Plasma synthetic jet actuators (PSJAs), capable of producing high-velocity pulsed jets at high frequency, are well suited for high-Reynolds-number subsonic and supersonic flow control. The effects of energy deposition and actuation frequency on the formation and evolution characteristics of plasma synthetic jets (PSJs) are investigated in detail by high-speed phase-locked particle imaging velocimetry (PIV). Increasing jet intensity with energy deposition is mainly contributed by the increasing peak jet velocity ($U_{p}$), while decreasing jet intensity with actuation frequency is attributed to both the reduced cavity density (primary factor) and the shortened jet duration (secondary factor). The total energy efficiency of the considered PSJA ($O(0.01\,\%)$) reduces monotonically with increasing frequency, while the time-averaged thrust produced by the PSJA is positively proportional to both the deposition energy and the frequency. A simplified theoretical model is derived and reveals a scaling power law between the peak jet velocity and the non-dimensional deposition energy (exponent$1/3$). The propagation velocity of the vortex ring attached at the jet front shows a non-monotonic behaviour of initial sharp increase and subsequent mild decay. The peak values for both the propagation velocity and the circulation of the front vortex ring are reached approximately two exit diameters away from the exit. Finally, analysis of the time-averaged flow fields of the issuing PSJ indicates that the axial decay rate of the centreline velocity is proportional to the actuation frequency whereas it is invariant with the energy deposition. The jet spreading rate of the PSJ is found to be higher than steady jets but lower than piezoelectric synthetic jets. Similarly, the entrainment coefficients of the PSJ are found to be twice as high as the values for comparable steady jets.