Nonlinear LC Circuit Research Articles

Higher Order Approximation in Two-Dimensional Nonlinear LC Circuit Equation by the Hirota Method

The higher order approximation for the two-dimensional nonlinear LC circuit equation is studied by the Hirota method. It is found that the Hirota method is effective not only for the completely integrable and weakly perturbed systems but also for the higher order approximation.

A method for obtaining a canonical Hamiltonian for nonlinear LC circuits

A method is presented for obtaining the Hamiltonian and the proper canonical variables that is both very general and simple. A tree of capacitors is used to write the state equations for an LC circuit with independent sources, and transformations to canonical variables are generated from the circuit topology by considering an alternate tree (cotree) containing as many inductors (capacitors) as possible. A method is given to obtain these transformations by inspection, based on fundamental loops and cutsets of both the capacitor and alternate trees. When the nonlinearities in the capacitors and inductors are small, a change of coordinates is given to achieve complete separation of the canonical variables of the unperturbed (linear) system. These results should allow the application of techniques from Hamiltonian dynamics, such as canonical perturbation theory, to circuits.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A Quasi-Continuous Description of a Nonlinear Transmission Line

We present a systematic approach to derive an equation describing the propagation of pulses along a transmission line made of a large number of nonlinear LC-circuits. This equation correctly captures the leading order effects due the discreteness of the line and describes a two-sided wave propagation, nonlinear wave-wave interactions, and wave-wall interactions. In contradistinction to previous works, our method is not limited to weakly nonlinear disturbances.

Toda Lattice with Mass Interface. I. Scattering of Soliton

We have experimentally investigated scattering of lattice soliton due to mass interface by using a nonlinear LC circuit which is equivalent to the Toda lattice. When a soliton travels into a lighter mass region, the transmitted wave breaks into multiple solitons plus a wave packet, and the reflected wave develops into a wave packet. If a soliton is propagated into the lattice with heavier mass, the reflected wave as well as the transmitted wave evolves into a soliton and a wave packet. The experimental result is qualitatively explained by the K-dV approximation of the lattice under the linearized boundary condition. The discrepancy between the observed soliton amplitude and the K-dV approximation becomes appreciable when the incident soliton amplitude is increased. This discrepancy comes from the breakdown of the linearized boundary condition at the interface.

Toda Lattice with Mass Interface. II. Coupled Toda Lattice

By connecting three lines of nonlinear LC circuit at one point, we investigate experimentally and numerically the propagation of soliton through the junction. We send solitons of equal amplitude from two of the three lines, let them collide at the junction and observe a transmitted soliton on the third line. It is confirmed that the amplitude of the transmitted soliton normalized by the incident soliton increases with the amplitude of the initial soliton. This increase is explained by nonlinear relation between soliton energy and amplitude. The present method is useful to obtain large soliton by the collision of small solitons.

An experimental look at the resonant behaviour of a nonlinear LC circuit

The resonant behaviour of a nonlinear LC circuit and two inductively coupled nonlinear LC circuits is examined experimentally. The amplitude and the phase response curves for primary and secondary resonances are shown.

Experiment on Lattice Soliton by Nonlinear LC Circuit –Observation of a Dark Soliton

We examine the propagation of a dark soliton in a nonlinear LC circuit which is equivalent to the Toda lattice. At first, we derive the nonlinear Schrodinger equation to describe a nonlinear wave of strong dispersion. It is shown theoretically that the wave of envelope is modulationally stable and propagates as a dark soliton, if the circuit is equivalent to the Toda lattice. In the experiment, it is confirmed that the dark soliton is generated from an initial wave and propagates stably in the circuit. The width of the soliton agrees with the theory.

Solitons in Nonlinear Transmission Line

Propagation of a soliton in nonlinear LC circuit has been investigated experimentally when the nonlinearity of a capacitor is contorolled externally. By connecting a linear capacitor parallel to the nonlinear capacitor, the nonlinearity is deviated from that of the Toda lattice. The velocity and the width of a soliton in such a transmission line are compared with the theory taking account of higher order nonlinearity and of the lowest order dispersion. We have obtained a good agreement between the theory and experiment.

Experiment on Soliton in Inhomogeneous Electric Circuit. I. Dissipative Case

We have experimentally investigated the propagation of a soliton in a nonlinear LC circuit with inhomogeneity and dissipation. In a homogeneous and dissipative circuit, the amplitude of a soliton decreases with the distance. The damping of a soliton is weakened if the inhomogeneity is introduced in a dissipative circuit, and the amplitude begins to increase in the circuit of strong inhomogeneity. The disintegration of soliton is not clearly observed in the present experiment. The experimental result is qualitatively explained by the Korteweg-de Vries equation with inhomogeneity and dissipation.

Experiment on Soliton in Inhomogeneous Electric Circuit. II. Dissipation-Free Case

We have experimentally investigated the propagation of a soliton in a nonlinear LC circuit with inhomogeneity and dissipation. In a homogeneous and dissipative circuit, the amplitude of a soliton decreases with the distance. The damping of a soliton is weakened if the inhomogeneity is introduced in a dissipative circuit, and the amplitude begins to increase in the circuit of strong inhomogeneity. The disintegration of soliton is not clearly observed in the present experiment. The experimental result is qualitatively explained by the Korteweg-de Vries equation with inhomogeneity and dissipation.

Experiment on Lattice Soliton by Nonlinear LC Circuit–Energy of Soliton

We have experimentally investigated the energy of soliton in a nonlinear transmission line which is equivalent to the Toda lattice. In the nonlinear circuit, the magnetic and electric energies are not equal with each other. The present experiment shows that the magnetic energy of a soliton is superior in quantity to the electric one, in agreement with theory. This corresponds to the fact that the kinetic energy is larger than the potential energy in a lattice soliton. It is also confirmed experimentally that the magnetic energy is converted to the electric energy at the instant of a head-on collision between two solitons. Such an energy exchange is not observed for an overtaking collision of two solitons.

Experiment on Recurrence in Nonlinear LC Circuit

The recurrence of an initial waveform in a nonlinear LC circuit which is equivalent to the Toda lattice is studied experimentally. When initial wave amplitude is small enough, we find that the recurrence length is independent of wave amplitude, but is determined only by the frequency. The amplitude of the second harmonic wave is also observed as a function of the distance. The results of small wave amplitude agree with the theory of weak nonlinearity. As the wave amplitude is increased, the recurrence length becomes shorter in proportion to A -1/2 , where A denotes the initial wave amplitude. Similarities and dissimilarities, between the recurrence in the discrete medium and that of the continuous system such as the Korteweg-de Vries equation are carefully examined.

Asymptotic Behaviour of Collisionless Shock in Nonlinear LC Circuit

Asymptotic property of collisionless shock has been examined experimentally using a nonlinear LC circuit which simulates the Toda lattice. The shock front velocity depends on the magnitude of a discontinuity as a collisional shock, but the collisionless shock is not stationary. That is, soliton like peaks are generated behind the shock front as an initial step voltage propagates in the circuit. The time interval between these peaks increases logarithmically in time but decreases in proportion to square root of an initial amplitude. The results are qualitatively explained by a solution of Korteweg-de Vries equation.

A simple description and demonstration of parametric echoes

A general formalism is developed that describes the generation of short- and long-range echoes in an ensemble of independent nonlinear oscillators such as exemplified by piezoelectric powder particles. The basic physical model is one of self-parametric interaction in which a parameter of the oscillating system is modulated by the oscillation in the system itself. The resulting formalism, which is similar to the one used in holography, correctly describes the observed echo phenomena. Moreover, it implies the possibility of more general signal storage, analogous to holography. To test the theory, a model has been built consisting of 20 nonlinear LC circuits. Experimental results confirm the existence of both short- and long-range echoes; some of the more detailed implications of the formalism that concern pulse shape are also borne out by the experiment.