Commun Nonlinear Sci Numer Simulat 46 (2017) 81–88 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns Mathematical modelling of fractional order circuit elements and bioimpedance applications Miguel Angel Moreles ∗ , Rafael Lainez CIMAT, Jalisco S/N, Valenciana, Guanajuato GTO 36240, Mexico a r t i c l e i n f o Article history: Received 12 May 2016 Revised 9 September 2016 Accepted 26 October 2016 Available online 29 October 2016 Keywords: Riemann–Liouville derivative Maxwell’s equations Fractional order circuits Caputo derivative a b s t r a c t In this work a classical derivation of fractional order circuits models is presented. Gener- alised constitutive equations in terms of fractional Riemann–Liouville derivatives are intro- duced in the Maxwell’s equations for each circuit element. Next the Kirchhoff voltage law is applied in a RCL circuit configuration. It is shown that from basic properties of Fractional Calculus, a fractional differential equation model with Caputo derivatives is obtained. Thus standard initial conditions apply. Finally, models for bioimpedance are revisited. © 2016 Elsevier B.V. All rights reserved. 1. Introduction In recent decades there has been a great interest on Fractional Calculus and its applications. A historical review of appli- cations is presented in Machado et al. [8]. Models involving fractional derivatives and operators have been found to better describe some real phenomena. In this work our interest is on fractional circuits models. Research on the topic is very active and lively. From the perspective of Systems Theory, an inviting introduction to Fractional Systems is presented in Ortigueira [3]. See also the work of Radwan and Salama [13]. A recent study on fractional circuits elements is that of Machado [9], where a case is made for elements even with complex valued derivatives. Focusing on the applications to fractional order circuits is the work of Elwakil [5]. Therein a motivation for fractional order circuits from Biochemistry and Medicine is described. Also, cardiac electrode tissue impedance and respiratory impedance with fractional circuit models are addressed respectively in Magin [10] and Ionescu, Derom and Keyser [7]. Their discussion on bioimpedance and listed references provide a rather complete and documented overview of the subject. Of particular interest to this work, is the survey on models from Biology and Biomedicine for fitting impedance data presented in Freeborn [4]. With regards to the Cole Model, the Laplace transform model of a fractional order circuit, it is pointed out that: ”while this model is effective at representing experimentally collected bioimpedance data, it does not provide an explanation of the underlying mechanisms”. Our objective is to provide some insight on this issue, namely, an explanation for fractional order circuits as models of electromagnetic phenomena in lossy media. Following a classical modelling approach, we derive fractional order cir- cuits models where fractional order derivatives are physically sound. As a first step, we derive models for fractional circuits ∗ Corresponding author. E-mail address: moreles@cimat.mx (M.A. Moreles). http://dx.doi.org/10.1016/j.cnsns.2016.10.020 1007-5704/© 2016 Elsevier B.V. All rights reserved.