In this paper, transient stability of power systems with structure preserving models is considered. A Hamiltonian function which
can be regarded as a Lyapunov function for the system is proposed. Based on this, the influence of flux decay dynamics, especially during
a fault, on transient stability is analyzed. With the increase of load power, the variation of stability boundary in the rotor angle/E'q plane
is shown. The Energy-based excitation control, aiming at injecting additional damping into the post-fault system may reduce the critical
clearing time (CCT). This can be demonstrated by the comparison of different flux decay dynamics in the fault-on condition, and the reason
is illustrated by the relationship between rotor angle/E'q and the stability boundary. An improved control strategy is proposed and applied to
increase the CCT. Simulation results verify that improvement is obtained both in transient stability and dynamic performance.