Nonlinear LC Circuit Research Articles

Investigation of synchronization for coupled quantum nonlinear electrical LC circuits with mutual inductance

Investigation of synchronization for coupled quantum nonlinear electrical LC circuits with mutual inductance

Dynamic modes of the phase number converter based on lc circuits with a common magnetic circuit

Phase multipliers are used in electronic equipment devices when it is not possible to connect to a power source with the required number of phases. Phase multipliers, which are used in single-phase circuits for powering three-phase consumers, have received the greatest application. From the point of view of circuitry, the simplest are multipliers of the number of phases based on nonlinear LC circuits, which have high reliability and relatively small dimensions when powering devices of low and medium power. However, phase multipliers based on LC circuits made on a common magnetic core have degraded dynamic characteristics due to the presence of magnetic bonds between phase-shifting circuits.

Analysis of nonlinear LC circuit by symplectic conservative perturbation method

PurposeThis paper aims to provide a symplectic conservation numerical analysis method for the study of nonlinear LC circuit.Design/methodology/approachThe flux linkage control type nonlinear inductance model is adopted, and the LC circuit can be converted into the Hamiltonian system by introducing the electric charge as the state variable of the flux linkage. The nonlinear Hamiltonian matrix equation can be solved by perturbation method, which can be written as the sum of linear and nonlinear terms. Firstly, the linear part can be solved exactly. On this basis, the nonlinear part is analyzed by the canonical transformation. Then, the coefficient matrix of the obtained equation is still a Hamiltonian matrix, so symplectic conservation is achieved.FindingsNumerical results reveal that the method proposed has strong stability, high precision and efficiency, and it has great advantages in long-term simulations.Originality/valueThis method provides a novel and effective way in studying the nonlinear LC circuit.

Designed Analogue Black Hole Solitons in Josephson Transmission Lines

In the nonlinear LC circuit dual to the Toda (voltage) soliton, the current (shock-wave-type) solitons spatially modulate the velocity of the electromagnetic waves in the circuit through the nonlinear inductance, resulting in the creation of analogue black holes. The analogue black hole solitons are spontaneously formed, but their configuration could not be controlled so far. In this paper, the on-demand scheme to design the configuration of analogue black holes is presented by using solitons and antisolitons in the nonlinear LC circuit. In addition, a novel photon source using analogue Hawking radiation is also proposed.

Application of Magnetic Energy Propagation Mode Excited on Nonlinear LC Circuit

Application of Magnetic Energy Propagation Mode Excited on Nonlinear LC Circuit

Analogue Hawking radiation from black hole solitons in quantum Josephson transmission lines

The nonlinear LC circuit dual to the Toda lattice (voltage) soliton is studied using the reductive perturbation method. It is found that the current flowing through the circuit obeys the defocusing modified Korteweg--de Vries equation with shock-wave-type (current) solitons. The current soliton spatially modulates the nonlinear inductance, equivalently the spatial modulations of electromagnetic wave velocity in the circuit, resulting in the generation of analogue black holes. Hawking radiation from the black holes is also discussed based on the tunneling models.

Nonlinear effect of forced harmonic oscillator subject to sliding friction and simulation by a simple nonlinear circuit

We study the nonlinear behaviors of mass-spring systems damped by dry friction using simulation by a nonlinear LC circuit damped by anti-parallel diodes. We show that the differential equation for the electric oscillator is equivalent to that of the mechanical system when a piecewise linear model is used to simplify the diodes' I–V curve. We derive series solutions to the differential equation under weak nonlinear approximation which can describe the resonant response as well as amplitudes of superharmonic components. The experimental results are consistent with the series solutions. We also present the phenomenon of hysteresis. A theoretical analysis along with numerical simulations is conducted to explore the stick-slip boundary. The correspondence between the mechanical and electric oscillators makes it easy to demonstrate the behaviors of this nonlinear oscillator on a digital oscilloscope. It can be used to extend the linear RLC experiment at the undergraduate level.

Defect induced asymmetric soliton transmission in the nonlinear circuit

Electrical diode, the first device to rectify the current flux, has significantly revolutionized fundamental science and advanced technology in various aspects of our routine life. Motivated by the one-way rectification effect, considerable effort has been dedicated to the study of the unidirectional transmission in other physical systems for the potential applications, such as the acoustic diode, thermal diode, etc. The nonlinear LC circuit, which has unique advantages in the measurement of energy with which the voltage and current can be achieved by digital oscilloscope conveniently, provides a simple and effective way of studying the nonlinear wave propagation in a dispersive medium. In this paper, we design a defective LC nonlinear circuit deliberately to realize asymmetric transmission of energy, and the energy carrier is nonlinear wave which is so-called soliton, instead of the linear wave in the pass band. The defect-induced localized wave is a kind of intrinsic bound-state wave mode that is evanescent away from the defect site but vibrates around the site with an intrinsic frequency fr. In the LC circuit, when the defect is close to the driver, with the frequency of driven signal in the forbidden band of system approaching to the intrinsic resonance frequency fr of the defect, the resonance induced by the defect enables the circuit to turn on, which is relevant to but somewhat different from what was uncovered by Leon et al. about the intrinsic instability of evanescent waves stirred up directly by a boundary drive. On the other hand, the system acts like an insulator, for the defect is far away from the drive. The defect changes the homogeneity of the line, which allows the soliton to be released in one direction by the local resonance, with the driver being at a lower amplitude. As a result, the introducing of defects significantly improves conversion efficiency from the driver energy into the soliton. To further understand this phenomenon in the defective LC nonlinear circuit, we numerically investigate the relationship among transmission energy, defect coefficient and driver amplitude. Finally, the combined defects are also considered to further adjust the LC nonlinear circuit.

Explicit and exact traveling wave solutions to the nonlinear LC circuit equation

Traveling wave in a nonlinear LC circuit with dissipation have been investigated theoretically. With the aid of the extended hyperbolic function method,developed by the authors in recent works to solve nonlinear partial differential equations exactly, the fourth order nonlinear wave equation with dissipation, which models shock wave propagation in a nonlinear LC circuit, have been analytically studied. Abundant explicit and exact traveling wave solutions to the fourth order nonlinear wave equation with dissipation are obtained. These solutions include exact shock wave solutions, singular traveling wave solutions, and periodic wave solutions in a rational form of trigonometric functions.

Birkhoffian formulation of the dynamics of LC circuits

We present a formulation of general nonlinear LC circuits within the framework of Birkhoffian dynamical systems on manifolds. We develop a systematic procedure which allows, under rather mild non-degeneracy conditions, to write the governing equations for the mathematical description of the dynamics of an LC circuit as a Birkhoffian differential system. In order to illustrate the advantages of this approach compared to known Lagrangian or Hamiltonian approaches we discuss a number of specific examples. In particular, the Birkhoffian approach includes networks which contain closed loops formed by capacitors, as well as inductor cutsets. We also extend our approach to the case of networks which contain independent voltage sources as well as independent current sources. Also, we derive a general balance law for an associated "energy function".

Resonant transmission of a soliton across an interface between two Toda lattices

The transmission of a single soliton is investigated numerically across an interface between two Toda lattices which are connected by a harmonic spring. We find that a resonant transmission of the soliton occurs when the spring constant of the harmonic spring is adjusted properly. Furthermore, when the amplitude of the incident soliton is large, the soliton transmission coefficient exhibits a local minimum which is due to an emergence of localized waves around the harmonic spring. We propose an experimental test of the results by using a nonlinear LC circuit.

Modified Ginzburg–Landau Equation and Benjamin–Feir Instability

In this paper, a modulated wave train in a nonlinear monoinductance LC circuit is studied. Using the method of multiple scales in the general form, we establish that the evolution of nonlinear excitations is governed by what we called the Modified Ginzburg–Landau Equation (MGLE). Benjamin–Feir instability for the MGLE is analyzed.

A Variational Principle for Nonlinear LC Circuits with Arbitrary Interconnection Structure

A variational interpretation of the dynamics of nonlinear LC circuits is given. An optimal control problem is associated to the circuit that asks for minimizing the time integral of magnetic coenergy minus electric energy subject to Kirchhoff's current law as a dynamic constraint. Via the Pontryagin maximum principle, various mathematical models are obtained for the circuit, both of a Lagrangian and a Hamiltonian nature.

Dressed shock waves in a nonlinear LC circuit

Nonlinear shock waves which appear in a nonlinear Toda lattice with internal dissipation have been studied. The tanh method is used as a perturbation approach together with the assumption of long wavelengths. Moreover, asymptotical behaviour is used to determine the velocity. Burgers-like wave solutions up to third order are derived. If the amplitude of the shock wave is given, the profiles are readjusted to match this boundary condition. The associated steepness and velocity of the solution can be written as a function of this boundary condition.

Remark on “Shock Wave in a Nonlinear LC Circuit”

Remark on “Shock Wave in a Nonlinear LC Circuit”

Reply to the Remark by W. Malfliet on “Shock Wave in a Nonlinear LC Circuit”

Reply to the Remark by W. Malfliet on “Shock Wave in a Nonlinear LC Circuit”

Shock Wave in a Nonlinear LC Circuit

A shock wave in a nonlinear LC circuit with dissipation has been investigated theoretically and experimentally. A shock wave solution of a stationary waveform traveling with a constant velocity and a higher-order correction have been obtained theoretically. In the experiment, it has been demonstrated that a shock wave with a constant amplitude propagates stably in a nonlinear LC circuit with dissipation. The velocity and width of the shock wave have been observed to be consistent with the theoretical prediction, provided that the amplitude is small enough to be consistent with the theoretical approximation.

Higher Order Solution of Nonlinear Waves. II. Shock Wave Described by Burgers Equation

A method of renormalization has been applied to a nonlinear differential equation which describes, in the continuum approximation, a weak shock wave propagating in a dissipative nonlinear LC circuit. In the lowest order of approximation, a well-known Burgers equation is derived. In the next order, a higher order equation which brings about correction to the Burgers equation is obtained. This equation contains secular terms and a term whose particular solution does not satisfy a boundary condition. A method of renormalization is introduced to eliminate such secular terms and an improper particular solution violating a boundary condition. It is shown that the renormalized solution can be described by the Burgers shock wave with the modifications in the width and the velocity.

Gap Solitons in Nonlinear Symmetric Electric Circuit

Soliton-like excitations with frequencies in the gap of a linear spectrum ar considered for a new type of nonlinear LC circuit recently introduced by S. Ishiwata et al . [J. Phys. Soc. Jpn. 59 (1990) 1163]. A nonlinear capacitor whose capacitance depends on the square root of the voltage in the shunt branch is composed of two diodes connected in series but inversely with each other. Numerical results are obtained which show the existence of a new class of localized solitary signals.

Experiment on Solitary Wave in a Symmetric Electric Circuit

A new type of nonlinear LC circuit has been introduced in order to investigate experimentally the propagation and interaction of solitary waves which is symmetric with respect to the inversion of voltage. A nonlinear capacitor in the shunt branch is composed of two diodes connected in series but inversely with each other. Both positive and negative solitary waves are stably propagated in the circuit. The velocity and width of a solitary wave deviate from the theoretical prediction, as the amplitude increases. This discrepancy comes from the fact that the dynamic response of nonlinear diodes is different from the static one. The phase shift resulting from a head-on collision of two solitary waves has also been observed.