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In this paper, techniques for analyzing transient stability of power systems with field flux decays via Lyapunov's direct method have been developed. First, Popov's stability criterion has been generalized to one for systems with multi-argument non-linearities, and a Lur'e type Lyapunov function has been constructed in a systematic method. Next, the transient stability region has been defined, and its variation has been investigated in detail. As a result, the second type of instability has been introduced. By a numerical example, stability characteristics of the power system have been illustrated, and the validity of the Lyapunov function has been proved.