In this paper, the design of a nonlinear excitation control of a synchronous generator is presented where the generator is connected to a single machine infinite bus (SMIB) system. An adaptive backstepping method is used to design the excitation controller with an objective of enhancing the overall dynamic stability of the SMIB system under different contingencies. In this paper, two types of contingencies are considered- i) unknown parameters and physical quantities during the controller design process and ii) controller performance evaluation under different system configurations such as three-phase short circuit faults. The adaption law, which is mainly based on the formulation of Lyapunov function, is used to estimate the unknown parameters which guarantee the convergence of different physical quantities of synchronous generators, e.g., the relative speed, terminal voltage, etc. The effectiveness of the proposed scheme is evaluated under different system configurations as mentioned in the second contingency and compared to that of an existing adaptive backstepping controller and a conventional power system stabilizer (PSS). Simulation results demonstrate the superiority of the proposed control scheme over the existing controllers.