Analytic Solution for the RL Electric Circuit Model in Fractional Order

and Applied Analysis 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 0.7 0.8 0.9 1


Introduction
Fractional calculus, involving derivatives of integrals of noninteger order, is the natural generalization of the classical calculus, which during recent years became a powerful and widely used tool for better modeling and control of processes in many areas of science and engineering [1][2][3].Many physical phenomena have been discussed by fractional calculus approach [4].In many applications, fractional calculus provides more accurate models of the physical systems than ordinary calculus does.Fundamental physical considerations in favor of the use of models based on derivatives of noninteger order are given in [5].Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes [6].This is the main advantage of fractional calculus in comparison with the classical integer-order models, in which such effects are in fact neglected.Fractional order models have been already used for modeling of electrical circuits (such as domino ladders and tree structures) and elements (coils, memristor, etc.).The review of such models can be found in [7].
Recently, a fractional differential equation has been suggested that combines the simple harmonic oscillations of an LC circuit with the discharging of an RC circuit.The behavior of this new hybrid circuit without sources has been analyzed [8].In the work of [9], the simple current sourcewire circuit has been studied fractionally using direct and alternating current source.It was shown that the wire acquires an inducting behavior as the current is initiated in it and gradually recovers its resisting behavior.Recently, Guia et al. [10] have analyzed time delay, rise time, and settling time of an RC circuit.In this paper, in the framework of fractional calculus, we are interested in the solution of an  circuit for different source terms.

Preliminary
The function () of the complex variable  defined by is called the Laplace transform of the function () [3].For the existence of the integral (1), the function () must be piecewise continuous and of exponential order .
The original () can be restored from the Laplace transform () with the help of the inverse Laplace transform [3] where  0 lies in the right half plane of the absolute convergence of the Laplace integral (1).In this work, we use the Caputo definition of the fractional derivative [3] where  ∈ R is the order of the fractional derivative and  − 1 <  ≤  ∈ N = {1, 2, 3, . ..},   () = (  /  )(), and Γ(⋅) is the Euler Gamma function.We consider the case  = 1; then 0 <  ≤ 1; that is, in the integrand (3), there is only first derivative.The Caputo definition of the fractional derivative is very useful in the time domain studies, because the initial conditions for the fractional order differential equations with the Caputo derivatives can be given in the same manner as for the ordinary differential equations with a known physical interpretation.
The formula for the Laplace transform of the Caputo fractional derivative (3) has the form [3] where   is the ordinary derivative.The inverse Laplace transform requires the introduction of the Mittag-Leffler function [3], which is defined as where Γ(⋅) is the Gamma function.When  = 1, from (5), we have Therefore, the Mittag-Leffler function includes the exponential function as a special case.

Formulation of Fractional Differential Equation Models for Flow of Electricity in Resistance-Inductance Circuit
The differential equation for the  circuit shown in Figure 1 is given by where  is the current and  is the inductance.The solution of  circuit is reported by Kreyszig [11] as In this paper, we develop the resistance-inductance circuit model in the form of fractional differential equation as If lim  → 1 (  /  ) = /, then (9) reduces to its original form (/) +  = ().
Solution of (9) Using the Initial Condition ((0) = , ( > 0)).Applying the Laplace transform on (9) using the initial condition (0) = , ( > 0), we have Now, applying the inverse Laplace transform and convolution, we have Here, we obtain the solution of different cases of the resistance-inductance circuit model ( 9) for different source terms.
Case 1 (when no electromotive force is applied (no source term), i.e., () = 0).In this case, (9) becomes and the initial condition is Solution.Rewrite (12a) as Taking the Laplace transform on both sides and using (12b), we get Taking the inverse Laplace transform on both sides, we obtain Case 2 (when constant electromotive force is applied, i.e., () =  0 ).In this case, (9) becomes and the initial condition is Rewrite (16a) as Taking the Laplace transform on both sides and using (16b), we get Taking the inverse Laplace transform on both sides, we have on applying the convolution theorem and using and from (15), we have In Figure 2, we observe the interesting behavior of current by using fractional calculus approach for different values of .When  = 0.1, the current decreases very sharply and it moves towards stability as time increases.On increasing the value of , the current increases for the specific time and afterwards attains its stability.Finally, when  = 1, then current shows its natural behavior.This exhibits the behavior of current for different values of  with respect to time () before it attends the natural behavior.