Cent. Eur. J. Phys.DOI: 10.2478/s11534-013-0236-y

Central European Journal of Physics

Analysis on the time and frequency domain for the RCelectric circuit of fractional order

Research Article

Manule Guía∗, Francisco Gómez, Juan Rosales

Departamento de Ingeniería Eléctrica. División de Ingenierías Campus Irapuato-Salamanca. Universidad de Guanajuato,Carretera Salamanca-Valle de Santiago, km. 3.5 + 1.8 km. Comunidad de Palo Blanco, 36885, Salamanca Guanajuato. México

Received 31 January 2013; accepted 11 April 2013

Abstract: This paper provides an analysis in the time and frequency domain of an RC electrical circuit described bya fractional differential equation of the order 0 < α ≤ 1. We use the Laplace transform of the fractionalderivative in the Caputo sense. In the time domain we emphasize on the delay, rise and settling times,while in the frequency domain the interest is in the cutoff frequency, the bandwidth and the asymptotes inlow and high frequencies. All these quantities depend on the order of differential equation.

PACS (2008): 45.10.Hj; 84.30.Bv; 84.32.Ff; 84.32.Tt

Keywords: fractional calculus • Caputo derivative • electrical circuits • Mittag-Leffler function© Versita sp. z o.o.

1. Introduction

Fractional calculus (FC), involving derivatives an integralsof non-integer order, is the natural generalization of theclassical calculus, which during recent years became apowerful and widely used tool for better modeling andcontrol of processes in many areas of science and engi-neering [1–8]. Many physical phenomena have “intrin-sic" fractional order description and so FC is necessary inorder to explain them [9]. In many applications FC pro-vide more accurate models of the physical systems thanordinary calculus do. Since, its success in descriptionof anomalous diffusion [10–13] non-integer order calculusboth in one and multidimensional space, it has becomean important tool in many areas of physics, mechanics,∗E-mail: guia@ugto.mx

chemistry, engineering, finances and bioengineering [14–17]. Fundamental physical considerations in favor of theuse of models based on derivatives of non-integer orderare given in [18, 19]. The Lagrangian and Hamilton for-mulation of dynamics and electromagnetic field in view offractional calculus has been reported in [20–25]. Frac-tional derivatives provide an excellent instrument for thedescription of memory and hereditary properties of variousmaterials and processes [26]. This is the main advantageof FC in comparison with the classical integer-order mod-els, in which such effects are in fact neglected. Anotherlarge field which requires the use of FC is the theory offractals [27, 28]. The development of the theory of fractalshas opened further perspective for the theory of fractionalderivatives, especially in modeling dynamical processes inself-similar and porous structures. Fractional-order mod-els have been already used for modeling of electrical cir-cuits (such as domino ladders, tree structures, etc.) andelements (coils, memristor, etc.). The review of such mod-

Analysis on the time and frequency domain for the RC electric circuit of fractional order

els can be found in [29–31].Recently, it has been suggested a fractional differentialequation that combines the simple harmonic oscillationsof an LC circuit with the discharging of an RC circuit. Thebehavior of this new hybrid circuit without sources hasbeen analyzed [32]. In the work [33] the simple currentsource-wire circuit has been studied fractionally using di-rect and alternating current source. It was shown that thewire acquires an inducting behavior as the current is ini-tiated in it and gradually recovers its resisting behavior.In this paper, in the framework of fractional calculus, weare interested in the analysis of the time delay, rise timeand settling time of an RC circuit. Also, in the frequencydomain the interest is in the cutoff frequency, the band-width and the asymptotes in low and high frequencies.2. PreliminaryIn this work we use the Caputo’s definition of the fractionalderivative [4]dγf (t)dtγ = C0Dγ

t f (t) = 1Γ(n− γ)∫ t

0f (n)(τ)(t − τ)γ−n+1 dτ, (1)

where γ ∈ R is the order of the fractional derivative andn − 1 < γ ≤ n ∈ N = {1, 2, 3, ...}, f (n)(τ) = dn

dtn f (τ) andΓ(·) is the Euler Gamma function. We consider the casen = 1, then, 0 < γ ≤ 1, i.e., in the integrand (1) there isonly first derivative. The Caputo’s definition of the frac-tional derivative is very useful in the time domain stud-ies, because the initial conditions for the fractional orderdifferential equations with the Caputo derivatives can begiven in the same manner as for the ordinary differentialequations with a known physical interpretation.The formula for the Laplace transform of the Caputo frac-tional derivative (1) has the form [4]

L[ dγdtγ f (t)] = sγF (s)− n−1∑

k=0 sγ−k−1f (k)(0), (2)

where f (k) is the ordinary derivative. The inverse Laplacetransform requires the introduction of the Mittag-Lefflerfunction, which is defined asEβ (t) = ∞∑

m=0tmΓ(βm+ 1) , (β > 0), (3)

where Γ(·) is the Gamma function. When β = 1, from (3),we haveE1(t) = ∞∑

m=0tmΓ(m+ 1) = ∞∑

m=0tmm! = et . (4)

Therefore, the Mittag-Leffler function includes the expo-nential function as a special case.3. Fractional RC circuitThe differential equation for the RC circuit showing in theFigure 1 is given by

E(t) = R dq(t)dt + q(t)C , (5)

where q(t) is the charge in the capacitor measured inCoulombs, R is the resistance in ohms, C is the capaci-tance in farads, E(t) is the source of excitation in volts andt is the time in seconds. Assuming, for transient analysis,

+ –VR

R

+–E(t)

+VC

–C

I

Figure 1. RC Circuit.

that E(t) = u(t) where u(t) is the unit step function, wecan write the equation (5) in a dimensionless form.d( q(t)

C

)d ( t

RC) + q(t)

C = u(t). (6)The equation (6) may be written as

dv̄c (̄t)dt̄ + v̄c (̄t) = u(̄t), (7)where v̄c (̄t) = q(̄t)

C is the voltage on the capacitor of theparallel plates type. Equation (7) is dimensionless, thenits corresponding fractional differential equation is ob-tained directly changing the order of the derivative. Ithas the form dαvc (̄t)dt̄α + vc (̄t) = u(̄t), (8)where 0 < α ≤ 1.

Manule Guía, Francisco Gómez, Juan Rosales

4. Solution and analysis of the RCcircuit in the time domainUsing the formula (2), the solution of the equation (8),with vc0 = 0, has the form

Vc(s) = 1s̄(s̄α + 1) , (9)

where s̄ = RCs. Applying the inverse Laplace transform,equation (9) takes the formvc(t) = [1− ∞∑

n=0 (−1)n ( tRC)nαΓ(nα + 1)

] u(t), [volts] (10)where we have returned to the starting variables R, C .In the Figure 2 we show some value of α . To say that

0.2

0.4

0.6

0.8

1.0

0.250.500.750.800.920.96α = 1.00

t̄

vc(t̄) [volts]

1 2 3 4 5

Figure 2. Plot of (10) for different values of α.

a RC circuit has been analyzed in the time domain, isrequired to determine its response in time indicating themost important times in which certain conditions are sat-isfied. Here are the definitions of these times to a systemof order one.1. Delay time: is the time it takes the response toreach 10% of its final value.2. Rise time: is the time taken for the response to gofrom 10% to 90% of its final value.3. Settling time: is the time at which the responsereaches 98% of its final value.

To make this analysis we need a test function. For sim-plicity, we take the unit step function u(t). Using (10) andthe settling time definition, we have0.98 = 1− ∞∑

n=0 (−1)n t̄nαssΓ(nα + 1) →→ f (̄tss) = ∞∑

n=0 (−1)n t̄nαssΓ(nα + 1) − 0.02 = 0, (11)

from thist̄(m+1)ss = t̄(m)

ss −f (̄t(m)

ss )f ′ (̄t(m)

ss ) , (12)and (m) means the m-th iteration of the Newton-Raphsonmethod.In Figure 3 is plotted the settling time t̄ss for 0.17 ≤α ≤ 1. From the graph can be seen that as the orderof the fractional derivative α tends to zero the settlingtime tends to infinity, that is, the settling time decreaseswith increasing the order of the fractional derivative, like-wise, the settling time sensitivity regarding the order ofthe derivative also decreases. No values are plotted tssto smaller values α , because these are very large. Ap-parently, there is an exponential growth settling time forsmaller values of the derivative order.

α

log10(t̄ss)

0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

t̄ss = tssRC

Figure 3. Plot of the settling time versus fractional order derivative,α.

Using (10) and the time delay definition, we have0.10 = 1− ∞∑

n=0 (−1)n t̄nαdΓ(nα + 1) →→ f (̄td) = ∞∑

n=0 (−1)n t̄nαdΓ(nα + 1) − 0.90 = 0, (13)from this

t̄(m+1)d = t̄(m)

d −f (̄t(m)

d )f ′ (̄t(m)

d ) . (14)In Figure 4 is plotted the time delay t̄d para 0.14 ≤ α ≤ 1.From the graph we can see that as the order of the frac-tional derivative α tends to zero the time delay tends tozero, that is, the time delay decreases with the decreases

Analysis on the time and frequency domain for the RC electric circuit of fractional order

of the order of derivative, likewise, the delay sensitivity ofthe order of derivative increases with decreasing the orderof the derivative. No values are plotted td for smaller val-ues α because these are very small. Apparently, it has anexponential decreases settling time for values under theorder of the derivative.α

log10(t̄d)0.2 0.4 0.6 0.8 1.0

-1.0

-3.0

-5.0

-7.0

t̄d = tdRC

Figure 4. Plot of the delay time versus fractional order derivative α.

In Figure 5 it is shown the rise time of the RC circuitas the order of the derivative varies from 0.11 to 1.0. Ascan be seen, the rise time increases as the order of thederivative decreases, becoming both more sensitive.

α

log10(t̄r)

0.2 0.4 0.6 0.8 1.00

2

4

6

8

t̄r = trRC

Figure 5. Plot of the rise time versus fractional order derivative α.

5. Analysis of the RC circuit in thefrequency domainIn order to analyze the RC circuit in the frequency domainwe use (9) multiplied by s̄ to obtain

H(s̄) = 11 + s̄α . (15)

The graphs representing the response in the frequency do-main are the Bode plots, but also can be used the Nyquistor Nichols plots. We use the Bode plots for the magni-tude and phase. Bode plots of magnitude and phase areconstructed from the relationH(jω) = H(s)|s=jω. (16)

From (16) we obtain the following two equations, fromwhich the Bode plots are constructed|H(jω)|dB = 20 log10(|H(jω)|), (17)argH(jω) = 180 argH(jω)

π . (18)From (15) we obtain

|H(jω̄)|dB = −10 log10[(1 + ω̄α cos ( πα2 ))2+ ω̄2αsen2( πα2 )] (19)

argH(jω̄) = − 180π arctan [ ω̄α sen( πα2 )1+ω̄α cos ( πα2 )], (20)

where ω̄ = ωRC . To construct the diagrams in decadeswe make ω̄ = 10n, obtaining|H|dB = −10 log10

[(1 + 10nα cos ( πα2 ))2 +102nαsen2( πα2 )] (21)argH = − 180

π arctan [ 10nα sen ( πα2 )1+10nα cos ( πα2 )]. (22)In Bode diagrams, the frequency response to the magni-tude and phase of the RC circuit are shown in Figure 6and Figure 7. From these plots, it can be see that for thecase of the magnitude response, Figure 6, the slope of thehigh frequency asymptote decreases as the order of thederivative decreases and the cutoff frequency moves to theleft. In the case of the phase Figure 7, it is noted a verylarge variation of the phase of the frequency rupture asthe order of the derivative decreases.In Figures 8 and 9 show a graphical behavior of the cutofffrequency and phase respectively as it varies α from zeroto one.6. ConclusionIn this paper we have studied the behavior of an RC elec-trical circuit in the framework of the fractional calculus.The explanation about the settling, delay and rise timeshas been analyzed, while in the frequency domain the

Manule Guía, Francisco Gómez, Juan Rosales

ω̄

|H|dB

-10

-20

-30

-40

10−1 100 101 102

α = 1.0

0.94

0.92

0.80

0.75

0.50

0.25

Figure 6. Bode diagram magnitude for some values of α.

0.25

0.50

0.75

0.80

0.92

0.94

α = 1.00

ω̄

arg(|H|)

-15

-30

-45

-60

-75

-90

10−1 100 101 102

Figure 7. Bode diagram phase for some values of α.

interest is in the cutoff frequency, the bandwidth and theasymptotes in low and high frequencies. We concludethat; the settling time decreases with increasing theorder of the fractional derivative, likewise, the settlingtime sensitivity regarding the order of the derivativealso decreases. The rise time increases as the order ofthe derivative decreases, becoming both more sensitive.The time delay decreases with the decreases of the

α

ω̄c

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0ω̄c = RCωc

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0ω̄c = RCωc

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0ω̄c = RCωc

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0ω̄c = RCωc

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0ω̄c = RCωc

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0ω̄c = RCωc

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0ω̄c = RCωc

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0ω̄c = RCωc

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0ω̄c = RCωc

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0ω̄c = RCωc

Figure 8. Behavior of the cutoff frequency for 0 < α ≤ 1.

α

φc0.2 0.4 0.6 0.8 1.0

-10

-20

-30

-40

-50φc = − 180

π arctan =(ω̄c)<(ω̄c)

Figure 9. Behavior of the phase for 0 < α ≤ 1.

order of derivative, likewise, the delay sensitivity of theorder of derivative increases with decreasing order of thederivative.In the frequency domain we have that the slope of thehigh frequency asymptote decreases as the order ofthe derivative decreases and the cutoff frequency movesto the left. In the case of the phase, Figure 7, it isnoted a very large variation of the phase of the cutofffrequency as the order of the derivative decreases. Fromthe behavior in the time domain is evident the effect thatthe order of the derivative can have on clock systems andsemiconductors circuits. On the other hand, in the steadystate behavior is observed the reduction in the bandwidthhaving as consequence a lower data transmission capacity.

Analysis on the time and frequency domain for the RC electric circuit of fractional order

We hope that this way of dealing with fractional electricalcircuit can be found applications in the power electronics,communication theory, control theory. Also in the model-ing of cells seen as an electrical RC circuit.AcknowledgmentsThe authors acknowledge fruitful discussions with Prof. D.Baleanu, and A. González. This research was supportedby PROMEP under the Grant: Fortalecimiento de CAs.,2011, UGTO-CA-27.References

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