Advances in Mathematical Physics

Volume 2016, Article ID 9740410, 8 pages

http://dx.doi.org/10.1155/2016/9740410

## On the Possibility of the Jerk Derivative in Electrical Circuits

^{1}CONACYT-Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, 62490 Cuernavaca, MOR, Mexico^{2}Departamento de Ingeniería Electrica, DICIS, Universidad de Guanajuato, Carretera Salamanca-Valle de Santiago, Km. 3.5 + 1.8 Km., Comunidad de Palo Blanco, Salamanca, GTO, Mexico^{3}Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, 62490, Cuernavaca, MOR, Mexico

Received 28 June 2016; Revised 15 August 2016; Accepted 22 September 2016

Academic Editor: Alexander Iomin

Copyright © 2016 J. F. Gómez-Aguilar 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

A subclass of dynamical systems with a time rate of change of acceleration are called Newtonian jerky dynamics. Some mechanical and acoustic systems can be interpreted as jerky dynamics. In this paper we show that the jerk dynamics are naturally obtained for electrical circuits using the fractional calculus approach with order . We consider fractional LC and RL electrical circuits with for different source terms. The LC circuit has a frequency dependent on the order of the fractional differential equation , since it is defined as , where is the fundamental frequency. For , the system is described by a third-order differential equation with frequency , and assuming the dynamics are described by a fourth differential equation for jerk dynamics with frequency .

#### 1. Introduction

Fractional calculus (FC) generalizes integer order derivatives and integrals; the mathematical formalisms were developed by Euler, Abel, Fourier, Liouville, Riemann, Grünwald, and Riesz, among many others [1–4]. The utility of the fractional derivatives and the fractional integrals resides on their feasibility for describing nonlocal properties because these consider the history and the nonlocal distributed effects of any physical system [5–8]. The classical electrical circuits consisting of resistors, capacitors, and inductors are conventionally described by integer order models. However, the electrical components have a nonconservative behavior, since they involve irreversible dissipative effects such as ohmic friction or internal friction; additionally, these components entail thermal memory and nonlinearities due to the effects of the electric and magnetic fields [9, 10]. The FC is applied to a variety of electrical circuit problems, such as domino ladders and tree structures, and to study a number of elements (coils, memristor, etc.) [11–15]. The use of fractional order operators allows us to generalize the propagation of electrical signals in devices, circuits, and networks [16–21].

With a basis on the fractional calculus concepts and previous works developed by the authors [22–27], this paper aims to describe fractional higher order power dissipation in resistive elements. This higher noninteger dynamics are known as the Newtonian jerky dynamics. These dynamics have been of interest in certain applications of mechanics, acoustics, and electrical circuits [28–37].

Here it is shown that if we have a fractional differential equation describing some physical process, then it is relatively easy to obtain the fractional jerky dynamics due the fractional order derivative . We consider fractional LC and RL electrical circuits with for different source terms. In the case of LC circuits the frequency is in general proportional to the fractional exponent of the fundamental frequency .

#### 2. Basic Concepts on Fractional Calculus

The Caputo fractional derivative for a function is defined as [3]where and is the order of the fractional derivative. From (1), it follows that the derivative of a constant is zero and the initial conditions of the fractional order differential equations are the same ones needed by the ordinary differential equations with a known physical interpretation [1].

The Laplace transform of the Caputo derivative (1) has the form [3]where is the Laplace transform of the function and . For different ranges of we have the following Laplace transform, corresponding to the fractional Caputo derivativeThe Mittag-Leffler function is defined as a power series [3]where indicates the real part; in general, is a complex quantity. For , we have ; therefore, the Mittag-Leffler function is a generalization of the exponential function. The generalization of (5) is given bywhere is the Gamma Euler function. The Laplace transform of the function iswhere is a constant. Consequently, the inverse Laplace transform is given byFrom (6) we have some common Mittag-Leffler functions [3]

#### 3. Electrical Circuits

##### 3.1. LC Electrical Circuit

A systematic way to construct fractional differential equations is given in [22]. In [23] it was shown that fractional LC circuits with constant source can be described by the equation

To be consistent with the dimensionality the parameter has dimensions of seconds. The parameter characterizes the fractional temporal structures giving an intermediate behavior between a conservative and dissipative system. The corresponding differential equations are of noninteger order and are related with a fractal space-time geometry. Therefore we have a new family of solutions [22]. The derivative considered is the Caputo type of order given by (1). The case with for was reported in [26].

In this work we study the case when and , using the Caputo derivative for LC and RL electrical circuits. We can rewrite (11) aswhereis the fractional angular frequency depending on , is the fundamental frequency of the system, and is defined in terms of a constant source . Due to dimensions of (second), we observe that the relation between the auxiliary parameter and the fractional order derivative is given by [23]In this case, (13) may be written only in terms of if , from (12) and (15) we encounter the conventional LC circuit with frequency , and then, for and also from (12) and (15), the system representation becomes a third-order differential equation with frequency . The third time derivative is called jerk (jerky dynamic) and, by definition, it is the first time derivative of the acceleration and has already been used before for assessing the comfort of motion, for example, in designing lifts and for slowly rotating rolling bearings [29]. If , the system is described by a fourth-order differential equation with frequency . Higher order derivatives permit described vibration analysis, control theory, and robotics applications [30–37].

We suppose the following initial conditions and ; then applying direct (4) and the inverse Laplace transforms to (12) we find a particular solution given bywhere is a dimensionless parameter. For we havewhich is the well-known result for the conventional case. Consider the electrical circuit LC with H, F, and volts. Plots for values of within are represented in Figures 1(a), 1(b), 1(c), and 1(d).