Mathematical modelling and parametric resonances of a nonlinear RLC series circuit

Abstract

In order to study nonlinear parametric oscillations of a RLC series circuit consisting of a nonlinear resistor, a nonlinear inductor and a nonlinear capacitor subjected to mixed-frequency voltage, a class of nonlinear ordinary differential equations is generated. From this class, a generalised mixed Rayleigh-Lienard oscillator used to describe the steady-state oscillations is derived. Applying the method of multiple scales to this specific equation, six resonances states are obtained whose four cases of resonances such as Ω = 3 ω0, Ω = 5 ω0, Ω = 7 ω0 and 2Ω = ω0 are studied. The modulation equations in the amplitude and the phase of these four cases of resonances are determined. The corresponding frequency-response equations and stability analysis of steady-state solutions are investigated. The frequency response curves and stability analysis of each steady-state solution are plotted. As results, bistability, hysteresis, bifurcation and jump phenomena are obtained. Finally, some effects of the nonlinear damping parameters and excitation voltage amplitudes are investigated numerically and results are presented graphically and discussed.