  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
