Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 5976301, 5 pages

http://dx.doi.org/10.1155/2016/5976301

## Fractional-Order Two-Port Networks

^{1}Engineering Mathematics and Physics Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt^{2}Department of Electrical and Computer Engineering, University of Sharjah, College of Engineering, P.O. Box 27272, UAE^{3}Nanoelectronics Integrated Systems Center (NISC), Nile University, Giza 12588, Egypt^{4}Department of Electrical and Computer Engineering (ECE), University of Calgary, AB, Canada T2N 1N4

Received 25 March 2016; Accepted 17 April 2016

Academic Editor: Riccardo Caponetto

Copyright © 2016 M. E. Fouda et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce the concept of fractional-order two-port networks with particular focus on impedance and admittance parameters. We show how to transform a impedance matrix with fractional-order impedance elements into an equivalent matrix with all elements represented by integer-order impedances; yet the matrix rose to a fractional-order power. Some examples are given.

#### 1. Introduction

Two-port networks are widely used in linear circuit analysis and design [1, 2]. The system under consideration is represented by a describing matrix which relates its input and output variables (voltages and currents). Such a representation enables the treatment of the system as a black box where the internal details become irrelevant. It also offers an extremely efficient computational technique which can be used to model series, parallel, or cascade interconnects of several systems. Standard Network Analyzers can be configured to measure several types of two-port network parameters including impedance, admittance, transmission, and scattering parameters.

Consider, for example, the impedance matrix representation of a system in which case we havewhere () are the voltages (currents) at the input port and output port, respectively, as shown in Figure 1. All elements in the impedance matrix are measured in and if the network is known to be symmetrical while if it is known to be reciprocal. However, with the increasing use of fractional-order impedance models, particularly in representing supercapacitors [3, 4], energy storage devices [5], oscillators [6], filters [7], and new electromagnetic charts [8], it is possible that the elements of are of fractional order. Consider the simple case of the grounded impedance , shown in Figure 2(a). Treated as a two-port network, this impedance is described by the impedance matrix Let be a supercapacitor operating in its Warburg mode, where ; . In this region of operation, the magnitude is proportional to , the phase angle is fixed at , and is the pseudocapacitance of the device [9, 10]. As a two-port network, this device would be described as Therefore all elements of the matrix are of fractional order. However, we can rewrite the above equation in the alternative formsince It is clear that the elements of are all integer-order impedances, each representing a capacitor of () Farad whereas the power of the matrix is fractional; that is, . In this paper we seek to generalize this procedure by obtaining the equivalent matrix and its fractional exponent such that . The procedure is not restricted to the impedance matrix and can be applied to any other type of two-port network parameters. The main advantage of this conversion is that an equivalent circuit of can be easily obtained with integer-order components. For example, if is reciprocal, then its equivalent circuit is that shown in Figure 2(b). However, it is not yet known how to use this equivalent circuit in association with the fractional exponent of the matrix to construct an overall equivalent model of the originally fractional-order two-port network.