Generalized Differential Transform Method for Solving RLC Electric Circuit of Non-Integer Order

N. Magesh; A. Saravanan

doi:10.1515/nleng-2017-0070

Article Open Access

Generalized Differential Transform Method for Solving RLC Electric Circuit of Non-Integer Order

N. Magesh and A. Saravanan

Published/Copyright: December 8, 2017

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From the journal Nonlinear Engineering Volume 7 Issue 2

Abstract

Systematic construction of fractional ordinary differential equations [FODEs] has gained much attention nowadays research because dimensional homogeneity plays a major role in mathematical modeling. In order to keep up the dimension of the physical quantities, we need some auxiliary parameters. When we utilize auxiliary parameters, the FODE turns out to be more intricate. One of such kind of model is non-homogeneous fractional second order RLC circuit. To solve this kind of complicated FODEs, we need proficient modern analytical method. In this paper, we use two different methods, one is modern and the other is traditional, namely generalized differential transform Method (GDTM) and Laplace transform method (LTM) to obtain the analytical solution of non-homogeneous fractional second order RLC circuit. We present the solution in terms of convergent series. Though GDTM and LTM are capable to produce the exact solution of fractional RLC circuit, great strength of GDTM over LTM is that differential transform of initial conditions occupy the coefficients of first two terms in series solution so that we arrive exact solution with few iterations and also, it does not allow the noise terms while computing the coefficients. Due to this, GDTM takes less time to converge than LTM and it has been demonstrated. Furthermost, we discuss the characteristics of non-homogeneous fractional second order RLC circuit through numerical illustrations.

Keywords: Caputo fractional derivative; Fractional RLC circuit; Generalized differential transform method; Laplace transform method; Series solution

MSC 2010: 34A08; 47N70; 01-08

1 Introduction

Fractional derivative offers an incredible tool for depicting the hereditary properties of distinct materials like polymers, rocks and in many other fields and generalizes the derivative of a function to non-integer order. These advantages of fractional derivatives motivated many mathematicians since it was born on 1695. The increasing number of applications of fractional derivatives point out that there is a much demand for enhanced mathematical models (see [1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 39]). The mathematical modeling based on the fractional derivatives directs to fractional differential equations. Physical and geometrical interpretation, many other applications and qualitative behaviors of fractional differentiation and integration have been discussed in [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26].

When we take a walk down memory lane of fractional calculus, there are many definitions of fractional derivative including classical fractional derivatives like Grünwald-Letnikov, Sonin-Letnikov, Riemann - Liouville, Caputo, Hadamard, Marchaud, Riesz, Riesz-Miller, Miller-Ross, Weyl, Erdélyi-Kober and the modern fractional derivatives like, Machado, Chen-Machado, Coimbra, Katugampola, Caputo-Katugampola, Hilfer, Davidson, Chen, Atangana-Baleanu, Pichaghchi (one can refer [26, 27, 28]). These definitions of fractional derivative are not unique. Though several definitions of fractional derivative available, Riemann - Liouville and Caputo’s approaches are used widely. To examine the characteristics of a fractional system it is essential to use a suitable definition of the fractional derivative. Between these two definitions, Caputo’s approach is the most popular in research community because of the initial conditions for FODEs are the same form as that of ODEs with integer derivatives and not require fractional order initial conditions like in Riemann - Liouville’s approaches. Furthermore, Caputo derivative of any constant is zero but Riemann - Liouville derivative of any constant is not equal to zero. In addition, the Laplace transform of the Caputo fractional derivative is a generality of the Laplace transform of integer order derivative, where n is replaced by α but the same does not happen for the Riemann-Liouville case. These are significant advantages of the Caputo’s approach over the Riemann-Liouville’s approach (refer [9, 10, 11, 12] and the references therein). So, we are not exempt from choosing Caputo’s approach.

Traditionally, when we make a mathematical model using fractional derivative the time derivative of integer order is replaced by fractional derivative, that is ddt is replaced by dαdtα, 0 < α ≤ 1, where α represents the order of the derivative (see [29, 30, 31, 32, 33, 34, 35, 36, 37, 38]). It is interesting to hear that the authors of [39, 40, 41, 42] said that the above substitution is not exactly right from a physical point of view because of the time derivative ddt has dimensions of inverse seconds (s⁻¹) while the time fractional derivative dαdtα has s^−α, that is, according to the statement of Joseph Fourier [43], any physically meaningful equation (and likewise any inequality and inequation) will have the same dimensions on the left and right sides (dimensional homogeneity). So, the mathematics has now needed to change to meet the prerequisites of physical reality. In order to keep up the dimension of time, the authors of [39, 40, 41, 42] have introduced the new auxiliary parameter σ as

[1σ1−αdαdtα]=1s,0<α≤1.(1.1)

(1.1) is dimensionally reliable iff the parameter σ has dimension of time [σ] = s. When α = 1 the expression (1.1) becomes an ordinary derivative operator ddt. So, they have suggested that ddt would be replaced by [1σ1−αdαdtα] in an ordinary differential equation to form a FODE. Based on this idea, a few works have been found in the literature [44, 45, 46].

RLC circuits have numerous applications as oscillator circuits. Radio receivers and TVs employ them for tuning to choose a narrow frequency range from encompassing radio waves. In this part the circuit is frequently referred to as a tuned circuit. A RLC circuit can be utilized as a band-pass filter, band-stop filter, low-pass filter or high-pass filter. The tuning application, for occasion, is a case of band-pass sifting. The RLC filter is portrayed as a second-order circuit, implying that any voltage or current in the circuit can be depicted by a second-order differential equation in circuit investigation. In [45], authors have discussed the analytical solution of the homogeneous RLC electrical circuit of non-integer order with the help of Laplace transform but they did not discuss the non-homogeneous case. Physically speaking, they discussed the free electrical vibrations of the RLC circuit but not of the forced vibrations. So, we are interested in finding the analytical solution of non-homogeneous RLC electrical circuit of non-integer order, that is, we find the steady state solution in an RLC circuit when the presence of the impressed voltage E(t).

To do that, we consider the simple electric circuits consisting of voltage sources (electro motive force E(t)), resistors (R), inductors (L) and capacitors (C) as given in 1. This RLC circuit is described by the non-homogeneous ordinary differential equation as [47, 48],

Ld2qdt2+Rdqdt+qC=E(t).(1.2)

Fig. 1

RLC - Circuit

According to the suggestions of [39, 40, 41, 42], we generalize (1.2) as

Lσ2(1−α)d2αqdt2α+Rσ1−αdαqdtα+qC=E(t),0<α≤1,(1.3)

with the initial conditions

q(0)=q0,q′(0)=i(0)=q1,(1.4)

where q₀ is the initial charge on the capacitor and q₁ is the initial current in the circuit. This differential equation is used to find the electric charge q(t) at any time t. Solution of (1.2) is a combination of transient and steady state solution. The parameters σ and α added the complexity to solve (1.3) by analytically. However, effective common schemes for solving them cannot be found yet in the most valuable works on fractional derivatives. Analytical solutions are the only way to address the characteristics of the physical model exactly. Further, once we get analytical solution of a physical model we do not worry about the qualitative properties like stability analysis, existence and uniqueness, etc., unlike in the numerical schemes [49]. So, we need efficient modern analytical method to solve (1.3) than existing methods in the literature. One of such modern methods is GDTM and the existing method in the literature is LTM.

In this paper, we solve (1.3) by GDTM and LTM. GDTM is based on one-dimensional differential transform (refer [50, 51, 52, 53, 54, 55, 56]), generalized Taylor’s formula [57, 58] and Caputo fractional derivative and it was first appeared in the literature in the year 2008 [59, 60, 61]. In these [59, 60, 61], authors have demonstrated the efficiency of GDTM for various models. LTM has been considered as a proficient technique in solving differential equations with integer-order. However, for differential equations with non-integer order, the LTM works well only for quite simple equations, because of the complicatedness of calculating inversion of Laplace transforms [62, 63, 64, 65, 66, 67]. Most of the works related to inverse Laplace transform involves the generalizations of factorial / exponential functions (Mittag-Leffler form). It is quite complicated to understand for non-mathematicians. Furthermore, in [68], authors have concluded that the LTM is suitable for constant coefficient fractional differential equations, but it demands for non-homogeneous terms, so not all constant coefficient non-integer order differential equation can be solved by the LTM. Moreover, GDTM has many advantages over LTM, that is, differential transform of initial conditions occupy the coefficients of first two terms in series solution so that we arrive analytical solution with few iterations and it does not allow the noise terms while computing the coefficients. Due to this, GDTM takes less time to converge than LTM and it has been shown. Furthermost, we demonstrate the behaviors of the non-homogeneous RLC circuit through simulation using MATLAB version R2012a (7.14.0.739).

2 Preliminaries

In this section, the basic definitions of the Caputo derivative, GDTM and Laplace transform of Caputo derivative are given as follows:

Definition 2.1

[9, 10, 11] The Caputo fractional derivative of q(t) of order α is defined as,

Daαq(t)=1m−α∫atqm(τ)dτ(t−τ)α+1−m,(2.1)

where m−1 < α ≤ m and m ∊ N and D^m is the usual integer differential operator of order m.

Definition 2.2

[59, 60, 61] The generalized differential transform of the k^th derivative of an analytic function q(t) in one variable is defined as,

Qα(k)=1Γ(αk+1)[dαkdtαkq(t)]t=0(2.2)

where 0 < α ≤ 1, k = 0,1,2,3,⋯, q(t) is the original function and Q_α(k) is the transformed function and the differential inverse transform of Q_α(k) is defined as follows:

q(t)=∑k=0∞Qα(k)tk.(2.3)

The fundamental properties of the GDTM are listed below [59, 60, 61]:

Table 1

Fundamental properties of the DTM

S. No	Original function	Transformed function
1	q(t) = g(t) ± h(t)	Q_α(k) = G_α(k) ± H_α(k)
2	q(t) = a g(t)	Q_α(k) = a G_α(k)
3	q(t) = g(t) × h(t)	Qα(k)=∑l=0kGα(l)Hα(k−l)
4	q(t)=D0αg(t)	Qα(k)=Γ(α(k+1)+1)Γ(αk+1)Gα(k+1)
5	q(t) = t^nα	Qα(k)=δ(k−n)whereδ(k)=1k=00k≠0

Definition 2.3

[9, 10, 11, 12] The Laplace transform of Caputo fractional derivative of q(t) is defined as,

L[Dαq(t)](s)=sαL[q(t)](s)−∑k=1nsα−kfk−1(0),(2.4)

where α > 0, n − 1 < α ≤ n, (n ∊ N), f(t) ∊ Cⁿ (0,∞), f⁽ⁿ⁾(t) ∊ L₁(0,b) for any b > 0.

For further details about fractional calculus, properties of GDTM and LTM one can refer [59, 60, 61, 62, 63, 64, 65, 66, 67] which are listed in the reference section.

3 Main result

In this section, we solve (1.3) by the GDTM and LTM.

3.1 GDTM

Applying the generalized differential transform on both sides of (1.3) and using appropriate properties of GDTM which are listed in the 2 then we get the recurrence relation as,

Lσ2(1−α)Γ[α(k+2)+1]Γ(αk+1)Qα(k+2)+Rσ1−αΓ[α(k+1)+1]Γ(αk+1)Qα(k+1)+1CQα(k)=Eα(k)(3.1)

where Q_α(k) and E_α(k) are transformed functions of q(t) and E(t) respectively. When we take k = 0 in (2.2) then we have,

Qα(0)=1Γ(1)[d0dt0q(t)]t=0,Qα(0)=q0.(3.2)

Table 2

List of symbols

α	order of the derivative
s	seconds
σ	auxiliary parameter
E(t)	electromotive force
R	resistors
L	inductors
C	capacitors
q(t)	electric charge
q₀	initial charge on the capacitor
q₁	initial current in the circuit
ω	resistance(ohms)
H	henry
mF	micro farad
C	coulomb
A	ampere

When we take k = 1 in (2.2) then we have,

Qα(1)=1Γ(α+1)[ddtq(t)]t=0,Qα(1)=q1Γ(α+1).(3.3)

Now, iterating (3.1) we get the following coefficients of Q_α(k) as,

Qα(2)=σ2(1−α)LΓ(2α+1)[Eα(0)−q0C−Rq1σ1−α],Qα(3)=σ2(1−α)LΓ(3α+1)[Γ(α+1)Eα(1)−q1C−Rσ1−αL(Eα(0)−q0C−Rq1σ1−α)],Qα(4)=σ2(1−α)LΓ(4α+1)[Γ(2α+1)Eα(2)−σ2(1−α)CL(Eα(0)−q0C−Rq1σ1−α)−Rσ1−αL[Γ(α+1)Eα(1)−q1C−Rσ1−αL(Eα(0)−q0C−Rq1σ1−α)]]⋯.(3.4)

Substituting (3.4), (3.3) and (3.2) in (2.3) we get the analytical solution q(t) up to O(t^4α) as,

q(t)=q0+q1Γ(α+1)tα+σ2(1−α)LΓ(2α+1)[Eα(0)−q0C−Rq1σ1−α]t2α+σ2(1−α)LΓ(3α+1)[Γ(α+1)Eα(1)−q1C−Rσ1−αL(Eα(0)−q0C−Rq1σ1−α)]t3α+σ2(1−α)LΓ(4α+1)[Γ(2α+1)Eα(2)−σ2(1−α)CL(Eα(0)−q0C−Rq1σ1−α)−Rσ1−αL[Γ(α+1)Eα(1)−q1C−Rσ1−αL(Eα(0)−q0C−Rq1σ1−α)]]t4α+⋯.(3.5)

3.2 LTM

Applying the Laplace transform on both sides of (1.3) and using appropriate properties which are given in [62, 63, 64, 65, 66, 67] we get,

Q(s)=a3E(s)+b1[s2α−1+a1sα−1]+b2[s2α−2+a1sα−2]P(sα)(3.6)

where a1=Rσ1−αL,a2=σ2−2αCL,a3=σ2−2αL,b₁ = q(0), b₂ = q’(0), P(s^α) = s^2α + a₁s^α + a₂ and Q(s) is the transformed function.

Finding inverse Laplace transform is not quite easy for non-integer order because the variable s should be transformed to the exponent of t and also we don’t have general procedure to compute it (see the articles cited in [62, 63, 64, 65, 66, 67]). Here, we first compute P⁻¹(s^α).

P−1(sα)=1s2α+a1sα+a2,P−1(sα)=s−α(sα+a1)[1+a2s−αsα+a1],P−1(sα)=s−α(sα+a1)[1+a2s−αsα+a1]−1,P−1(sα)=s−α(sα+a1)[1−(a2s−αsα+a1)+(a2s−αsα+a1)2−⋯],P−1(sα)=s−α(sα+a1)[∑k=0∞(−1)k(a2s−αsα+a1)k],P−1(sα)=∑k=0∞(−a2)k[s−αk−α(sα+a1)k+1],P−1(sα)=∑k=0∞(−a2)k[s−αk−αsα(k+1)(1+a1sα)k+1],P−1(sα)=∑k=0∞(−a2)ks−2αk−2α(1+a1sα)−(k+1).

Using binomial result (1+x)−n=∑k=0∞(−1)k(n+k−1k)xk, we get,

P−1(sα)=∑k=0∞∑r=0∞(−a2)kk!(−a1)rr!(k+r)!s−αr−2αk−2α.(3.7)

Now, taking inverse Laplace transform on both sides of (3.6) then we have,

q(t)=a3φ−1[E(s)P−1(sα)]+b1φ−1[s2α−1P−1(sα)]+a1b1φ−1[sα−1P−1(sα)]+b2φ−1[s2α−2P−1(sα)]+a1b2φ−1[sα−2P−1(sα)],

which implies that the analytical solution of (1.3) as,

q(t)=a3φ−1[E(s)∑k=0k=∞∑r=0r=∞(−a2)k(−a1)r(k+r)!s−αr−2αk−2αk!r!]+∑k=0k=∞∑r=0r=∞(−a2)k(−a1)r(k+r)!k!r!{b1tαr+2αkΓ(αr+2αk+1)+a1b1tαr+2αk+αΓ(αr+2αk+α+1)+b2tαr+2αk+1Γ(αr+2αk+2)+a1b2tαr+2αk+α+1Γ(αr+2αk+α+2)},(3.8)

where E(s) is the Laplace transform of E(t) and φ⁻¹ represents inverse Laplace transform of Q(s).

4 Numerical Results and discussion

If energy stored within the circuit is released into it, currents will start to flow but resistance elements will break up the energy and finally the currents pass away to zero and, usually, no energy will remain. The circuit’s behavior under these conditions is termed its natural behavior or transient response, which corresponds to the solution of homogeneous part of a differential equation. The characteristics of the circuit is analyzed by the discriminant R2−4LC. If the discriminant is less than zero then the circuit is under damped, greater than zero then the circuit is over damped and equal to zero then the circuit is critically damped. Further, in [45] the authors clearly mentioned that the relationship between the σ and α. For under-damped case as α=[1LC−R24L2]12σ and for over-damped case as α=[R24L2−1LC]12σ.

4.1 Under-damped case

When E(t) = 0, L = 0.25H, R = 10Ω, C = 0.001mF, q₀ = 2C and q₁ = i(0) = 0A, the circuit is under-damped.

Case: 1

If α=14 and σ=1240, then

GDTM Solution:

q(t)=2−(4×2000240×4×15)t12Γ(32)+(4×40×2000240×4×15×(240)34)t34Γ(74)+4240×4×15(8000000240×4×15−3200000240×4×15)t+⋯.(4.1)

q(t)=2−(253×15)t12Γ(32)+(259×(15)14)t34Γ(74)+2518t+⋯.(4.2)

LTM Solution:

q(t)=∑k=0k=∞∑r=0r=∞(−4000(240)32)k(−40(240)34)r(k+r)!k!r!×{2tr4+k2Γ(r4+k2+1)+80(240)34tr4+k2+14Γ(r4+k2+14+1)},(4.3)

which implies that

q(t)=2−(253×15)t12Γ(32)+(259×(15)14)t34Γ(74)+2518t+⋯.(4.4)

From (4.1) and (4.2), we observe that GDTM do not allow the noise term while calculating the coefficients of O(t) in each and every term in the series solution whereas from (3.8) and (4.4) we can see that LTM admits the noise terms and they have been canceled so that we are spending more time to get each components of the series solution. Further, GDTM finds the coefficients of first two terms of the series solution using the initial conditions and the rest of the components are found by the recurrence relation in the straight forward manner but the same are not guaranteed by the LTM since noise terms are recurring in each and every iterations. This is the main advantage of GDTM over LTM. Moreover, from (4.2) and (4.4), we observe that both the methods are capable to provide the same analytical solution. So, we give the analytical solutions of the following cases without describing the steps much in detail.

Case: 2

If α=12 and σ=1120 then GDTM and LTM solutions (3.5) and (3.8) yield,

q(t)=2−(2003)t+(40003×30)t32Γ(52)+(40003)t2Γ(3)+⋯.(4.5)

Case: 3

If α=34 and σ=180 then GDTM and LTM solutions (3.5) and (3.8) provide,

q(t)=2−(20005)t32Γ(52)+(40000(5)34)t94Γ(134)+240000t3Γ(4)+⋯.(4.6)

Case: 4

If α = 1 then GDTM and LTM solutions (3.5) and (3.8) give,

q(t)=2−8000t22!+320000t33!+⋯.(4.7)

This solution can be represented as,

q(t)=2e−20t(cos⁡60t+13sin⁡60t),(4.8)

which is the exact solution of integer order ordinary differential equation. In this under-damped case, the transient solution contains the factor e^−20t, so q(t) → 0 as t → ∞ and also the charge on the capacitor oscillates as it decays, that is, the capacitor is charging and discharging as t → ∞. Moreover, from Figure 2 the charge oscillates about 0 before finally decaying to 0.

Fig. 2

The underdamped response of the circuit when L = 0.25H, R = 10Ω, C = 0.001mF, q₀ = 2C, q₁ = i(0) = 0A.

4.2 Over-damped case

When L = 0.25H; R = 20Ω; C=1300mF; E(t) = 100V, q₀ = 4C and q₁ = i(0) = 0A, the circuit is over-damped.

Case: 1

If α=14 and σ=180 then GDTM and LTM solutions (3.5) and (3.8) yield,

q(t)=4−(4400(80)32)t12Γ(32)+(4400(80)54)t34Γ(74)−(71516)t+⋯.(4.9)

Case: 2

If α=12 and σ=140 then GDTM and LTM solutions (3.5) and (3.8) give,

q(t)=4−110t+(4400(10)12)t32Γ(52)−14300t2Γ(3)+⋯.(4.10)

Case: 3

If α=34 and σ=380 then GDTM and LTM solutions (3.5) and (3.8) give,

q(t)=4−4400×(380)12t32Γ(52)+352000×(380)34t94Γ(134)+858000t3Γ(4)+⋯.(4.11)

Case: 4

If α = 1 then GDTM and LTM solutions (3.5) and (3.8) give,

q(t)=4−4400×t2Γ(3)+352000×t3Γ(4)−22880000t4Γ(5)+⋯.(4.12)

This solution can be represented as closed form as,

q(t)=336e−20t−116e−60t+13.(4.13)

It is observed from Figure 3 that the solution is aperiodic exponential decay function with no oscillations and the system returns to equilibrium exponentially and also it has a very large settling time. In addition, it is noticed that when a constant electro motive force is applied to a system in which the energy is finally stored in an ideal capacitor, one-half of the energy input to the system is transformed into heat. Thus, under the condition of a constant applied force, a capacitor is capable of storing, at best, only 50 percent of the applied energy in reversible form.

$Fig. 3 The over-damped response of the circuit when L = 0.25H; R = 20Ω; C=1300$\begin{array}{} C=\frac{1}{300} \end{array} $mF; E(t) = 100v, q0 = 4C, q1 = i(0) = 0A.$

Fig. 3

The over-damped response of the circuit when L = 0.25H; R = 20Ω; C=1300mF; E(t) = 100v, q₀ = 4C, q₁ = i(0) = 0A.

4.3 Critically-damped case

From a physical point of view this case is insignificant because of the quantities R, L and C are all experimentally measured quantities-there is no possibility that these measurements could be such that R2=4LC exactly. Any small deviation from equality will result in either of the previous two cases. So, it requires a separate discussion.

5 Conclusions

Behavior of RLC circuit has been described well by oscillating system which describes the charge on the capacitor as a fractional non-homogeneous second order differential equation. It is not quite easy to solve systematically constructed FODE because of the presence of auxiliary parameter in it. So we used two different methods, one is modern and the other is traditional, namely GDTM and LTM respectively. We have compared the results obtained by GDTM and LTM and it reveals that GDTM and LTM are capable for providing the exact solution of FODE. Moreover, GDTM provides the exact solution of FODE without allowing noise terms unlike LTM. In addition, generalized differential transform of initial conditions occupied the first two components of the power series solution so that we arrived the exact solution in few iterations. This is the main advantage of GDTM over LTM. Taking into account of these advantages of GDTM, we conclude that it is quite better than LTM to solve systematically constructed FODE.

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Received: 2017-3-31

Accepted: 2017-10-25

Published Online: 2017-12-8

Published in Print: 2018-6-26

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Articles in the same Issue

https://doi.org/10.1515/nleng-2017-0070

Keywords for this article

Caputo fractional derivative; Fractional RLC circuit; Generalized differential transform method; Laplace transform method; Series solution

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