A Harmonic Balance Methodology for Circuits with Fractional and Nonlinear Elements

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).