Analysis of electrical circuits including fractional order elements

This paper deals with the analysis of electrical circuits with classical one-port elements including two novel defined one-port fractional order elements: fractional-order resistive-capacitive RC-α and fractional-order inductive RL-α element. The definitions and analytical relations between current, voltage and power of introduced fractional elements are provided. An example of fractional element realization via ladder electrical circuit composed of classical resistors, capacitors and/or inductors is presented. Several examples are analyzed to illustrate the behavior of electrical circuit with fractional order elements for different values of fractional order α including differentiator/integrator circuits as well as complex circuits without accumulated energy.

[1]  YangQuan Chen,et al.  ANALOGUE FRACTIONAL-ORDER GENERALIZED MEMRISTIVE DEVICES , 2009 .

[2]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[3]  I. Podlubny,et al.  Analogue Realizations of Fractional-Order Controllers , 2002 .

[4]  Edmundo Capelas de Oliveira,et al.  On the Generalized Mittag-Leffler Function and its Application in a Fractional Telegraph Equation , 2012 .

[5]  Budimir Lutovac,et al.  Some electromechanical systems and analogies of mem-systems integer and fractional order , 2016, 2016 5th Mediterranean Conference on Embedded Computing (MECO).

[6]  F. Mainardi,et al.  Recent history of fractional calculus , 2011 .

[7]  D. Zorica,et al.  Generalized time-fractional telegrapher’s equation in transmission line modeling , 2017 .

[8]  Gary W. Bohannan,et al.  Analog Realization of a Fractional Control Element-Revisited , 2002 .

[9]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[10]  Gangquan Si,et al.  The fractional-order state-space averaging modeling of the Buck–Boost DC/DC converter in discontinuous conduction mode and the performance analysis , 2015 .

[11]  J. Sabatier,et al.  Crone control of a nonlinear hydraulic actuator , 2002 .

[12]  Khaled N. Salama,et al.  Fractional-Order RC and RL Circuits , 2012, Circuits Syst. Signal Process..

[13]  Khaled N. Salama,et al.  Passive and Active Elements Using Fractional ${\rm L}_{\beta} {\rm C}_{\alpha}$ Circuit , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.

[14]  Alain Oustaloup,et al.  CRONE control of continuous linear time periodic systems: application to a testing bench. , 2003, ISA transactions.

[15]  Faqiang Wang,et al.  Fractional order Buck–Boost converter in CCM: modelling, analysis and simulations , 2014 .

[16]  Zoran D. Jelicic,et al.  Optimization of distributed order fractional PID controller under constraints on robustness and sensitivity to measurement noise , 2014, ICFDA'14 International Conference on Fractional Differentiation and Its Applications 2014.