to any higher order non-linear dynamic system, capability to handle non-linearities in equations, and complete elimination of all the manual calculations, are shown.