On the Possibility of the Jerk Derivative in Electrical Circuits

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 Ingenieŕıa 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


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][2][3][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][6][7][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][12][13][14][15].The use of fractional order operators allows us to generalize the propagation of electrical signals in devices, circuits, and networks [16][17][18][19][20][21].
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  − 1 <  < .We consider fractional LC and RL electrical circuits with 1 ⩽  < 2 for different source terms.In the case of LC circuits the frequency  is in general proportional to the fractional exponent of the fundamental frequency  0 .

Basic Concepts on Fractional Calculus
The Caputo fractional derivative  0    for a function () is defined as [3] where  = 1, 2, . . .∈ N 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  ∈ N.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 R indicates the real part; in general,  is a complex quantity.For  = 1, we have   ; therefore, the Mittag-Leffler function is a generalization of the exponential function.The generalization of ( 5) is given by where Γ(⋅) is the Gamma Euler function.The Laplace transform of the function where  is a constant.Consequently, the inverse Laplace transform is given by From (6) we have some common Mittag-Leffler functions [3]

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  0 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  = 1 for 0 <  ⩽ 1 was reported in [26].
In this work we study the case when  = 2 and 1 <  ⩽ 2, using the Caputo derivative for LC and RL electrical circuits.We can rewrite (11) as where is the fractional angular frequency depending on ,  0 = 1/ √  is the fundamental frequency of the system, and  = ( 2(1−) /) 0 is defined in terms of a constant source  0 .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  = 1, from ( 12) and ( 15) we encounter the conventional LC circuit with frequency  =  0 , and then, for  = 3/2 and also from ( 12) and (15), the system representation becomes a third-order differential equation with frequency  2 (3/2) = (2/3) 3 0 .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  = 2, the system is described by a fourth-order differential equation with frequency (2) =  2 0 /2.Higher order derivatives permit described vibration analysis, control theory, and robotics applications [30][31][32][33][34][35][36][37].We suppose the following initial conditions (0) =  0 and q (0) =  0 ; then applying direct (4) and the inverse Laplace transforms to (12) we find a particular solution given by

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where t =  0  is a dimensionless parameter.For  = 1 we have which is the well-known result for the conventional case.
Suppose now that we have an alternating source; thus where  = ( 2(1−) /) 0 ,  is given by (15), and   is the frequency of the source.Applying Laplace transform (4) with the initial conditions (0) =  0 and q (0) =  0 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  = 1 we substitute (10) and  2,2 (− 2 0  2 ) = (1/ 0 ) sin  0  on (21) to obtain  Taking the same values for , , and  0 as before, plots for different values of  within 1 <  ⩽ 2 are represented in Figures 2(a), 2(b), 2(c), and 2(d).
Table 1 shows the critical values of () for different values of .

RL Electrical Circuit.
The fractional differential equation corresponding to a RL circuit with constant source  0 is given by [23]   2(1−) and we can rewrite it as where  = (/) 1− and  = ( 0 /) 2(1−) .Using the initial conditions (0) =  0 and q (0) =  0 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  = (/)   1− and  = ( 0 / 2 )(/) 2  2(1−) .Substituting these expressions in the inverse Laplace transform, the solution is Consider the electrical circuit RL with  = 100 Ω,  = 10 H, and  0 = 10 volts.Plots for values of  within 1 <  ⩽ 2 are represented in Figures 3(a   Considering an alternating source we have rewriting where  and  are the same defined for the RL circuit with constant source.Using the initial conditions (0) =  0 and q (0) =  0 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 .
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  = (/)   1− for arbitrary values of .

Conclusions
In this paper, we studied the behavior of third-and fourthorder 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 [1,2).Results show that the LC electrical circuit behavior is strongly dependent on the fractional order, which was varied within the interval [1, 2) and determines the frequency according to (15).The fractional time constant   = (/)   1− 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  ∈ (1, 2) 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.

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

Table 2 :
Values of the constant time  = (/)   1− for arbitrary values of .