Soft Temporal Switching of Transmission Line Parameters: Wave-Field, Energy Balance, and Applications

Time-varying guiding structures introduce an additional degree of freedom, besides spatial variation, that enables better control over the guided wave in a device. Periodically, time-modulated structures which are usually considered enable wave control over narrowband signals. However, for ultrawideband short-pulse signals, time variation in the form of temporal discontinuities is required. Such a setup has recently been proposed as a means to overcome the Bode-Fano bound on impedance matching. While hard (abrupt) temporal discontinuities are relatively simple to analyze by employing continuity of magnetic flux and electric charge, soft (gradual) temporal switching of the guiding structure parameters is more challenging. This article explores the case of a short-pulse dynamics in a 1-D, metamaterial transmission line (TL), medium with general smooth time variation of its parameters. In this time-varying TL, wave-field solutions are obtained by a Wentzel-Kramers-Brillouin (WKB) approach which is more common in the context of gradual spatial variations. Using this methodology, a leading order transmitted and reflected waves due to time variation are derived, followed by a discussion of the energy balance in such switched media. A canonical example of capacitor discharge into a long time-varied TL is given. These results may be used as analysis/synthesis tools for time-varying wave devices in electromagnetics and acoustics.