The transient stability of a synchronous machine is analysed by utilizing Lyapunov's direct method. Lyapunov functions of Lur'e-type are developed, which include the effects of first and second-order speed governor. The used Lyapunov functions have been developed essentially by a systematic procedure based on the nonlinear control theory, Kalman's transformation and the second Lyapunov theorem. The paper evaluates the maximum region of asymptotic stability in the state-plane of a swinging synchronous machine while the computed critical clearing times are compared with those obtained by numerical integration. A numerical example is given.