Network Theorems for Fractional-Order Circuits

In this paper, we provide a circuit-theoretic analysis of linear fractional-order circuits based on the use of Tellegen's theorem. The advantages of the circuit-theoretic analysis over the more common control-theoretic one are illustrated with general theorems on the driving-point impedances and admittances of multi-type, fractional-order circuits made of an arbitrary, finite, connected mesh of fractional-order elements. The explicit use of Kirchhoff's circuit laws leads to sharp results on the stability and resonant behavior of such networks. In particular, it results in more intuitive conditions on the range of fractional orders needed to guarantee network stability or resonant behavior.