Circadian clock is an essential molecular regulatory mechanism that coordinates daily biological processes. Although the underlying design principles of eukaryotic circadian clock have been investigated in great detail, the circadian mechanism in cyanobacteria, the only prokaryote that possesses circadian clock, is not fully understood. In this study, we focus on elucidating the underlying systems property that drives the oscillation of the cyanobacterial clockwork. We apply combined methods of time scale separation, phase space analysis and bifurcation analysis to a model of circadian clock proposed by us recently. The original model is reduced to a three-dimensional slow subsystem by time scale separation. Phase space analysis of the reduced subsystem shows that the null-surface of the Serine-phosphorylated state (S state) of KaiC is a bistable surface and that the features of the phase portrait indicate that the kernel mechanism of the clockwork is a relaxation oscillator induced by positive and negative feedback loops. Bifurcation diagrams together with phase space analysis show that the S state of KaiC is a key component for the protein regulatory network of the cyanobacterial circadian clock.