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In this paper, we study the scattering and diffraction phenomena in time-modulated metamaterials of metallic nature by means of Floquet equivalent circuits. Concretely, we focus on a time-periodic screen that alternates between "metal" and "air" states. We generalize our previous approaches by introducing the concepts of "macroperiod" and "duty cycle" to the time modulation. This allows to analyze time-periodic metallic metamaterials whose modulation ratios are, in general, rational numbers. Furthermore, with the introduction of the duty cycle, perfect temporal symmetry is broken within the time modulation as the time screen could remain a different amount of time in metal and air states. Previous statements lead to an enrichment of the diffraction phenomenon and to new degrees of freedom that can be exploited in engineering to control the reflection and transmission of electromagnetic waves. Finally, we present some analytical results that are validated with a self-implemented finite-difference time-domain (FDTD) approach. Results show that the scattering level and diffraction angles can be controlled independently by means of the duty cycle and the modulation ratio, respectively. Thus, novel time-based pulsed sources and beamformers can be efficiently designed