Floquet analysis of linear dynamic RLC circuits

Mohamed El-Borhamy,Ismail Sobhy,Essam Eddin M Rashad

doi:10.1515/phys-2020-0136

Abstract

AbstractIn this article, the linear dynamic analysis of AC generators modeled as RLC circuits with periodically time-varying inductances via Floquet’s theory is considered. Necessary conditions for the dynamic stability are derived. The harmonic balance method is employed to predict the transition curves and stability domains. An approximate expression for the Floquet form of solution is constructed using Whittaker’s method in the neighborhood of transition curves. Numerical verifications for the obtained theoretical results are considered. In accordance with the experimental results, a satisfactory agreement is relatively achieved with the closed experimental literature of the problem.

Highlights

There have been extensive studies to examine qualitative properties in problems of mathematical physics governed by linear differential equations that include the effects of periodic parametric excitations, time delays and fractional derivatives [2,3,31,33,39]
Floquet [11] developed an interesting theory for the part of linear differential equations with periodic coefficients obtaining a general form of solution applied in every physical process [4,9,19,35,38]
Ignatyev [18] discussed the asymptotic stability of zero solution of second-order differential equation and Duc et al [7] proved that the zero solution is asymptotically stable if one of the hypotheses introduced in the study by Ignatyev [18] does not exist

Summary

Introduction

There have been extensive studies to examine qualitative properties in problems of mathematical physics governed by linear differential equations that include the effects of periodic parametric excitations, time delays and fractional derivatives [2,3,31,33,39]. This part of differential equations and its different variants of linear dynamical systems have mostly appeared in the description of numerous physical problems such as in the field of electrical circuits and small oscillation systems [5,8].

Modeling system

Floquet analysis

Prediction of transition curves

Construction of normal Floquet and periodic solutions

Stability of periodic solutions

Conclusion