Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 2869731, 7 pages

https://doi.org/10.1155/2018/2869731

## Modulation of Pulse Train Using Leapfrogging Pulses Developed in Unbalanced Coupled Nonlinear Transmission Lines

Correspondence should be addressed to Koichi Narahara

Received 18 October 2017; Accepted 14 January 2018; Published 12 February 2018

Academic Editor: Ben T. Nohara

Copyright © 2018 Koichi Narahara. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The leapfrogging pulses in two unbalanced electrical nonlinear transmission lines (NLTLs) with capacitive couplings are investigated for efficient modulation of a pulse train. Due to the resonant interactions, the nonlinear solitary waves in the NLTLs exhibit complementary behaviors of amplitudes and phases called leapfrogging. For maximizing resonance, both solitary waves should have a common average velocity. Sharing the common velocity, the characteristic impedance can still be freely designed for two coupled solitary waves. In this study, we characterize the leapfrogging pulses developed in unbalanced NLTLs having distinct characteristic impedance. Through the soliton perturbation theory and numerical time-domain calculations, it is found that both the leapfrogging frequency and the voltage variations of pulse amplitudes increase as the difference in the characteristic impedance becomes large. These properties can improve the on/off ratio of modulated pulse train.

#### 1. Introduction

In coupled nonlinear systems, the resonant energy exchange can occur between supported nonlinear solitary waves. Through the energy transfer from the leading solitary wave to the trailing one, the leading wave is attenuated, whereas the amplitude of the trailing wave becomes larger than that of the leading wave. In weakly dispersive cases, the velocity of a long-wavelength nonlinear solitary wave increases as its amplitude increases; therefore, the trailing wave overtakes the leading wave. Then, the direction of energy transfer is reversed so that the original order of the two waves is restored. This overtaking is repeated, resulting in oscillatory behavior called leapfrogging [1]. Leapfrogging solitary waves have mainly been investigated on separated pycnoclines [2–6]. When two horizontal pycnoclines are vertically separated by a small amount, leapfrogging occurs between spatially localized disturbances in two pycnoclines. For weak couplings, the leapfrogging solitary waves are well modeled by the coupled Korteweg-de Vries (KdV) equations. Recently, we investigated two identical transmission lines with regularly spaced Schottky varactors coupled via capacitors, called coupled nonlinear transmission lines (NLTLs), and successfully observed leapfrogging phenomena for the nonlinear solitary waves developed in them [7]. Because of easiness in designing both nonlinearity and dispersion separately, the electronic system can characterize leapfrogging waves efficiently.

In addition, leapfrogging can be used to manage traveling electrical pulses. Originally, NLTLs have been used in ultrafast electronic circuits such as a subpicosecond electrical shock generator and a short-pulse amplifier [8–10]. For example, the leapfrogging can be used for the detection of temporal separation between two short pulses inputted to coupled NLTLs. It has been shown that leapfrogging pulses with relatively large amplitudes exhibit nontrivial properties as their initial relative delay varies. The phase and amplitude of leapfrogging pulses depend on the initial delay between incident pulses, such that the pulse amplitude at the output port varies with the initial delay. Accordingly, the temporal delay between two inputted pulses is converted to the pulse amplitude at the output [7]. Another potential of electrical leapfrogging pulses results from their management by the biasing voltage to the varactors. The leapfrogging frequency depends on the biasing voltage, so that the line length required for the pulses on the lines to become maximal also depends on the biasing voltage. Conversely, the pulse amplitude on one of the lines can be managed by the biasing voltage at outputs that are separated from the inputs by a fixed length. The major output is thus switched from one line to the other by varying the bias voltage. Hence, the leapfrogging in coupled NLTLs can provide a novel switching method for the incident pulse by which the incident pulse is selectively output to one of the two ports of the coupled NLTLs [7]. The same mechanism can be used to modulate the inputted pulse train by the biasing voltage. Figure 1 illustrates this. The pulse train inputted to the lines is modulated by to be outputted as . In order to obtain fine modulation efficiency, the modulating signal is applied, such that the pulse amplitude at the output becomes maximal at and it becomes minimal at . The on/off ratio of is uniquely determined by the leapfrogging-pulse dynamics and can become larger than . This means that only small swing of can give sufficient modulation. Including these examples, the key is to maximize the amplitude variation of the leapfrogging pulse.