Closed-Form Oscillatory Condition in Electrical Circuits Containing Two Fractional Order Elements

Abstract

Oscillatory condition in LTI dynamic systems is generally expressed as possessing purely imaginary solutions by their characteristic equations. Dealing with the class of fractional order systems, such a condition is equivalently restated as owing complex roots with specific arguments by a polynomial defined based on the system characteristic equation. The degree of this polynomial can be unboundedly high. Consequently, such statement for the oscillatory condition in fractional order systems, which is based on arguments of the roots of a polynomial with an unbounded degree, cannot be viewed as a closed-form expression. To tackle this challenge, this brief introduces an approach to obtain a closed-form oscillatory condition for electrical circuits containing two fractional order elements. It is shown that the obtained results are helpful in analytically finding the oscillatory curves in parametric regions. Moreover, it is discussed whether the existence of more than one frequency in the frequency content of steady-state oscillations produced by the considered class of fractional order circuits is possible.