A Conversion Matrix Approach for Modeling Time-Modulated Circuit Elements with Periodically Varying Modulation Index

Space-time-modulated systems are known to exhibit extraordinary physical phenomena, such as parametric amplification, non-reciprocal transmission and reception, harmonic generation, temporal aiming, anti-reflection coatings, etc. In its simplest form, such material systems are implemented in practice by utilizing lumped discrete time-varying circuit components. Amongst numerous techniques existing in the literature, the Conversion Matrix Method (CMM) is a popular choice for analyzing time-varying circuits when the circuit parameters (such as inductance, capacitance, resistance) conform to a periodic variation. In this work, we introduce a new class of time-varying circuits whose amplitude variation itself is a slow periodic function of time. This implies that such a form of temporal modulation results in a slowly varying amplitude-modulated envelope function. We demonstrate the usefulness of such circuits in their ability to distribute the electromagnetic energy into a newer set of harmonics, which is otherwise not possible with conventional time-modulated circuits. In addition, these capabilities offer a designer with enhanced degrees of freedom to synthesize and engineer several different harmonics of interest. Further, we extend the conventional CMM formalism and propose a new matrix representation to characterize the behavior of these newly introduced forms of temporal modulation.