Nonlinear Transmission Line: Shock Waves and the Simple Wave Approximation

In recent years, nonlinear transmission lines (NLTLs) have been investigated for high-power radiofrequency (RF) generation. High-power waves produced using NLTLs can be applied in mobile defense platforms and satellite communications as they can reach tens of MW of peak power. Their principle of operations is based on the nonlinearity of the lumped elements (Ls and/or Cs) used in these lines. In this work, a NLTL using commercial ceramic capacitors as nonlinear elements and linear inductors was simulated and built. Experimental and simulated results show good agreement validating the model of ceramic capacitors. The frequency obtained from soliton generation at the output of the line was about 4 MHz with peak power of the order of 1 kW.

Multidimensional non-linear transmission lines (NLTLs) have been studied and explored due to their outstanding capability of pulse generation and compression. In this connection, various types of NLTLs are theoretically investigated and developed including non-linear electrical transmission lines (NLETLs), gyromagnetic NLTLs, dielectric NLTLs, and magnetic NLTL. This study is set to provide comprehensive theoretical foundations, design perspectives, and recent developments of NLTLs for shorter and faster pulse generation and compression. Among them, NLETLs are described in detail with advanced models with comparison to the current state-of-the-art versions. Finally, the past studies of popular NLTL topologies are summarized and also the ongoing technological trends and future development opportunities in the field are briefly described.

The nonlinear transmission line (NLTL) equations are significant nonlinear evolution equations (NLEEs) in nonlinear electrical transmission line (NLETL) regulation. The method is employed to compute some traveling wave patterns of the NLTL equation. The new extended direct algebraic method (NEDAM) is a viable and successful mathematical method to construct the traveling wave patterns of science and engineering problems. The NEDAM is effectively utilized to obtain the traveling wave structures of a considered model in the form of trigonometric and hyperbolic functions containing parameters. The Lie symmetry technique is used to analyze the NLTL equation and compute the Infinitesimal generators. Moreover, we have shown graphically obtained wave profiles by using the different suitable values of the parameters involved. Further, the nonlinear transmission line equation is described through nonlinear self-adjointness, and conserved quantities are computed for each vector.

PurposeThe paper explores the spectral solution of a partial differential equation governing nonlinear transmission lines (NLTLs). It aims to present an efficient simulation technique for a nonlinear fractional-order transmission line model and to show the effect of fractional derivatives on the evolution of solitons and wave packets of increasing spatial frequency.Design/methodology/approachThe paper is concerned with the solution of the equation governing NLTLs for both integer and fractional time derivatives. The spectral method shall be considered with Hermite and Malmquist-Takenaka orthogonal functions and integer and fractional implementations of the implicit trapezoidal rule. While the Hermite functions and Malmquist-Takenaka functions are suitable for solitons, the Malmquist-Takenaka functions are selected as these are superior for approximating wave packets of increasing frequency (Iserles, Luong and Webb, 2023). Furthermore, they possess properties that are favourable from a computational viewpoint.FindingsResults will show that the proposed approach is effective for simulating NLTLs with both integer and fractional time derivatives. In particular, fractional derivatives are shown to have a significant effect on the evolution of wave packets on a nonlinear transmission line.Originality/valueThe method is novel in using Malmquist-Takenaka basis functions for a spectral solution of a nonlinear transmission line equation with both integer and fractional time derivatives and for examining the effect of fractional derivatives on wave packets.

A well-known feature of a propagating localized excitation in a discrete lattice is the generation of a backwave in the extended normal mode spectrum. To quantify the parameter-dependent amplitude of such a backwave, the properties of a running intrinsic localized mode (ILM) in electric, cyclic, dissipative, nonlinear 1D transmission lines, containing balanced nonlinear capacitive and inductive terms, are studied via simulations. Both balanced and unbalanced damping and driving conditions are treated. The introduction of a unit cell duplex driver, with a voltage source driving the nonlinear capacitor and a synchronized current source, the nonlinear inductor, provides an opportunity to design a cyclic, dissipative self-dual nonlinear transmission line. When the self-dual conditions are satisfied, the dynamical voltage and current equationsof motion within a cell become the same, the strength of the fundamental, resonant coupling between the ILM and the lattice modes collapses, and the associated fundamental backwave is no longer observed.

In the first half of the article, interaction between the small amplitude travelling waves (“sound”) and the shock waves in the transmission line containing both nonlinear capacitors and nonlinear inductors is considered. Herein, the “sound” wave coefficient of reflection from (coefficient of transmission through) the shock wave is calculated. These coefficients are expressed in terms of the speeds of the “sound” waves relative to the shock and the wave impedances. In the second half of the article, the dissipation in the system is explicitly included into consideration, introducing Ohmic resistors shunting the inductors and also in series with the capacitors. This allows us to justify the conditions on the shocks, postulated in the first half of the article. This also allows us to describe the shocks as physical objects of finite width and study their profiles, same as the profiles of the waves closely connected with the shocks—the kinks. The profiles of the latter, and in some particular cases, the profiles of the former are obtained in terms of elementary functions.

In this study, we explore the dynamics of breathers and positons in a nonlinear electrical transmission line modeled by the modified Noguchi circuit, governed by the Kundu-Eckhaus equation. Utilizing the reductive perturbation method and a specific transformation, we analyze the influence of different time-dependent linear potentials on these nonlinear wave structures. The analysis is conducted for three representative cases: i) a constant potential, which modifies the orientation and amplitude of breathers and positons, ii) a periodically modulated potential, which transforms them into crescent-shaped structures with unique spatial characteristics, and iii) an exponentially varying potential, which induces asymmetric crescent-shaped waveforms. Additionally, we show that linear potentials significantly influence breather and positon dynamics in the modified electrical transmission line by altering their position and positon amplitude-constant potentials maintain peaks at the origin, periodic potentials shift breathers forward and positons backward, while exponential potentials move breathers backward and positons forward. Our findings highlight the critical role of external modulation in shaping wave propagation, localizing waves, and altering their amplitude, demonstrating its potential for controlling wave dynamics in nonlinear transmission lines. Unlike previous studies that focused on rogue waves, this work provides new insights into the evolution of breathers and positons under external perturbations. The results may have significant implications for applications in electrical transmission networks.

Rational W-shape solitons on a nonlinear electrical transmission line with Josephson junction

Analogue systems are used to test Hawking radiation, which is hard to observe in actual black holes. One such system is the electrical transmission line, but it suffers the inevitable issue of excess heat that collapses the successfully generated analogue black holes. Soliton provides a possible solution to this problem due to its stable propagation without unnecessary energy dissipation in nonlinear transmission lines. In this work, we propose analogue Hawking radiation in a nonlinear LC transmission line including nonlinear capacitors with a third-order nonlinearity in voltage. We show that this line supports voltage soliton that obeys the nonlinear Schrödinger equation by using the discrete reductive perturbation method. The voltage soliton spatially modifies the velocity of the electromagnetic wave through the Kerr effect, resulting in an event horizon where the velocity of the electromagnetic wave is equal to the soliton velocity. Therefore, Hawking radiation bears soliton characteristics, which significantly contribute to distinguishing it from other radiation.

This paper discusses the use of parametric amplification to address the losses in left-handed nonlinear transmission line (NLTL) metamaterials. The paper introduces a new amplifying NIM, where the energy in a pump wave at one frequency is transferred to the energy in a weak signal wave at another frequency, amplifying it. An 8.3 db parameteric amplification was observed in a LH NLTL media when both the pump and the signal waves fall into the LH region. This approach can also be scaled from its current L-band form into THz, infrared, or ultimately visible form. Extending the results to a two-dimensional structure and taking advantage of the orthogonality between the pump and the signal waves that is possible in a two-dimensional system would enable embedded amplification to occur within the NIM, thus overcoming the losses that dominate their performance

We present a description of the spatial evolution of quantized discrete-mode operators along a lossy nonlinear transmission line. The nonlinearity is formed by hundereds or even thousands of Josephson junctions which are placed periodically along a microwave transmission line. Dissipation is added to the system Hamiltonian by coupling the nonlinear transmission line to an Ohmic bath. Using the Hamiltonian of the open quantum system, Heisenberg equations of motion for the discrete mode operators can be derived in terms of quantum Langevin equations. The temporal equations of motion are then translated to the spatial domain to investigate the performance of a nonlinear four-wave-mixing process, while signals propagate along the transmission line.

The wave reflection and transmission in a nonlinear LC transmission line terminated with a resistive load are studied numerically. It is found that the soliton signals in the strongly nonlinear transmission line are almost perfectly absorbed by the linear resistor terminating the transmission line.

The generation and transmission of relatively short duration, broadband, peak pulse power bursts has numerous applications in the communications and defense sectors. In the past, nonlinear transmission lines (NLTLs) have been used to create such broadband pulses. While most NLTL designs are based on nonlinear capacitors or inductors arranged in a ladder network, the method presented here replaces the transmission line with a novel ferroelectric filled waveguide. When a large transient voltage is input to the guide the resulting electric field will cause the polarization field to move into the saturation region. This reduces the effective dielectric permittivity and thus the group velocity of the peak power portion of the wave is faster than all other portions of the pulse. This results in the middle portion of the pulse overtaking the leading edge and “piling-up” energy at the front edge of the pulse, creating what appears to be a temporal compression of the leading edge. The temporal compression results in increased harmonic spectral content. The simulated NLTLs can be fabricated using Substrate Integrated Waveguides (SIW) in low temperature cofired ceramics (LTCC). Closely spaced vias form trenches in the waveguide that are used to create space for the nonlinear dielectric and the trenches are filled with ferroelectric materials using a sol-gel method. The trench dimensions and type of ferroelectric fill material for each layer are determined using a genetic algorithm optimization routine that produces a maximum rise time compression of the input pulse.

The objective of this project is to develop a preprocessor for speech recognition systems operating in noisy environments. The preprocessor, consisting of a nonlinear inhomogeneous transmission line, will be realized in software, although realization in hardware in FY91 should be possible. More specifically we will:1) Develop a nonlinear transmission line preprocessor that accurately simulates the mechanics of the mammalian inner ear at all sound pressure levels.2) Preprocess speech with the nonlinear transmission line and show that there is a substantial improvement in the signal to noise ratio.3) Assess the desirability and feasibility of implimenting either a digital or analog transmission line on a chip and using it as a preprocessor in the CMU, BBN, or MIT DARPA funded speech recognition systems.

We study wave propagation in a nonlinear transmission line with dissipative elements. We show analytically that the telegraphers' equations of the electrical transmission line can be modeled by a pair of continuous coupled complex Ginzburg-Landau equations, coupled by purely nonlinear terms. Based on this system, we investigated both analytically and numerically the modulational instability (MI). We produce characteristics of the MI in the form of typical dependence of the instability growth rate on the wavenumbers and system parameters. Generic outcomes of the nonlinear development of the MI are investigated by dint of direct simulations of the underlying equations. We find that the initial modulated plane wave disintegrates into waves train. An apparently turbulent state takes place in the system during the propagation.