Nonlinear Transmission Line Research Articles

Nonlinear harmonic excitations in collisional, asymmetrically-driven capacitive discharges

The standing wave effect, which may lead to center-high density profiles in high frequency capacitive discharges, can be enhanced by nonlinearly excited harmonics. In this work, a nonlinear transmission line model, which solves for the electromagnetic fields in the time domain, is coupled to a two-dimensional bulk plasma fluid model to study nonlinear effects in asymmetric cylindrical capacitive argon discharges. An analytical collisional or collisionless (ion) sheath model is used to determine the stochastic and ohmic sheath heating and the nonlinear dependence of sheath voltage on sheath charge. We first examine a base case of a 20 mTorr argon discharge driven with an electron power P e = 40 W at a frequency f = 60 MHz, using collisionless and collisional sheath models. For the collisionless sheath model, the nonlinearly excited harmonics near the series and spatial resonance frequencies significantly enhance the on-axis power deposition and lead to a sharp peak of electron density at the discharge center. The collisional sheath model gives a smaller sheath width, leading to lower series and spatial resonance frequencies and a smaller source voltage for the fixed electron power. As a result, lower harmonics with broader spatial profiles and decreased magnitude are excited, reducing the center-high plasma nonuniformity. Then, we examine the discharge in a pressure range of 20–100 mTorr at fixed P e = 40 W and f = 60 MHz, using the collisional sheath model. As pressure increases, the harmonics gradually damp out, and the enhancement of on-axis power deposition becomes less significant. At the same time, more power is localized near the powered electrode edge due to decreased skin depth and smaller energy diffusion. As a result, the density peak shifts from the radial center to the powered electrode edge.

Nonlinear Permeability Measurements for Nickel Zinc Ferrite and Nickel Zinc Ferrite/Barium Strontium Titanate Composites From 1 to 4 GHz

Nonlinear transmission lines (NLTLs), which exhibit permittivity as a function of electric field and/or permeability as a function of magnetic field strength, are of increasing importance for sharpening pulses to less than 100 ps and serving as radio frequency (RF) sources; however, NLTLs often are not easily modified to achieve different output parameters. One method under investigation involves combining nonlinear dielectric [barium strontium titanate (BST)] and/or magnetic [nickel zinc ferrite (NZF)] inclusions to tune the NLTL properties by adjusting the inclusion loading fractions. This article focuses on measuring the nonlinear permeability and magnetic loss tangent of composites comprising various volume loadings of NZF or NZF and BST inclusions encapsulated in a silicon matrix. We measured the relative permeability from 1 to 4 GHz using a coaxial airline while biasing the samples in an external dc magnetic field from 0 to 171 kA/m. The permeability decreased from 1 to 4 GHz for each volume fraction but increased with increasing magnetic field strength at low-magnetic field strengths with sufficient NZF volume loading. The magnetic loss tangent of the composites increased with increasing frequency and/or NZF volume fraction but was suppressed by increasing the external magnetic field strength. Adding BST to NZF composites did not cause a significant change in permeability compared to NZF alone, based on an analysis of variance (ANOVA) and multiple comparison test. These results elucidate the frequency dependence of NZF volume loading at microwave frequency and provide initial information for simulating NLTLs and examining more comprehensive RF system behavior.

On the soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission line

This paper focuses on finding soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission lattice. Our investigation is based on the fact that for a realistic system, the electrical characteristics of a capacitor (and an inductor via skin effect) should include a fractional-order time derivative. In this respect for the model under consideration, we derive a fractional nonlinear partial differential equation for the voltage dynamics by applying the Kirchhoff’s laws. It is realized that the behavior of new soliton solutions obtained is influenced by the fractional-order time derivative as well as the coupling values. The fractional order also modifies the propagation velocity of the voltage wave notwithstanding their structure and tends to set up localized structure for low coupling parameter values. However, for a high value of the coupling parameter, the fractional order is less seen on the shapes of the new solitary solutions that are analytically derived. Several methods such as the Kudryashov method, the $$(G'/G)$$ -expansion method, the Jacobi elliptical functions method and the Weierstrass elliptic function expansion method led us to derive these solitary solutions while using the modified Riemann–Liouville derivatives in addition to the fractional complex transform. An insight into the overall dynamics of our network is provided through the analysis of the phase portraits.

Operation analysis of the wideband high-power microwave sources based on the gyromagnetic nonlinear transmission lines.

The wideband High-Power Microwave (HPM) sources, which combine the advantages of narrowband and ultrawideband sources, have drawn much attention. As a kind of wideband source, the gyromagnetic nonlinear transmission lines (GNLTLs) can directly modulate the incident pulses into radio frequency pulses without relying on the interaction between e-beam and microwaves. Due to the special working mechanism of gyromagnetic precession, the center frequency of the GNLTL can also be adjusted in a certain range. Based on classical magnetism and a simplified model of the GNLTL, this paper semi-quantitatively and theoretically analyzed the generation mechanism of HPM and illustrated the influences of the variations of parameters on the output microwaves. Then, a simple simulation based on 1-dimensional transmission line modeling method was carried out to study the performance of the GNLTL quantitatively, with the coupling of 1D telegraphist equations and the 3D Landau-Lifshitz-Gilbert equation. Simulation results preliminarily verified the conclusions derived from the theoretical analysis, and some working characteristics of the GNLTL were also obtained. This paper may help to understand the special working mechanism of the GNLTL and provide certain guidance for related simulations and experiments.

On the generation of electrical compact pulses: theory and simulations

In this work, a self-sustained generator of pulse signal voltages with compact shape is introduced and investigated. It consists of a coupling between the modified Colpitts oscillator (MCO) and a nonlinear electrical transmission line (NLTL) with the appropriated inter-site nonlinearities, exhibiting pulse signal voltages with compact shape, with interesting properties compared to those of pulse solitons. From the state equations governing the dynamics of the MCO part, and through the bifurcation diagram, we have first shown that the system may exhibit an harmonic solution, in certain domain of the control parameter, useful for exciting the NLTL part. In the parameter where the MCO experiences harmonic oscillations, the conditions for a synchronized dynamics of the two main parts of the device are established. Within these conditions, the system generates a train of compact voltage pulse signals. The generator performances namely the pulse width, the repetition rate and the period of pulse repetition rate are derived analytically, and the results indicate their dependence on the characteristic parameters of the circuit. The results of numerical and Pspice simulations have confirmed the accuracy of the analytical investigations and the capacity of the circuit to work as a generator of these compact pulse signal voltages.

A Nonlinear Transmission Approach to Compressing Rise and Fall Time in Picosecond Pulse Generation

This article presents the generation scheme of picosecond (ps) pulses based on both step recovery diode (SRD) topology and nonlinear transmission line (NLTL) scheme with a focus on simultaneous rise and fall time compression. First, the SRD topology is suggested and investigated for manifesting the capability of pulse compression. The SRD solution is then integrated with different NLTL schemes for alienating and achieving rise or fall time compression in the ps regime. It is observed that single NLTL integration only sharpens the rising or falling edge at the same time. To overcome this challenge, two NLTLs having different diode polarities are then integrated with SRD topology-based ps pulse generator to compress rise and fall time simultaneously. The theory of NLTL is developed in each case, and based on it, the shortest rise and fall time is selected for the final design. Several parameters that are responsible for waveform distortion are also investigated in detail as well. Finally, we have fabricated four different pulse generators in which the first three only compress the rise time whereas the fourth one is responsible for both rise and fall time compressions. The overall pulse duration for the first three generators falls within 80–95 ps having different ringing levels and they are based on rise time compression only. The pulse duration for the fourth pulse generator is almost 25 ps having a detailed ringing level of −14.24 dB and full-width at half-maximum (FWHM) of 10.1 ps based on both rise and fall time compressions. Finally, all these pulse generators are compared with the state of the arts in the literature where they utilize different techniques for sharpening and the superiority of our proposed designs is provided. These generators will find their key role and will be the best candidates for ultra-wide band (UWB) radars for electromagnetic imaging and sensing applications, ultrafast electronic techniques including edge sharpening, future oscilloscopes, comb, or pulse generators, and geophysical exploration.

Multi-Sasa Satsuma rogue events and multi-wave trains generation in a nonlinear left-handed transmission line

We demonstrate at the first time that the left-handed transmission line with varactor diodes can support higher-order effects. So, we establish at the first time, a nonlinear Schrodinger equation including second-, third- and fourth- order dispersions and self- and quintic-phase modulations. We use collective coordinate’s theory in order to give a good characterization of the light pulse. Moreover, we introduce a new Ansatz function called “type I Ansatz function” with seven coordinates compared to the conventional Ansatz function with six coordinates. The additional coordinate called “nonlinear phase” will give supplemental details on internal deformation which occurs during the propagation of the soliton light pulse. We find that the action of quintic-phase modulation on soliton light pulse leads to the generation of Peregrine solitons and several narrow wave trains. The introduction of third-order dispersion induces the appearance of multi-Sasa–Satsuma rogue events. Further, we introduce fourth-order dispersion and we assist to the generation of Parallel wave trains associated with strange wave’s field. Considering an increase in frequency combined to the action of all those effects, the precedent Parallel wave trains are changed into conventional Sasa–Satsuma rogue waves. Furthermore, we present physical conditions and theoretical frequencies leading to the generation of those specific extreme events. The new introduced Ansatz function improves the comprehension of rogue wave’s mechanism of generation.

A nonlinear transmission line based on periodically placed silicon high voltage switches

The generation of high-power nanosecond radio frequency pulses is obtained by means of sequential commutation of silicon high voltage switches periodically placed in a nonlinear transmission line. The coaxial transmission line had the inner conductor made in the form of a corrugated periodic structure. The dispersive characteristics of the transmission line were optimized for RF generation by an electromagnetic shock front using numerical simulation. In the experiment, nanosecond RF pulses at frequency near 1 GHz were obtained for 2 kV incident pulses. The peak-to-peak amplitude of the excited oscillations reached about 30% of the amplitude of the incident pulse.

On the propagation of alphabetic-shaped solitons to the (2 + 1)-dimensional fractional electrical transmission line model with wave obliqueness

This manuscript deals with the (2 + 1)-dimensional nonlinear electrical transmission line (NETL) equation of fractional-order to explain the dynamics of waves in the nonlinear network lines. The fractional transform with conformable derivative and wave obliqueness is applied to the model equation for converting it into an ordinary differential equation (ODE). To explore alphabetic-shaped solitons, an ansatz and the auxiliary equation methods are utilized to the ODE. The utilized transformation is then put back to the received solutions of the ODE. The solutions of the model equations representing alphabetic solitons are attained in terms of fractionality and wave obliqueness. It is seen that some of the beforehand generated solutions are the particular case of the newly derived solutions. The effects of fractionality, obliqueness, and free parameters on the attained solutions are tested and are demonstrated graphically with physical descriptions. Also, the behavior of the voltage wave propagation of the attained solutions are described. It is found that the alphabetic wave phenomena are changed with the change of obliqueness and fractionality, and the free parameters have substantial effects on the attained solutions. All the attained solitons propagate in the xy-plane as time (t) increases and the soliton propagation is found to be controlled by the free parameters, fractionality, and wave obliqueness. All the reported alphabetic-shaped solitons are found to be new in the sense of conformable derivative, wave obliqueness, and the applied methods.

Stable solutions to the nonlinear RLC transmission line equation and the Sinh–Poisson equation arising in mathematical physics

AbstractThe Sinh–Poisson equation and the RLC transmission line equation are important nonlinear model equations in the field of engineering and power transmission. The modified simple equation (MSE) procedure is a realistic, competent and efficient mathematical scheme to ascertain the analytic soliton solutions to nonlinear evolution equations (NLEEs). In the present article, the MSE approach is put forward and exploited to establish wave solutions to the previously referred NLEEs and accomplish analytical broad-ranging solutions associated with parameters. Whenever parameters are assigned definite values, diverse types of solitons originated from the general wave solutions. The solitons are explained by sketching three-dimensional and two-dimensional graphs, and their physical significance is clearly stated. The profiles of the attained solutions assimilate compacton, bell-shaped soliton, peakon, kink, singular periodic, periodic soliton and singular kink-type soliton. The outcomes assert that the MSE scheme is an advance, convincing and rigorous scheme to bring out soliton solutions. The solutions obtained may significantly contribute to the areas of science and engineering.

Field-Line Coupling Method for the Simulation of Gyromagnetic Nonlinear Transmission Line Based on the Maxwell-LLG System

The gyromagnetic nonlinear transmission line (GNLTL) is a compact and solid-state high-power microwave source, which can be treated as a special coaxial transmission line partially filled with the ferrite material. Due to the complex interaction process of the electromagnetic fields with the dynamic damped precessional motion of the magnetization vector in the ferrite, it is of great challenge to theoretically predict the output of the GNLTL. Here, we present a novel field-line coupling simulation method to predict the output waveform of the GNLTL by simultaneously solving the coupled system of the Landau-Lifshitz-Gilbert equation and Maxwell's equations in the 2-D cylindrical coordinate system. In particular, the simulation model is validated by comparing the computation results with the experimental results obtained by Bragg et al. in the HPM experiment. The influences of the GNLTL length and bias magnetic field are analyzed. The proposed computation method can provide a solution for improving the performance of the GNLTL.

Optical solutions of the (2 + 1)-dimensional hyperbolic nonlinear Schrödinger equation using two different methods

This paper studies the (2 + 1)-dimensional hyperbolic nonlinear Schrödinger equation. The first integral of this equation, the phase portraits and the effective potentials are provided. Two different methods are applied to find exact analytical solutions. These methods are the arbitrary nonlinear parameters and the new Jacobi elliptic function expansion method. To give a behavior of the equation studied, some representations are done. In the context of mono-mode optical fibers and in many other domains like nonlinear transmission lines, Bose-Einstein capacitors and so on, the results obtained may be used. We have also established that the solutions obtained here are different from those encounter in the literature concerning the same model.

Dark solitons and modulational instability of the nonlinear left-handed transmission electrical line with fractional derivative order

This paper unearths solitary wave in a simplified electrical model practical of the nonlinear left-handed transmission electrical line having series capacitance and shunt inductance. The detailed study of the practical model digs out that by taken a good parameter of the lattice, the obtained equation from the evolution of the wave envelope can be reduced to a one-dimensional fractional nonlinear Schrödinger equation, which gives the pipe of the dark solitons propagation. By simulating transient circuits, the evolution of the soliton and modulational instability gain were demonstrated in the areas of perturbation frequency, pulsation, nonlinearity and signal strength with a continuous wave type input signal. It has also been shown that increased perturbation, nonlinearity and signal strength in simplified metamaterial models of short nonlinear transmission lines lead to the formation of Schrödinger dark solitons and increased instability gain in a continuous pulse structure. The establishment of Schrödinger solitons and the progressive distance of a point, i.e. the fractional parameter α = 1, could reduce the width of the signal bandwidth and the breaking of the symmetry of the instability gain in the simplified electrical model practical of the nonlinear left-handed transmission electrical line, so that the evolution of the system cannot be controlled due to the memory effect, which can find important practical applications in communication systems.

Traveling and localized solitary wave solutions of the nonlinear electrical transmission line model equation

Analytical solutions describing the kink–antikink type, hyperbolic, trigonometric, doubly periodic, localized bright and dark soliton like, breather and rational periodic train solutions to the discrete nonlinear electrical transmission line in $$(2+1)$$ -dimensions have been investigated with the Kudryashov method, the fractional linear transform method, and the simplest equation method with its modified and extended versions. Consequently, many new and more general explicit forms of traveling waves are retrieved under parametric conditions. The 3D, 2D, and density profile of some of the obtained results are numerically simulated by selecting suitable values of the various parameters of the NLTL model equation. Furthermore, all the obtained results were checked and verified by putting back into the considered nonlinear transmission line model equation with Maple and Mathematica software. The acquired results show the validity and reliability of the proposed schemes to the studied nonlinear electrical transmission line model.

A note on “Envelope periodic solutions for a discrete network with the Jacobi elliptic functions and the alternative (G′/G)-expansion method including the generalized Riccati equation”

Employing Jacobi elliptic functions and the so-called alternative ( $$ G^{\prime}/G $$ )-expansion method including the generalized Riccati equation, Tala-Tebue et al. (Eur Phys J Plus 129:136, 2014) derived exact solutions for a discrete nonlinear electrical transmission line in (2 + 1)-dimensions. We show that the obtained solutions of the model given in Tala-Tebue et al. (2014) are all incorrect due to the incorrect reduced ODE Eq. (14) in that paper.

A variety of new traveling and localized solitary wave solutions of a nonlinear model describing the nonlinear low- pass electrical transmission lines

Abstract In this work, we focus on investigating the traveling and other localized solitary wave propagation in nonlinear low-pass electrical transmission lines model practicing the new modified sub-ODE method, the unified Riccati equation expansion method, and the fractional linear transform method. A variety of traveling and solitary wave solutions are emerging comprising of bright, dark, kink, anti-kink, hyperbolic function, and doubly periodic Jacobian elliptic function solutions. The applied three integration schemes are reliable and stalwart for acquiring the new kink, bright, dark, periodic and non-singular soliton solutions of the wave propagation in nonlinear low-pass electrical transmission lines. Also, we give the geometric description of some of the obtained solutions for the considered model by computing the most important geometric quantities viz the Gaussian and the mean curvatures. To display the extant physical significance of the considered model equation, some two, three-dimensional figures and density profiles of the acquired solutions are illustrated for the specific choice of arbitrary parameters. The effects of the variation of nonlinear parameters of the nonlinear low-pass electrical transmission lines model on the evolution of soliton solutions are demonstrated. In addition, all derived solutions were verified by substituting back into the considered equation with the aid of Mathematica software. Furthermore, a comparison of our results for the considered model with the obtained solutions in the literature is also provided.

Bifurcation analysis with chaotic motion of oblique plane wave for describing a discrete nonlinear electrical transmission line with conformable derivative

The present work discusses the basic features of bifurcation properties with chaotic motion of oblique plane wave in the discrete nonlinear electrical transmission lines having conformable derivative evolution. The planar dynamical systems are obtained to understand such physical issues for this model. In addition, the analytical plane wave solutions of nonlinear mathematical leading equation are determined by using two different mathematical methods. Both methods are supported only by the shock plane wave solutions. The conditions for other types of oscillatory plane wave solutions are also determined. The effect of obliqueness on the bifurcation properties, chaotic motion and shock plane waves structures are focused graphically with discussions by considering both of locality and non-locality in the system. It is found that the parameter of non-local operator is only affected on the nonlinear shock wave phenomena, whereas all basic features of discrete nonlinear electrical transmission line are changed with the changes of obliqueness.

Soliton solutions to voltage analysis in nonlinear electrical transmission lines and electric signals in telegraph lines

Soliton solutions of couple of nonlinear evolution equations (NLEEs) ascending for voltage analysis in nonlinear electric transmission lines and electric signals in telegraph lines have been investigated through the modified simple equation (MSE) method. We establish general closed form soliton solutions associated with transmission line parameters and integral constants, for definite values of which some of the existing solutions in the literature are rediscovered and some new solutions are originated. The 3D and contour plots of the obtained solutions are depicted to illustrate the voltage characteristics in the transmission line and in the optical fibers. The obtained solutions might play important role in science and engineering fields. It is noteworthy to observe that the MSE method is easy, compatible and powerful mathematical tool to extract soliton solutions.

Construction of traveling and solitary wave solutions for wave propagation in nonlinear low-pass electrical transmission lines

In this study, our aim to constructed the traveling and solitary wave solutions for nonlinear evolution equation describe the wave propagation in nonlinear low-pass electrical transmission lines by implemented the modification of mathematical method. We obtained the new and more general solutions in rational, trigonometric, hyperbolic type which represent to kink and anti-kink wave solitons, bright-dark solitons and traveling waves. The physical interpretation of some results demonstrated by graphically with symbolic computation. We are hopefully determined results have numerous applications in optical fiber, geophysics, fluid dynamics, laser optics, engineering, and many other various kinds of applied sciences. The complete investigation prove that proposed technique is more reliable, efficient, straightforward, and powerful to investigate various kinds of nonlinear evolution equations involves in geophysics, fluid dynamics, nonlinear plasma, chemistry, biology, and field of engineering.

Competent closed form soliton solutions to the nonlinear transmission and the low-pass electrical transmission lines

The nonlinear transmission line and the low-pass electrical transmission line equations are very important nonlinear evolution equations in electrical transmission line management. The modified simple equation (MSE) technique is a compatible and effective mathematical tool to extract soliton solutions of science and engineering problems. In this article, the MSE technique is introduced and implemented to extract broad-ranging exact wave solutions to the formerly stated equations and accomplish analytic soliton solutions including trigonometric and hyperbolic functions with parameters. Whenever the parameters accept appropriate values, soliton solutions are formulated from the wave solutions. We describe the solutions through 3D and 2D graphs. The shapes of the obtained solutions include kink soliton, peakon, periodic soliton, singular periodic soliton, bell shape soliton, compacton and singular kink type soliton. The results show that the MSE technique is a further operative and powerful mathematical tool for extracting the soliton solutions.