Solitary-wave dynamics in an R2 LC nonlinear transmission line with voltage bias

It was recently established that when an LC transmission line with a Schottky varactor is connected to a voltage terminal, a pulse is formed whose amplitude grows exponentially leading eventually to its instability after some finite propagation time. In the present study an R2LC nonlinear transmission line, consisting of a lumped circuit with unit cells composed of a linear inductance and a linear resistance in the primary branch, and a Schottky varactor shunted with a linear resistance in the secondary branch, all driven by a voltage terminal, is examined. Our objective is to investigate the effects of the competition between the two resistances, on the soliton amplification and propagation characteristics in general, along the nonlinear transmission line. The study is carried out in two steps: first, an analytical study is carried out within the framework of the reductive perturbation theory. This first step leads to a perturbed Korteweg–de Vries equation, in which the two resistances as well as the voltage terminal are regarded as perturbations. It is found that the electrical pulse amplitude is either exponentially amplified or exponentially depressed, depending on a specific relationship between the two resistances and the voltage terminal. Next the discrete line equations are directly solved numerically, assuming the analytically obtained pulse as input signal. This second analysis suggests a complex behaviour of the electrical pulse amplitude, in response to the influence of the two resistances. A particularly interesting behaviour observed in numerical simulations is the decay of a single-pulse soliton with propagation, and its burst into two-pulse and eventually multi-pulse soliton patterns after some finite propagation time. The proposed nonlinear transmission-line model can find widespread applications in complex transmission networks requiring high-power signals, as for instance in wideband and microwave digital signals, wireless, radar and sensor array processings.