Analysis of fractional electrical circuits in transient states

Using the Caputo definition of fractional derivative-integral it is shown that the mesh method can be also applied to analysis of fractional linear electrical circuits in transient state. Using the mesh method it is shown that the reciprocity theorem is also valid for fractional linear circuits in transient state.

The classical Thevenin theorem and Norton theorem are extended for fractional linear electrical circuits.
ANALIZA LINIOWYCH OBWODÓW ELEKTRYCZNYCH NIECAŁKOWITEGO RZĘDU W STANIIE NIEUSTALONYM
Korzystając z pochodno-całki Caputo rzędu niecałkowitego uogólniono metodę oczkową na rozgałęzione obwody elektryczne rzędu niecałkowitego w stanach nieustalonych. Wykazano, że: 1) zasada wzajemności jest również prawdziwa dla liniowych obwodów elektrycznych niecałkowitego rzędu, 2) twierdzenie o zastępczym źródle napięciowym i twierdzenie o zastępczym źródle prądowym są również prawdziwe dla liniowych obwodów elektrycznych niecałkowitego rzędu w stanie nieustalonym
1. ITRODUCYION
The mesh method and the node method are the basic methods of analysis of linear electrical circuits [1, 4, 6]. In this analysis the reciprocity theorem, the classical Thevenin theorem and Norton theorem play an important role.
Recently a dynamical development of the fractional linear systems theory can be observed
[11-13]. An overview of state of art in positive systems theory is given in the monograph
[10]. The stability of fractional linear discrete-time and continuous-time systems has been investigated in [2, 3, 5, 8, 9].
In this paper it will be shown that the mesh method can be applied to analysis of fractional linear circuits in transient state and the classical reciprocity theorem, Thevenin theorem and Norton theorem will be extended for fractional linear circuits in transient state.
The paper is organized as follows. In section 2 the basic Caputo definition of the fractional derivative-integral and the fractional state equation and its solution are recalled. (…)

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