Nonlinear Electrical Networks Research Articles

Conformable derivative in a nonlinear dispersive electrical transmission network

Conformable derivative in a nonlinear dispersive electrical transmission network

Mathematical modeling of chirped modulated waves along a multi-coupled nonlinear electrical transmission line with dispersive elements

This work presents a mathematical modeling of chirped modulated waves propagating along a two-dimensional nonlinear electrical transmission line consisting of nonlinear dispersive and purely nonlinear capacitive elements. Combining the rotating-wave approximation technique with the perturbation method, we derive a two-dimensional extended nonlinear Schrödinger equation with self-steepening and self-frequency shift, that predicts the formation and propagation of a very rich variety of soliton-like waves including bright, dark, kink, and double-kink soliton-like waves with nonlinear chirp in a multi-coupled nonlinear electrical transmission network with dispersive elements. Using this model equation, parameter domains are delineated in which these modulated waves exist. The parameter domains are found to be strongly dependent on the dispersive elements of the network system. Through exact solutions of the amplitude equation, we investigate the dynamics of chirped modulated waves along the network system under consideration. Our results show that the nonlinear chirp associated with each of the found soliton-like pulses is directly proportional to the wave intensity and is saturated at some finite value as the propagation coordinate approaches its asymptotic value. Also, we show that increasing the number of the network lines decreases the propagating speed of the soliton-like pulses. Most importantly, theoretical analysis show that chirping of found network pulses can be controlled by varying the self-steepening term and self-frequency shift. The validity of analytical results and the robustness of these chirped soliton-like voltage signals which may have important applications in signal processing systems are confirmed by numerical simulations. The results of our study may be launched in long distance telecommunication lines involving higher-order nonlinearities of the fiber.

Modulated solitons and transverse stability in a two-dimensional nonlinear reaction diffusion electrical network

We investigate the propagation of modulated solitons in a two-dimensional (2D) nonlinear reaction diffusion electrical network with the intersite circuit elements (both in the propagation and transverse directions) acting as nonlinear resistances. Model equations for the circuit are derived and they reduce from the reductive perturbation technique to the 2D nonlinear dissipative Schrödinger equation governing the propagation of the small dissipative amplitude signals in the network. This equation does’nt have conserved quantities and it admits as solutions the 2D dissipative pulse and dark solitons, according to the sign of the product of dispersive and nonlinearity coefficients, with amplitude which narrows as the time increases. The exactness of the analytical analysis is confirmed by numerical simulations. Then by using the method of constants variation, the train of periodic pulse and dark solitons are also found, with their existence constraints also connected to the sign of the product of dispersive and nonlinearity coefficients, as found for standard pulse and dark solitons. Then the modulational instability (MI) criterion in system is found and is connected to the existence of modulated solitons.

Chirped modulated wave excitations in an electrical model of microtubules

Inspired by standard electrophysiological models of microtubules (MTs), we consider a relatively large one-dimensional spatial array of nonlinear electrical transmission network with a negative nonlinear resistance. Employing the reductive perturbation approach in the semi-discrete approximation, we show that the evolution of ionic waves in this electrical circuit model of microtubules is described by a nonlinear Schrödinger (NLS) equation with a dissipative/amplification term in the presence of an external linear potential. The modulational instability analysis of the amplitude equation shows that the derived NLS equation suitably describes the dynamics of our lattice of biochemical units as an excitable medium. Based on exact and approximate solutions of the amplitude equation, we investigate analytically the existence of modulated chirped solitonic waves along microtubules. We suggest possible biological implications of these nonlinear modulated waves. Our system is found to amplify significantly the amplitude of the input signal, confirming thus known experimental results on MTs. Nevertheless, a proper choice of various parameters of the negative nonlinear resistance is required for further validation of our theoretical results.

Periodic orbit analysis of three dynamical systems for a nonlinear electrical dissipative transmission network

Periodic orbit analysis of three dynamical systems for a nonlinear electrical dissipative transmission network

M-shaped and other exotic solitons generated by cubic-quintic saturable nonlinearities in a nonlinear electrical transmission network with higher-order dispersion effects

This investigation presents the generalized cubic-quintic Guizburg-Landau (GCQGL) equation with higher-order dispersion effects in a nonlinear electrical transmission network. Based on the collective coordinates' theory with a conventional Gaussian Ansatz, the study reveals that the waves which propagate in this medium are exotic solitons such as M-shaped solitons. The investigation presents the stability of the dark soliton subjected to the quintic-phase modulation. Besides, the introduction of the cubic- and quintic-saturable nonlinearities induce the generation of twin narrow solitons, M-shaped soliton and Sasa-Satsuma soliton. Moreover, the introduction of the third-order dispersion (TOD) and the fourth-order dispersion (FOD) provokes the generation of twin large solitons associated to symmetric radiation lobes and twin narrow solitons associated to the damping effect. Some exact field solutions of the GCQGL equation are also illustrated. Moreover, some theoretical frequencies are found where significant results are exposed. In addition, physical conditions associated to the saturation lead to a particular internal perturbation. This internal excitation is measured in detail by the collective coordinates in order to build-up exotic solitons. This technique improves the comprehension of some exotic waves' mechanism of generation in nonlinear electrical transmission network.

Exotical solitons for an intrinsic fractional circuit using the sine-cosine method

This work focuses on seeking soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission network obtained through the simplest sine-cosine method. The studied model is governed by a fractional nonlinear partial differential-difference equation in (2 + 1) spatio-temporal dimensions, and the method used to get exact solutions is simple, concise and well-known. We achieve for the model studied here that, the sought solutions specifically by means of the sine-cosine method are functions of all the capacitor's nonlinearities (quadratic and cubic) if and only if, we use the fourth-order spatial dispersion (FOSD)during the continuous media approximation. In contrast, in the absence of the FOSD term, the solutions only exist if, either the quadratic or the cubic nonlinearity is considered separately. In addition, the obtained solutions shapes are exotical, unexpected and novel. These findings (singular bright solitary waves, pulse, U-shaped and M-shaped waves trains) get many applications; for instance, codifying data in the allowed or forbidden band for the signal's transmission in the waveguides.

Hybrid behavior of a two-dimensional Noguchi nonlinear electrical network

In this work, the propagation of modulated waves in a nonlinear two-dimensional discrete electrical line is studied. Based on the linear dispersive law, we demonstrate that the network can adopt Hybrid’s behavior due to the fact that, without changing its appearance structure or its parameters values, it can become alternatively, a purely right-handed line, a purely left-handed line or a composite right-left-handed line depending on coupling transverse parameters and Using the reductive perturbation method, a (2+1)-dimensional nonlinear Schrodinger equation governing the dynamics of the small amplitude signals in the network is derived and the impact of the above transverse parameters on the sign of the product of their coefficients is examined. Likewise, backward and forward solitary wave propagations in the system is predicted and analyzed quantitatively and qualitatively. The effects of relevant transverse network parameters on the characteristic parameters of these waves are investigated. We find out that these transverse parameters considerably affect the characteristics and the nature of the waves that are propagated throughout the system.

Engineering chirped Lambert W-kink signals in a nonlinear electrical transmission network with dissipative elements

The dynamics of modulated waves in a nonlinear discrete transmission network with dissipative elements are investigated analytically. The generalized cubic-quintic Ginzburg-Landau (GCQ-GL) equation with intrapulse Raman scattering (IRS) terms governing spatially damped slowly modulated wave propagation is derived. Considering modulated Stokes wave propagating in the network, we investigate their baseband modulational instability (MI) and show that there are significant changes for the bandwidth frequency where the network may exhibit baseband MI. We show that the MI growth rate increases with the propagating frequency. The analytical chirped Lambert W-kink soliton-like solution of the derived GCQ-GL equation is presented and used to analyze the dissipative effects on chirped Lambert W-kink signals propagating through our network system. We show that the amplitude decreases both in space and time, while the width and the velocity remains constant when the Lambert W-kink signal propagates along the dissipative network. The effects of the cubic-Raman contributions on (i) the propagating frequency, (ii) the baseband MI, as well as (iii) the dynamics of Lambert W-kink signals are also investigated. In particular, our results reveal that a decrease in the cubic-Raman contributions enhances the baseband MI. The results of this paper may be useful for theoretical study of the propagation of ultrashort pulses in a single-mode optical fiber, as well as for experimental realization of undistorted transmission of optical pulses in optical fibers and further understanding their optical transmission properties.

THE CONTROLLABILITY OF RLCM NETWORKS OVER F(Z) AND THEIR APPLICATIONS

The subject of controllability and observability in dynamical systems is one of the important matters in linear system theory. The controllability and observability in linear systems over the field R of real numbers were heavily studied. If the values of resistors, capacitors, and self and mutual inductors in RLCM network are regard as independently variable parameters, the network is defined over the field F(z) of all rational functions in its physical parameters. Some concepts and results on the reducibility and separability for this network are presented; the controllability problem and application are discussed. In this paper the author will discuss the relationship between separability and reducibility of RLCM networks over F(z), the relationship between the physical structure of RLCM networks and structural controllability. The results in this paper can be used to study different kinds of linear or nonlinear electrical networks (over F(z)).

Comparative Analysis of State-Space and Companion-Circuit Methodologies for the Periodic Steady-State Solution in Time-Domain of Nonlinear Electric Networks

This contribution reports a comparative analysis of two methodologies for the periodic steady-state solution of linear and nonlinear electric networks in time-domain (TD), based on the state-space model (SSM) and companion-circuit analysis (CCA), respectively. Newton methods based on a conventional Numerical Differentiation (ND) and Enhanced Numerical Differentiation (END) process, respectively, are used. The application of the SSM and CCA methods is done using a brute force (BF) procedure and the application of advanced numerical techniques (ANT) through the forward Euler (FE) and Trapezoidal Rule (TR) numerical integration, respectively. Related analysis show comparisons of each case study in terms of efficiency, speed, computer effort and accuracy between the combination of SSM, CCA and Newton methods, e.g. SSM-ND, SSM-END, CCA-ND and CCA-END, respectively. In particular, the analysis is shown in terms of time-steps per period, CPU processing time and a computer platform used are shown. The results obtained from the reported case studies are validated against the Power Systems Computer Aided Design/Electromagnetic Transients Program including Direct Current (PSCAD/EMTDC®) response, widely accepted by the power industry.

Structural design of a piezoelectric meta-structure with nonlinear electrical Bi-link networks for elastic wave control

This study reports a structural optimization work of a piezoelectric meta-structure with nonlinear electrical switched Bi-link electronic networks for adaptive wave propagation manipulation and vibration control applications. The investigated meta-structure consists in a periodic arrangement of elementary cells shaping the beam substructure, each of these comprising two piezoelectric elements with connected electrical terminals through semi-passive nonlinear electrical interface, which therefore forms the so-called Bi-link electrical connection. In order to optimize the investigated structure for wave propagation control, the effects of adjusting the geometric/material parameters and locations of the two piezoelectric elements in one periodic cell are analyzed and summarized in detail. Four kinds of structural optimization methods including eighteen structural optimization sub-methods for the open circuit condition and the SSDI Bi-link condition are concluded and discussed in this research. Results show that the performance of these new resonant-type band gaps can be manipulated to meet the targeted aim in terms of wave propagation characteristics in different frequency domains by proper tuning of the investigated parameters.

A Time-Domain Companion-Circuit Newton-Based Methodology Applied to the Harmonic Analysis of Electrical Networks With PV Systems

This contribution proposes an efficient methodology in time domain (TD) based on companion-circuit analysis (CCA) and a Newton method of extrapolation to the Limit Cycle (periodic steady state solution) based on a Numerical Differentiation (ND) process. Its formulation is amenable to the solution approach followed by standardized digital simulators widely accepted by the power industry, such as EMTP and/or EMTDC. The particular application reported in this research is oriented to the dynamic and fast periodic steady state solution of nonlinear single-phase electrical networks incorporating photovoltaic (PV) systems under harmonic distortion conditions. Fundamentally, the proposed methodology is based on the representation of linear and nonlinear elements (e.g., switching elements), using Norton equivalents. This solution method is based on the determination of discrete differential equations sets, obtained by nodal analysis. The solution process is iteratively obtained by using LU decomposition. In general terms, the application is based on the combination of the CCA and ND methods (CCA-ND), for the applying in different case studies associated with the assessment of harmonics distortion, in terms of individual and total harmonic distortion at the point of common coupling (PCC) between the PV system and the electrical system are reported. It is shown that the methodology is precise, reliable and efficient. The documented results have been successfully validated against those obtained PSCAD/EMTDC® simulator, widely accepted by the power industry.

Management of modulated wave solitons in a two-dimensional nonlinear transmission network

Based on a modified one-dimensional Noguchi electrical transmission network containing a linear dispersive element CS with a voltage source and a one-dimensional series capacitor transmission network, we build a two-dimensional nonlinear discrete electrical network which allow the wave propagation in both the longitudinal and the transverse direction. These transmission lines are coupled to one another in the transverse (longitudinal) direction by a linear capacitor C2 (a linear inductor L1 in parallel with the linear capacitance Cs). The linear dispersion relation of the network system is derived and the effects of the transverse coupling element C2 on the linear waves are established. Using the continuum limit approximation and assuming that the perturbation voltage is small enough compared with the equilibrium value, we show that the dynamics of small-amplitude pulses in the network can be governed by a two-dimensional modified Zakharov–Kuznetsov (ZK) equation with a voltage source term. Analyzing the wave propagation in a reduced direction, we show that a best choice of the coupling capacitance C2 and the linear dispersive element CS can lead to the propagation at the same frequency of two distinct waves propagating in different reduced propagation directions. The transverse stability of plane solitary waves is investigated and the effects of the dispersive element CS on the transverse instability are presented. Through the analytical exact bright solitary wave solution of the derived ZK equation, we investigate analytically the effects of the linear dispersive element CS, the effects of the management parameter, and the effects of the reduce propagation direction on the characteristic parameters (amplitude, width, and velocity) of bright solitary waves propagating through our network system. We find that the management parameter of the ZK equation can be used to manipulate the motion of pulse voltages through the network system.

Existence and dynamics of solitary waves in a two-dimensional Noguchi nonlinear electrical network

In this work, we investigate solitary waves in a nonlinear two-dimensional discrete electrical lattice. It is made of several of the well-known Noguchi electrical transmission lines, that are transversely or longitudinally coupled to one another by an inductor ${L}_{2}$ or ${L}_{1}$ and a capacitor ${C}_{2}$ or ${C}_{1}$ mounted in parallel. The linear dispersion law of the network is given and the effects of the transverse coupling elements ${L}_{2}$ and ${C}_{2}$ on the allowed bandwidth frequencies are examined. Using the continuum limit approximation, we show that the dynamics of the small amplitude signals in the network can be governed by a (2+1)-dimensional generalized modified Zakharov-Kuznetsov equation. The fixed points of our model equation are examined and the bifurcations of its phase portrait are presented, as functions of the wave velocity of the signals that are to propagate in lattice. Likewise, we derive exact explicit solutions that are possible under different wave velocities and for physically realistic values of the network's parameters. These include pulse, kink, and anti-kink wave solutions and correspond to some special level curves of the first integral of the model equation. We find out that the transverse coupling parameters considerably affect the characteristics of the waves that are propagated throughout the system. Direct numerical simulations are also performed on the exact equations of the network and the results are in agreement with the analytical predictions.

Solitonlike pulses along a modified Noguchi nonlinear electrical network with second-neighbor interactions: Analytical studies.

A modified lossless nonlinear Noguchi transmission network with second-neighbor interactions is considered. In the semidiscrete limit, we apply the reductive perturbation method and show that the dynamics of modulated waves propagating through the network are governed by an NLS equation with linear external potential. Classes of exact solitonic solutions of this network equation are derived, proving possible transmission of both bright and dark solitonlike pulses through the network. The effects of both the coupling second-neighbor parameter L_{3} and the strength λ of the linear potential on the dynamics of modulated waves through the network are investigated. One of the main results of our work is that with the introduction of the second neighbors in the network, two solitary signals, either two bright solitary signals or one bright and one dark solitary signal, may simultaneously propagate at the same frequency through the network.

Periodic structure with interconnected nonlinear electrical networks

This article aims at investigating the filtering abilities of periodic structures with nonlinear interconnected synchronized switch damping on inductor electrical networks. Periodic structures without electrical networks themselves naturally have the function of filtering since the structure response breaks into pass bands and stop bands when the structure is excited by an external force with multiple or varying frequencies. Introduction of linear electrical networks in the periodic structure makes stop bands of the structure wider than that of the structure without electrical networks. However, nonlinear piezoelectric electrical networks may have better effect on the mechanical wave attenuation than linear piezoelectric electrical networks in terms of frequency band. Therefore, this article proposes a piezoelectric periodic structure with nonlinear interconnected synchronized switch damping on short-circuit/synchronized switch damping on inductor electrical network. A transfer matrix formulation including the interconnected electrical network is also proposed for deriving the characteristics of elastic wave propagation. The results show that the proposed technique permits enhancing the damping abilities in particular frequency bands compared to electrically independent periodic cells, which, combined with structural tailoring, would allow achieving high damping performance.

Vibration reduction for smart periodic structures via periodic piezoelectric arrays with nonlinear interleaved-switched electronic networks

Smart periodic structures covered by periodically distributed piezoelectric patches have drawn more and more attention in recent years for wave propagation attenuation and corresponding structural vibration suppression. Since piezoelectric materials are special type of energy conversion materials that link mechanical characteristics with electrical characteristics, shunt circuits coupled with such materials play a key role in the wave propagation and/or vibration control performance in smart periodic structures. Conventional shunt circuit designs utilize resistive shunt (R-shunt) and resonant shunt (RL-shunt). More recently, semi-passive nonlinear approaches have also been developed for efficiently controlling the vibrations of such structures. In this paper, an innovative smart periodic beam structure with nonlinear interleaved-switched electric networks based on synchronized switching damping on inductor (SSDI) is proposed and investigated for vibration reduction and wave propagation attenuation. Different from locally resonant band gap mechanism forming narrow band gaps around the desired resonant frequencies, the proposed interleaved electrical networks can induce new broadly low-frequency stop bands and broaden primitive Bragg stop bands by virtue of unique interleaved electrical configurations and the SSDI technique which has the unique feature of realizing automatic impedance adaptation with a small inductance. Finite element modeling of a Timoshenko electromechanical beam structure is also presented for validating dispersion properties of the structure. Both theoretical and experimental results demonstrate that the proposed beam structure not only shows better vibration and wave propagation attenuation than the smart beam structure with independent switched networks, but also has technical simplicity of requiring only half of the number of switches than the independent switched network needs.

Dynamics of modulated waves in a lossy modified Noguchi electrical transmission line.

We study analytically the dynamics of modulated waves in a dissipative modified Noguchi nonlinear electrical network. In the continuum limit, we use the reductive perturbation method in the semidiscrete limit to establish that the propagation of modulated waves in the network is governed by a dissipative nonlinear Schrödinger (NLS) equation. Motivated with a solitary wave type of solution to the NLS equation, we use both the direct method and the Weierstrass's elliptic function method to present classes of bright, kink, and dark solitary wavelike solutions to the dissipative NLS equation of the network. Through the exact solitary wavelike solutions to the dissipative NLS equation, we investigate the effects of the dissipative elements of the network on wave propagation. We show that the wave amplitude decreases and its width increases when the dissipative element of the network increases. It has been also found that the dissipative element of the network can be used to manipulate the motion of solitary waves through the network. This work presents a good analytical approach of investigating the propagation of solitary waves through discrete electrical transmission lines and is very important for studying modulational instability.

Transverse compactlike pulse signals in a two-dimensional nonlinear electrical network.

We investigate the compactlike pulse signal propagation in a two-dimensional nonlinear electrical transmission network with the intersite circuit elements (both in the propagation and transverse directions) acting as nonlinear resistances. Model equations for the circuit are derived and can reduce from the continuum limit approximation to a two-dimensional nonlinear Burgers equation governing the propagation of the small amplitude signals in the network. This equation has only the mass as conserved quantity and can admit as solutions cusp and compactlike pulse solitary waves, with width independent of the amplitude, according to the sign of the product of its nonlinearity coefficients. In particular, we show that only the compactlike pulse signal may propagate depending on the choice of the realistic physical parameters of the network, and next we study the dissipative effects on the pulse dynamics. The exactness of the analytical analysis is confirmed by numerical simulations which show a good agreement with results predicted by the Rosenau and Hyman K(2,2) equation.