Advances in Mathematical Physics
Volume 2016 (2016), Article ID 9740410, 8 pages
http://dx.doi.org/10.1155/2016/9740410

Research Article

On the Possibility of the Jerk Derivative in Electrical Circuits

J. F. Gómez-Aguilar,¹ J. Rosales-García,² R. F. Escobar-Jiménez,³ M. G. López-López,³ V. M. Alvarado-Martínez,³ and V. H. Olivares-Peregrino³

¹CONACYT-Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, 62490 Cuernavaca, MOR, Mexico
²Departamento 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
³Centro 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 derivative

The 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 by

where is the Gamma Euler function. The Laplace transform of the function is

where is a constant. Consequently, the inverse Laplace transform is given by

From (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) as

where

is 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 by

where is a dimensionless parameter. For we have

which 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).

(a)

(b)

(c)

(d)

Figure 1: Simulation of (16) for (a) (classical case); (b) ; (c) ; and (d) .

Suppose now that we have an alternating source; thus

where , is given by (15), and is the frequency of the source. Applying Laplace transform (4) with the initial conditions and we have

for the third term in (19), we select the highest power of as a common factor from the denominator, and then we expand the denominator in an alternating geometric series [38]; as a result we have

Applying the inverse Laplace transform, finally, the solution is

For the particular case of we substitute (10) and on (21) to obtain

Taking the same values for , and as before, plots for different values of within are represented in Figures 2(a), 2(b), 2(c), and 2(d).

(a)

(b)

(c)

(d)

Figure 2: Simulation of (21) for (a) (classical case); (b) ; (c) ; and (d) .

Table 1 shows the critical values of for different values of .

Table 1: Critical values of () for different values of .

Values of ()

Value

Value

1
1
1.1
0.990514

1.2
0.964193

1.3
0.924309

1.4
0.874075

1.5
0.816497

1.6
0.754272

1.7
0.689741

1.8
0.624859

1.9
0.561205

1.95

0.530233

2.0
0.50

3.2. RL Electrical Circuit

The fractional differential equation corresponding to a RL circuit with constant source is given by [23]

and we can rewrite it as

where and . Using the initial conditions and and applying the direct Laplace transform, we have

Solving for , we have

For the last terms in (26) we take the highest power of as a common factor from the denominator and then we expand in an alternating geometric series [38]; given that

then and . Substituting these expressions in the inverse Laplace transform, the solution is

Consider the electrical circuit RL with ,  H, and volts. Plots for values of within are represented in Figures 3(a), 3(b), 3(c), and 3(d).

(a)

(b)

(c)

(d)

Figure 3: Simulation of (28) for (a) (classical case); (b) ; (c) ; and (d) .

Considering an alternating source we have

rewriting

where and are the same defined for the RL circuit with constant source. Using the initial conditions and and applying the direct Laplace transform, we have

For the last term in (31) we take the highest power of as a common factor from the denominator and then we expand in an alternating geometric series [38]; then we apply the inverse Laplace transform

here, we used relation (27) and the aforementioned expressions defining and .

Taking the same values for , and , plots for values of within are represented in Figures 4(a), 4(b), 4(c), and 4(d).

(a)

(b)

(c)

(d)

Figure 4: Simulation of (32) for (a) (classical case); (b) ; (c) ; and (d) .

For both simulations, the numerical solutions show a change in the amplitude of the charge and variations in the phase exhibits fractality in time to different scales and shows the existence of heterogeneities in the electrical components (resistance, inductance). These behaviors depend on the fractional derivative order and modified the constant time of the electrical circuits. The numerical solution exhibits temporal fractality at different scales as well as the existence of material heterogeneities in the electrical components.

Table 2 shows different values of the constant time for arbitrary values of .

Table 2: Values of the constant time for arbitrary values of .

Values of constant time

Value

Value

1
0.1
1.1
0.0786793

1.2
0.0608364
1.3
0.0463252

1.4
0.0347976
1.5
0.0258199

1.6
0.0189465
1.7
0.0137621

1.8
0.00990335
1.9
0.00706516

1.95
0.00594931
2.0
0.005

4. Conclusions

In this paper, we studied the behavior of third- and fourth-order differential equations modeling LC and RL electrical circuits. These high order equations are naturally obtained using fractional calculus for derivative orders chosen in the interval . Results show that the LC electrical circuit behavior is strongly dependent on the fractional order, which was varied within the interval and determines the frequency according to (15). The fractional time constant is modified in the RL electrical circuit with the order of the fractional derivative. This constant time tends to move forward in time as the exponent changes the dynamics and the relaxation times of the system (they are faster). For all figures, in the range (1 ; 2) the charge exhibits fractional jerky dynamics; furthermore, when the order of the derivative is equal to 2 the charge exhibits Jounce dynamics. Our results indicate that the fractional order has an important influence on the charge and the general solution of the fractional circuit depends on the parameter . These solutions represent a new family of solutions for the charge. In the range the charge is unstable, the charge exhibits an increment of the amplitude, and the behavior becomes anomalously dispersive (the charge increases with increasing order of ). The electrical circuits studied and their respective analogue mechanical system can be used to analyze the vibration levels of machinery, serial mechanisms, robotics, oscillating circuits modeling, and instability of electrical and mechanical circuits, to evaluate reconfigurable machines or to make mobility analysis or algebraic formulations of motion equations.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All the authors have equally contributed in this paper and read the final version before the submission.

Acknowledgments

José Francisco Gómez Aguilar acknowledges the support provided by CONACYT: Cátedras CONACYT para Jovenes Investigadores 2014.

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