Abstract
This paper discusses the ability to obtain periodic steady-state solutions for fractional nonlinear circuit problems. For a class of nonlinear problems with fractional derivatives (based on the Caputo or Riemann–Liouville definitions), a methodology is proposed to derive equations representing the dependencies between the harmonics of the sought variables. Two approaches are considered for how to address the apparent nonlinear dependencies: one based on symbolic computation and the other a numerical approach based on the analysis of time functions. An example problem with fractional and nonlinear elements is presented to illustrate the usefulness of the proposed methodology. Two error criteria are introduced to verify the accuracy of the obtained results. The methodology is mainly designed to provide referential solutions in analyses of the numerical method called SubIval (the subinterval-based method for computation of the fractional derivative in initial value problems).
Highlights
Fractional CalculusFractional calculus is an increasingly popular field due to its many potential applications
The current study is simplified to the case where: (a) the circuit is in a periodic steady state; (b) nonlinearities consist of only strictly increasing odd functions; (c) each time function of the solution consists of odd harmonics, where the highest is represented by the integer hmax; (d) the source time functions consist only of odd harmonics, where the first time harmonic is dominant and the highest harmonic is below hmax
Smaller error values are obtained for larger hmax, which indicates that the methodology works for the integer-order case and can be applied to classic periodic steady-state nonlinear problems in circuit theory
Summary
Fractional calculus is an increasingly popular field due to its many potential applications. Analyses are performed concerning the usefulness of fractional derivatives and/or integrals in:. – circuit analyses [17,19], including the application of fractional capacitors [13, 15,28,56] (e.g., in modeling supercapacitors) and fractional coils (in modeling ferromagnetic coils [44]);. Many theoretical considerations have been conducted concerning the behavior of fractional-order systems (e.g., stability and controllability analyses [27]). The most important theoretical aspect is, the ability to solve problems where fractional derivatives and integrals appear, as this is essential for all other analyses. This paper considers circuit analyses, those with periodic steady-state sources and fractional or nonlinear elements, even elements that are both fractional and nonlinear. The motivation for the research is given
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