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Research ArticleEquivalent Circuits Applied in
ElectrochemicalImpedance Spectroscopy and Fractional
Derivativeswith and without Singular Kernel
J. F. Gómez-Aguilar,1 J. E. Escalante-Martínez,2 C.
Calderón-Ramón,2
L. J. Morales-Mendoza,3 M. Benavidez-Cruz,2 and M.
Gonzalez-Lee3
1CONACYT-Centro Nacional de Investigación y Desarrollo
Tecnológico, Tecnológico Nacional de México,Interior Internado
Palmira S/N, Colonia Palmira, 62490 Cuernavaca, MOR,
Mexico2Facultad de Ingenieŕıa Mecánica y Eléctrica, Universidad
Veracruzana, Avenida Venustiano Carranza S/N,Colonia Revolución,
93390 Poza Rica, VER, Mexico3Facultad de Ingenieŕıa Electrónica y
Comunicaciones, Universidad Veracruzana, Avenida VenustianoCarranza
S/N, Colonia Revolución, 93390 Poza Rica, VER, Mexico
Correspondence should be addressed to J. F. Gómez-Aguilar;
[email protected]
Received 2 February 2016; Accepted 26 April 2016
Academic Editor: Alexander Iomin
Copyright © 2016 J. F. Gómez-Aguilar et al. This is an open
access article distributed under the Creative Commons
AttributionLicense, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is
properlycited.
We present an alternative representation of integer and
fractional electrical elements in the Laplace domain for
modelingelectrochemical systems represented by equivalent
electrical circuits. The fractional derivatives considered are of
Caputo andCaputo-Fabrizio type. This representation includes
distributed elements of the Cole model type. In addition to
maintainingconsistency in adjusted electrical parameters, a
detailed methodology is proposed to build the equivalent circuits.
Illustrativeexamples are given and the Nyquist and Bode graphs are
obtained from the numerical simulation of the corresponding
transferfunctions using arbitrary electrical parameters in order to
illustrate the methodology. The advantage of our representation
appearsaccording to the comparison between our model and models
presented in the paper, which are not physically acceptable due to
thedimensional incompatibility. TheMarkovian nature of the models
is recovered when the order of the fractional derivatives is
equalto 1.
1. Introduction
Electrochemical Impedance Spectroscopy (EIS) is widelyused to
investigate the interfacial and bulk properties ofmaterials,
interfaces of electrode-electrolyte, and the inter-pretation of
phenomena such as electrocatalysis, corrosion,or behavior of
coatings onmetallic substrates.This techniquerelates directly
measurements of impedance and phase angleas functions of frequency,
voltage, or current applied. Thestimulus is an alternating current
signal of low amplitudeintended to measure the electric field or
potential differ-ence generated between different parts of the
sample. Therelationship between the data of the applied stimulus
andthe response obtained as a function of frequency provides
the impedance spectrum of samples studied [1]. Transferfunction
analysis is a mathematical approach to relate aninput signal (or
excitation) and the system’s response. Theratio formed by the
pattern of the output and the input signalmakes it possible to find
the zeros and poles, respectively. Toanalyze the behavior of the
transfer function in the frequencydomain, several graphical methods
were used, such as Bodeplots that provide a graphical
representation of themagnitudeand phase versus frequency of the
transfer function andthe Nyquist diagrams that are polar plots of
impedancemodulus and phase lag. It is very common in the
literatureto analyze impedance results by a physical model; this
modelis expressed by a mathematical representation and usuallyis
represented by equivalent electrical circuits composed of
Hindawi Publishing CorporationAdvances in Mathematical
PhysicsVolume 2016, Article ID 9720181, 15
pageshttp://dx.doi.org/10.1155/2016/9720181
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2 Advances in Mathematical Physics
elements such as resistors, capacitors, inductors, the
ConstantPhase Element (CPE), and Warburg elements (which is
aspecial case of the CPE). Although the resulting model isnot
necessarily unique, it describes the system with greatprecision in
the range of frequencies studied [1].
Fractional calculus (FC) is the investigation and treat-ment of
mathematical models in terms of derivatives andintegrals of
arbitrary order [2–5]. In the last years, theinterest in the field
has considerably increased due to manypractical potential
applications [6–13]. In the literature, anumber of definitions of
the fractional derivatives have beenintroduced, namely,
theHadamard, Erdelyi-Kober, Riemann-Liouville, Riesz, Weyl,
Grünwald-Letnikov, Jumarie, and theCaputo representation [2–5].
For example, for the Caputorepresentation, the initial conditions
are expressed in terms ofinteger-order derivatives having direct
physical significance[14], this definition is mainly used to
include memory effects.Recently, Caputo and Fabrizio in [15]
present a new defini-tion of fractional derivative without a
singular kernel; thisderivative possesses very interesting
properties, for instance,the possibility to describe fluctuations
and structures withdifferent scales. Furthermore, this definition
allows for thedescription of mechanical properties related to
damage,fatigue, and material heterogeneities. Properties of this
newfractional derivative are reviewed in detail in [16].
Measurements of properties of materials, interfaces
ofelectrode-electrolyte, corrosion, tissue properties of
proteinfibers, semiconductors and solid-state devices, fuel
cells,sensors, batteries, electrochemical capacitors, coatings,
andelectrochromic materials have shown that their impedancebehavior
can only be modeled by using Warburg elementsin conjunction with
resistors, dispersive inductors, or CPEelements [17]. In this
context, FC allows the investigationof the nonlocal response of
electrochemical systems, thisbeing the main advantage when compared
with classicalcalculus. Some researches concerning EIS introduce
FC; forexample, the authors of [18] study the electrical impedance
ofvegetables and fruits from a FC perspective; the experimentsare
developed for measuring the impedance of botanicalelements and the
results are analyzed using Bode andNyquist diagrams. In [19] a
technique is presented to extractthe parameters that characterize a
dispersion Cole-Coleimpedance model. Oldham in [20] describes how
fractionaldifferential equations have influenced the
electrochemistry.Other applications of fractional calculus in
electrochemicalimpedance are given in [21–26].
Unlike thework of the authorsmentioned above, inwhichthe pass
from an ordinary derivative to a fractional one isdirect,
Gómez-Aguilar et al. in [12] analyze the ordinaryderivative
operator and try to bring it to the fractional formin a consistent
manner. Following this idea we present analternative representation
of integer and fractional electricalelements in the Laplace domain
formodeling electrochemicalsystems represented by equivalent
electrical circuits; theorder of the fractional equation is 0 <
𝛾, 𝛽 ≤ 1. Inthis representation an auxiliary parameter 𝛼 is
introduced;this parameter characterizes the existence of the
fractionaltemporal components and relates the time constant of
thesystem.
The paper is organized as follows: Section 2 explains thebasic
concepts of the FC, Section 3 presents the examplesconsidered and
the interpretation of typical diagrams, and theconclusions are
given in Section 4.
2. Introduction to Fractional Calculus
The use of Caputo Fractional Derivative (CD) in Physics
isgaining importance because of the specific properties:
thederivative of a constant is zero and the initial conditions
forthe fractional order differential equations can be given in
thesamemanner as for the ordinary differential equations with
aknown physical interpretation [4].
The CD is defined as follows [4]:
𝐶
0𝐷𝛾
𝑡𝑓 (𝑡) =
1
Γ (𝑛 − 𝛾)∫
𝑡
0
𝑓(𝑛)
(𝛼)
(𝑡 − 𝛼)𝛾−𝑛+1
𝑑𝛼, (1)
where 𝑑𝜑/𝑑𝑡𝜑 = 𝐶𝑎𝐷𝜑
𝑡is a CD with respect to 𝑡, 𝜑 ∈ 𝑅 is
the order of the fractional derivative, and Γ(⋅) represents
thegamma function.
The Laplace transform of the CD has the following form[4]:
𝐿 [𝐶
0𝐷𝛾
𝑡𝑓 (𝑡)] = 𝑆
𝛾𝐹 (𝑆) −
𝑚−1
∑
𝑘=0
𝑆𝛾−𝑘−1
𝑓(𝑘)
(0) . (2)
The Caputo-Fabrizio fractional derivative (CF) is definedas
follows [15, 16]:
CF0D𝛾
𝑡𝑓 (𝑡) =
𝑀(𝛾)
1 − 𝛾∫
𝑡
0
�̇� (𝛼) exp [−𝛾 (𝑡 − 𝛼)
1 − 𝛾] 𝑑𝛼, (3)
where 𝑑𝛾/𝑑𝑡𝛾 = CF0𝐷𝛾
𝑡is a CF with respect to 𝑡 and 𝑀(𝛾) is
a normalization function such that 𝑀(0) = 𝑀(1) = 1; inthis
definition the derivative of a constant is equal to zero,but,
unlike the usual Caputo definition (1), the kernel doesnot have a
singularity at 𝑡 = 𝛼.
If 𝑛 ≥ 1 and 𝛾 ∈ [0, 1], the CF fractional
derivative,CF0D(𝛾+𝑛)
𝑡𝑓(𝑡), of order (𝑛 + 𝛾), is defined by
CF0D(𝛾+𝑛)
𝑡𝑓 (𝑡) =
CF0D(𝛾)
𝑡(CF0D(𝑛)
𝑡𝑓 (𝑡)) . (4)
The Laplace transform of (3) is defined as follows [15, 16]:
𝐿 [CF0D(𝛾+𝑛)
𝑡𝑓 (𝑡)]
=1
1 − 𝛾𝐿 [𝑓(𝛾+𝑛)
𝑡] 𝐿 [exp(−𝛾
1 − 𝛾𝑡)]
=𝑠𝑛+1
𝐿 [𝑓 (𝑡)] − 𝑠𝑛𝑓 (0) − 𝑠
𝑛−1𝑓(0) ⋅ ⋅ ⋅ − 𝑓
(𝑛)(0)
𝑠 + 𝛾 (1 − 𝑠).
(5)
For this representation in the time domain it is suitable to
usethe Laplace transform [15, 16].
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Advances in Mathematical Physics 3
From this expression we have
𝐿 [CF0D𝛾
𝑡𝑓 (𝑡)] =
𝑠𝐿 [𝑓 (𝑡)] − 𝑓 (0)
𝑠 + 𝛾 (1 − 𝑠), 𝑛 = 0,
𝐿 [CF0D(𝛾+1)
𝑡𝑓 (𝑡)] =
𝑠2𝐿 [𝑓 (𝑡)] − 𝑠𝑓 (0) − �̇� (0)
𝑠 + 𝛾 (1 − 𝑠),
𝑛 = 1.
(6)
3. Electrochemical Impedance and theInterpretation of Typical
Diagrams
Regarding the equivalent circuits, there is a diversity ofmodels
used. The most common is to adjust the system to asimple model or
one that includes a bilayer structure, eitherwith RC groups in
parallel or in series model, although thereare cases where it is
appropriate to include the Warburgimpedance element to consider
possible diffusive processeson the surface. Generally, using a
complex equivalent circuitis not necessary to obtain a good
characterization of thereal system; commonly a simple electrical
circuit is the firstchoice and increases its complexity when
knowledge of theelectrochemical behavior of the system also
increases.
The Cole impedance model is based on replacing theideal
capacitor in the Debye model [27, 28]. Cole modelis represented by
a series resistor 𝑅𝑠, a capacitor 𝐶𝑝, anda resistor in parallel 𝑅𝑝.
In general it reflects the electricalresistance of the interface
sample-electrode and maintains anegligible value with respect to𝑅𝑝;
𝛾 is the order of the powerthat best fits the model obtained, 𝛾 ∈
(0; 1), giving an idealcapacitor when it is 1. Using the algebraic
representation ofthe circuit can be said to represent the total
impedance as
𝑍𝑇 (𝑠) = 𝑅𝑠 +
𝑅𝑝
1 + (𝑠𝑅𝑝𝐶𝑝)𝛾 , (7)
where 𝑠 = 𝑗𝜔.On electric structures RC type, bias resistor 𝑅
represents
the charge transfer resistance and capacitance 𝐶 of doublelayer;
the CPE is a component that models the behavior ofa double layer
capacitor in actual electrochemical cells, thatis, an imperfect
capacitor, and the impedance is representedas
𝑍𝑇= 𝐴. (8)
Equation (8) describes the deviation from ideal
capacitors.Considering 𝛾 = 1 and the constant 𝐴 = 1/𝐶 (the inverse
ofcapacitance), this equation describes a capacitor. For a CPE,the
exponent 𝛾 is less than one [29].
In electrochemical systems the diffusion can createan impedance
called Warburg impedance [26], commonlyused to describe phenomena
such as diffusion, adsorp-tion, or desorption of electroactive
substances at interfacesmetal/coating; this impedance depends on
perturbationfrequency: at high frequency a small Warburg
impedanceresults and at low frequency a higher Warburg impedanceis
generated. The Warburg impedance parameter indicates
the existence of diffusive processes that can be related tothe
release of the dissolved species. This parameter onlyreports the
blocking ability of the passive layer, so it is notpossible to know
the nature of the species which are dissolvedby electrochemical
impedance spectroscopy technique. Theequation for the infinite
thickness of theWarburg impedanceis given by
𝑍𝑇= (𝜔)
−1/2(1 − 𝑗) , (9)
where is aWarburg coefficient.On aNyquist plot the
infiniteWarburg impedance appears as a diagonal line with a slope
of0.5; on a Bode plot, the Warburg impedance exhibits a phaseshift
of 45∘ [29].
Equation (9) is valid if the diffusion layer has an
infinitethickness. Quite often this is not the case. If the
diffusion layeris bounded, the impedance at lower frequencies no
longerobeys (9). For the Warburg impedance with finite thickness,we
get the form
𝑍𝑇 = (𝜔)−1/2
(1 − 𝑗) tanh [𝛿 (𝑗𝜔
𝐷)] , (10)
where𝛿 is theNernst diffusion layer thickness;𝐷 is an
averagevalue of the diffusion coefficients of the diffusing
species.Thisequation is more general and is called finite Warburg
[29].
3.1. Equivalent Representation Based on Fractal Capacitors.One
of the problems of the fractional representation is thecorrect
sizing of the physical parameters involved in thedifferential
equation, to be consistent with dimensionalityand following [12] we
introduce an auxiliary parameter 𝛼 inthe following way:
𝑑
𝑑𝑡→
1
𝛼1−𝛾
𝐶
0𝐷𝛾
𝑡, 𝑛 − 1 < 𝛾 ≤ 𝑛, (11)
or
𝑑
𝑑𝑡→
1
𝛼1−𝛾
CF0D𝛾
𝑡, 𝑛 − 1 < 𝛾 ≤ 𝑛, (12)
where 𝑛 is an integer and when 𝛾 = 1. Expressions (11) and(12)
become a classical derivative; the auxiliary parameter 𝛼has the
dimension of time (seconds). This nonlocal time iscalled the cosmic
time in the literature [30]. Another physicaland geometrical
interpretation of the fractional operators isgiven in
Moshrefi-Torbati and Hammond [31]. Parameter 𝛼characterizes the
fractional temporal structures (componentsthat show an intermediate
behavior between a conservativesystem and dissipative; such
components change the timeconstant of the system) of the fractional
temporal operator[32]. In the following we will apply this idea to
construct thefractional equivalent circuits and examples are
analyzed.
3.2. Polarizable Electrode. The model of polarizable elec-trode,
also known as faradaic reaction, provides a simpledescription of
the impedance of an electrochemical reactionon electrode
surfaces.The equivalent circuit is represented byFigure 1.
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4 Advances in Mathematical Physics
CV(t)
i(t)
Rs
Rp
+
−
Figure 1: Equivalent electrical circuit for the polarizable
electrodemodel.
Considering initial conditions equal to zero, the equiv-alent
impedance is found by the following equation in thecomplex
frequency domain:
𝑍𝑇 (𝑠) =𝑉 (𝑠)
𝐼 (𝑠). (13)
Applying Kirchhoff laws to the circuit of Figure 1, we have
𝑉 = 𝑅𝑠𝑖 + 𝑉𝐶, (14)
𝑖 = 𝑖𝑅+ 𝑖𝐶. (15)
Before applying the Laplace transform of (15) the
followingconsiderations must be taken into account:
𝑖𝑅 =𝑉𝐶
𝑅𝑝
, (16)
𝑖𝐶= 𝐶
𝑑𝑉𝐶
𝑑𝑡. (17)
Consider (11); in the Caputo sense (1), (17) becomes
𝑖𝑅=
𝑉𝐶
𝑅𝑝
, (18)
𝑖𝐶 (𝑡) = 𝐶 +𝐶
𝛼1−𝛾
𝐶
0𝐷𝛾
𝑡𝑉𝐶 (𝑡) . (19)
Substituting (19) into (15), we obtain
𝑉 (𝑡) = 𝑅𝑠𝑖 (𝑡) + 𝑉𝐶 (𝑡) , (20)
𝑖 (𝑡) =𝑉𝐶
𝑅𝑝
+𝐶
𝛼1−𝛾
𝐶
0𝐷𝛾
𝑡𝑉𝐶 (𝑡) . (21)
Applying the Laplace transform (2) to (21), we obtain
𝑉 (𝑠) = 𝑅𝑠𝐼 (𝑠) + 𝑉𝐶 (𝑠) , (22)
𝐼 (𝑠) =𝑉𝐶 (𝑠)
𝑅𝑝
+𝐶
𝛼1−𝛾𝑠𝛾𝑉𝐶 (𝑠) . (23)
Finally from (23) we have the fractional impedance of
thecircuit
𝑍𝑇 (𝑠) = 𝑅𝑠 +
𝑅𝑝
1 + (𝑅𝑝𝐶/𝛼1−𝛾) 𝑠𝛾
, (24)
where 𝑠 = 𝑗𝜔.Consider (12), in the Caputo-Fabrizio sense (3),
(17)
becomes
𝑖𝑅 =𝑉𝐶
𝑅𝑝
, (25)
𝑖𝐶 (𝑡) = 𝐶 +𝐶
𝛼1−𝛾
CF0D𝛾
𝑡𝑉𝐶 (𝑡) . (26)
Substituting (26) into (15), we obtain
𝑉 (𝑡) = 𝑅𝑠𝑖 (𝑡) + 𝑉𝐶 (𝑡) , (27)
𝑖 (𝑡) =𝑉𝐶
𝑅𝑝
+𝐶
𝛼1−𝛾
CF0D𝛾
𝑡𝑉𝐶 (𝑡) . (28)
Applying the Laplace transform (5) to (28) we obtain
𝑉 (𝑠) = 𝑅𝑠𝐼 (𝑠) + 𝑉𝐶 (𝑠) , (29)
𝐼 (𝑠) =𝑉𝐶 (𝑠)
𝑅𝑝
+𝐶
𝛼1−𝛾[
𝑠
𝑠 + 𝛾 (1 − 𝑠)]𝑉𝐶 (𝑠) . (30)
Finally from (30) we have the fractional impedance of
thecircuit
𝑍𝑇 (𝑠) = 𝑅𝑠 +
𝑅𝑝
1 + (𝑅𝑝𝐶/𝛼1−𝛾) (1/ (1 − 𝛾 + 𝛾/𝑠))
, (31)
where 𝑠 = 𝑗𝜔 and 𝛼1−𝛾 represent the fractional componentsof the
system. Equations (24) and (31) are the result ofapplying the
fractional temporal operator of Caputo andCaputo-Fabrizio type in
(17) for the current in the capacitor;this general representation
includes an arbitrary constant, 𝛼,which can be considered its own
electrochemical parameter.In the particular case of 𝛼 = 𝑅
𝑝𝐶, (24) is reduced to the Cole
model (7). Equations (24) and (31) presented here preservethe
dimensionality of the studied system for any value of theexponent
of the fractional derivative. On the other hand, if𝛾 = 1 in (24)
and (31), we obtain an ideal RC circuit and theMarkovian nature of
the model is recovered.
Consider that the values of the parameters of the circuitshown
in Figure 1 correspond to 𝑅
𝑠= 100Ω, 𝐶 = 1−4 F, and
𝑅𝑝= 200Ω and 𝑅
𝑝= 300Ω. Figures 2(a), 2(b), 2(c), and 2(d)
show the Nyquist and Bode plots for (24).Now consider that the
values of the parameters of the
circuit shown in Figure 1 correspond to 𝑅𝑠 = 100Ω, 𝐶 =1−4 F, and
𝑅𝑝 = 200Ω and 𝑅𝑝 = 300Ω. Figures 3(a), 3(b),3(c), and 3(d) show the
Nyquist and Bode plots for (31).
Several studies use equivalent circuits considering
purecapacitances for adjusting the impedance spectra and
thusdescribe phenomena as deterioration of materials due to
itsporosity or the effects of exposure and surface preparation
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Advances in Mathematical Physics 5
Nyquist diagram
0
10
20
30
40
50
60
70
80
90
100
Imag
inar
y pa
rt
150 200 250 300100Real part
𝛾 = 1
𝛾 = 0.95
𝛾 = 0.9
𝛾 = 0.85
(a)
Bode diagram
40
45
50
Mag
nitu
de (d
B)
0
−10
−20
−30
Phas
e (de
g)
101 102 103 104100
Frequency (rad/s)
100 102 103 104101
Frequency (rad/s)
𝛾 = 1
𝛾 = 0.95
𝛾 = 0.9
𝛾 = 0.85
(b)
Nyquist diagram
0
50
100
150
Imag
inar
y pa
rt
150 200 250 300 350 400100Real part
𝛾 = 1
𝛾 = 0.95
𝛾 = 0.9
𝛾 = 0.85
(c)
Bode diagram
40
45
50
55M
agni
tude
(dB)
0
−10
−20
−30
−40
Phas
e (de
g)
101 102 103 104100
Frequency (rad/s)
101 102 103 104100
Frequency (rad/s)
𝛾 = 1
𝛾 = 0.95
𝛾 = 0.9
𝛾 = 0.85
(d)
Figure 2: Nyquist and Bode diagram for themodel of polarizable
electrode, Caputo derivative approach, in (a) and (b):𝑅𝑠= 100Ω,𝐶 =
1−4 F,
and 𝑅𝑝= 200Ω for 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85; for (c)
and (d), 𝑅
𝑠= 100Ω, 𝐶 = 1−4 F, and 𝑅
𝑝= 300Ω for 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9,
and 𝛾 = 0.85.
of substrates on the impedances [33–36]. The
electricalcapacitance has information on the conductive properties
ofmaterials and their chemical composition. Recently,
severalauthors have replaced this pure capacitance by a CPE,because
this has been considered an improvement in thesettings of the
theoretical equations regarding experimentalresults [37–39], for
example, in predicting the useful life ofcoatings and surface
roughness heterogeneity resulting fromthe presence of impurities,
fractality, dislocations, adsorp-tion of inhibitors, or formation
of porous layers [40–47].Replacing the capacitor 𝐶 by a CPE adds a
pseudocapacitiveconstant to the circuit and causes the attenuation
of the
imaginary part of the impedance causing a flattening ofthe
semicircles in the Nyquist diagrams (this is their mainfeature);
see Figures 2(a) and 2(c) for the Caputo approachand Figures 3(a)
and 3(c) for the Caputo-Fabrizio approach.These figures allow
seeing the effect of varying the values ofthe pseudocapacitive
constants: in the range 𝛾 ∈ (0.85; 1)we have the behavior of a CPE
and when 𝛾 = 1 the valuescorrespond to pure capacitances. In this
range, the orderof the derivative transforms the capacitor 𝐶
𝑝into a CPE.
When increasing the value of the pseudocapacitive constants,this
also increases the imaginary part of the impedance andthis changes
the time constants of the system [32], which
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6 Advances in Mathematical Physics
Nyquist diagram
𝛾 = 1
𝛾 = 0.999
𝛾 = 0.998
𝛾 = 0.997
0
10
20
30
40
50
60
70
80
90
100
Imag
inar
y pa
rt
150 200 250 300100Real part
(a)
Bode diagram
𝛾 = 1
𝛾 = 0.999
𝛾 = 0.998
𝛾 = 0.997
40
45
50
Mag
nitu
de (d
B)
0
−10
−20
−30
Phas
e (de
g)
101 102 103 104100
Frequency (rad/s)
100 102 103 104101
Frequency (rad/s)
(b)
Nyquist diagram
𝛾 = 1
𝛾 = 0.999
𝛾 = 0.998
𝛾 = 0.997
150 200 250 300 350 400100Real part
0
50
100
150
Imag
inar
y pa
rt
(c)
Bode diagram
𝛾 = 1
𝛾 = 0.999
𝛾 = 0.998
𝛾 = 0.997
101 102 103 104100
Frequency (rad/s)
40
45
50
55M
agni
tude
(dB)
0
−10
−20
−30
−40
Phas
e (de
g)
100 102 103 104101
Frequency (rad/s)
(d)
Figure 3: Nyquist and Bode diagram for the model of polarizable
electrode, Caputo-Fabrizio derivative approach, in (a) and (b): 𝑅𝑠=
100Ω,
𝐶 = 1−4 F, and 𝑅
𝑝= 200Ω for 𝛾 = 1, 𝛾 = 0.999, 𝛾 = 0.998, and 𝛾 = 0.997; for (c)
and (d), 𝑅
𝑠= 100Ω, 𝐶 = 1−4 F, and 𝑅
𝑝= 300Ω, for 𝛾 = 1,
𝛾 = 0.999, 𝛾 = 0.998, and 𝛾 = 0.997.
correspond to creating fractal structures for particular
valuesof 𝛾. In theCaputo-Fabrizio approach it is noted that the
valueof the resistance series is modified, showing the existence
ofheterogeneities in this component. In Figures 2(b) and 2(d)for
the Caputo approach and Figures 3(b) and 3(d) for
theCaputo-Fabrizio approach, the Bode plots exhibit changes inthe
cutoff frequency, hence the phase shift and the decrease ofthe
magnitude. For the range of frequency 10 to 10000 rad/s ashift in
themagnitude and phase is shown, which implies thatthe proposed
circuit better defines these frequencies. We seethat, by increasing
the frequency from the cutoff frequency,the magnitude decreases,
while for frequencies below the
cutoff magnitude it is almost constant. This means that
thecurrent through the circuit is very high with
decreasingfrequency and is very low when the frequency increases.
Thecurrent flowing through these resistors at low
frequenciespresents dissipative effects that correspond to the
nonlinearsituation of the physical process (realistic behavior that
isnonlocal in time), for example, the ohmic friction, whichraises
the temperature and therefore the kinetic energy ofthe molecules of
the system [48]. Concerning the phase, wehave that by increasing
frequency the displacement currentand polarization are very much
increased. In this context,the need for CPE has been attributed by
some authors to
-
Advances in Mathematical Physics 7
i(t) R1
R2
R3
C1C2
Figure 4: Equivalent electrical circuit for the heterogeneous
reac-tion model.
the presence of roughness, corrosion products, changes in
themorphology of the material, or heterogeneous surfaces.
3.3. Heterogeneous Reaction. This model describes a
hetero-geneous reaction which occurs in two stages with
absorptionof intermediates products and absence of diffusion
limita-tions.This circuit has been commonly used tomodel a
porouselectrode coating or a defective electrolyte interface and
hasrecently been applied in the assessment of passive
metal-electrolyte or metal-electrolyte interfaces with hard
coating[49, 50]. The equivalent circuit is represented by Figure
4.
This model has also been used in the study of coatedmetals. In
this case, 𝐶
1represents the capacitance of the
coating; this value is smaller than a capacitance representedby
a CPE;𝑅
2represents the pore resistance; it is the resistance
of ion conducting paths developed in the coating.These pathsmay
be physical pores filled with electrolyte; this electrolytesolution
can be very different than the bulk solution outsideof the coating;
𝐶2 (double layer capacitance) and 𝑅3 (chargetransfer reaction)
represent the interface between this pocketof solution and the bare
metal.
Following the samemethodology fromprevious example,the
fractional impedance in the Caputo sense of this circuit is
𝑍𝑇 (𝑠) = 𝑅1
+𝑅3
(𝑅3𝐶1/𝛼1−𝛾
𝛾 ) 𝑠𝛾 + 1/ (𝑅
2/𝑅3+ 1/ ((𝑅
3𝐶2/𝛼1−𝛽
𝛽) 𝑠𝛽 + 1))
.(32)
In the Caputo-Fabrizio sense the fractional impedance isgiven
by
𝑍𝑇 (𝑠) = 𝑅1 +
𝑅3
(𝑅3𝐶1/𝛼1−𝛾
𝛾 ) (1/ (1 − 𝛾 + 𝛾/𝑠)) + 1/ (𝑅2/𝑅3 + 1/ ((𝑅3𝐶2/𝛼1−𝛽
𝛽) (1/ (1 − 𝛽 + 𝛽/𝑠)) + 1))
, (33)
where 𝑠 = 𝑗𝜔, in (32) and (33), 𝛼1−𝛾𝛾
and 𝛼1−𝛽𝛽
representsthe fractional components of the system. This general
rep-resentation includes an arbitrary constant, 𝛼, which can
beconsidered its own electrochemical parameter. In the case of𝛼𝛾the
physical parameters involved are 𝑅
3𝐶1and for 𝛼
𝛽they
are 𝑅3𝐶2, the time constant of the system. Equations (32)
and (33) presented here preserve the dimensionality of
thestudied system for any value of the exponent of the
fractionalderivative; when 𝛾 = 𝛽 = 1, we recover the Markovian
natureof the model.
Consider that the values of the parameters of the circuitshown
in Figure 4 correspond to 𝑅
1= 50Ω, 𝑅
2= 100Ω,
𝑅3= 200Ω, 𝐶
1= 1−3 F, 𝐶
2= 1−2 F, and 𝐶
2= 3−3 F. Figures
5(a), 5(b), 5(c), and 5(d) show the Nyquist and Bode plots
for(32).
Now consider that the values of the parameters of thecircuit
shown in Figure 4 correspond to 𝑅1 = 50Ω, 𝑅2 =100Ω, 𝑅
3= 200Ω, 𝐶
1= 1−3 F, 𝐶
2= 1−2 F, and 𝐶
2= 3−3 F.
Figures 6(a), 6(b), 6(c), and 6(d) show the Nyquist and
Bodeplots for (33).
Some authors make changes to the ideal capacitorsincluding
elements of Warburg [49, 50]; these elements arerepresented in our
circuitmaintaining the order of the deriva-tives fixed at a value
of 𝛾 = 𝛽 = 1/2. For Caputo and Caputo-Fabrizio approach, Figures
7(a) and 7(c), respectively, showthe resulting Nyquist diagrams
considering the order of thederivative for both capacitors at 1/2.
From these figures itcan be seen that for high frequency the
impedance is smallbecause the reagents must move away from the
surface.
The low-frequency disturbances allow the movement ofdistant
molecules.
The circuit shown in Figure 4 has also been used withsuccess in
the description of titanium alloys, considering car-bon coatings,
silicon oxide, and titanium oxide; in such appli-cations it is
generally considered that the coating presentsa porous outer
sublayer and a denser internal sublayer[49, 51]. In Figure 7(c) it
is observed that the resistance valuedecreases, which can provide
valuable information on thecharacteristics of a coating film.
3.4. Limited Diffusion, Warburg Element. This model des-cribes
the polarization of an electrode considering limitingthe diffusion.
This circuit models a cell where polarization isdue to a
combination of kinetic and diffusion processes, 𝑅𝑠 isthe resistance
of the electrolyte solution, CPE is the imperfectcapacitor, 𝑅
𝑝is the electron-transfer resistance, and𝑊 is the
Warburg element due to diffusion of the redox couple to
theinterface from the bulk of the electrolyte [52]. The
equivalentcircuit is represented by Figure 8.
Following the samemethodology fromprevious example,the
fractional impedance in the Caputo sense of this circuit is
𝑍𝑇 (𝑠)
= 𝑅𝑠
+
𝑅𝑝
(𝑅𝑝𝐶𝑝/𝛼1−𝛾
𝛾 ) 𝑠𝛾 + 1/ (1 + 1/ (𝑅
𝑝𝐶𝑠/𝛼1−𝛽
𝛽) 𝑠𝛽)
.
(34)
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8 Advances in Mathematical Physics
Nyquist diagram
100 150 200 250 300 35050Real part
0
20
40
60
80
100
120
Imag
inar
y pa
rt
𝛾 = 1
𝛾 = 0.95
𝛾 = 0.9
𝛾 = 0.85
(a)
Bode diagram
−30
−20
−10
0
Phas
e (de
g)
35
40
45
50
Mag
nitu
de (d
B)
𝛾 = 1
𝛾 = 0.95
𝛾 = 0.9
𝛾 = 0.85
100 102 103 104 105101
Frequency (rad/s)
100 102 103 104 105101
Frequency (rad/s)
(b)
Nyquist diagram
100 150 200 250 300 35050Real part
0
20
40
60
80
100
120
Imag
inar
y pa
rt
𝛾 = 1
𝛾 = 0.95
𝛾 = 0.9
𝛾 = 0.85
(c)
Bode diagram
−30
−20
−10
0
Phas
e (de
g)
35
40
45
50
Mag
nitu
de (d
B)
𝛾 = 1
𝛾 = 0.95
𝛾 = 0.9
𝛾 = 0.85
101 102 103 104 105100
Frequency (rad/s)
100 102 103 104 105101
Frequency (rad/s)
(d)
Figure 5:Nyquist andBode diagram for the heterogeneous
reactionmodel, Caputo derivative approach, in (a) and (b):𝑅1=
50Ω,𝑅
2= 100Ω,
𝑅3= 200Ω, 𝐶
1= 1−3 F, and 𝐶
2= 1−2 F for 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85; for (c) and
(d), 𝑅
1= 50Ω, 𝑅
2= 100Ω, 𝑅
3= 200Ω,
𝐶1= 1−3 F, and 𝐶
2= 3−3 F for 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85. The
capacitor 𝐶
2is considered ideal, 𝛽 = 1.
In the Caputo-Fabrizio sense the fractional impedance isgiven
by
𝑍𝑇 (𝑠) = 𝑅𝑠 +
𝑅𝑝
(𝑅𝑝𝐶𝑝/𝛼1−𝛾
𝛾 ) (1/ (1 − 𝛾 + 𝛾/𝑠)) + 1/ (1 + 1/ (𝑅𝑝𝐶𝑠/𝛼1−𝛽
𝛽) (1/ (1 − 𝛽 + 𝛽/𝑠)))
, (35)
where 𝑠 = 𝑗𝜔, in (34) and (35), and 𝛼1−𝛾𝛾
and 𝛼1−𝛽𝛽
representthe fractional components of the system. This general
rep-resentation includes an arbitrary constant, 𝛼, which can be
considered its own electrochemical parameter. In the case of𝛼𝛾
the physical parameters involved are 𝑅𝑝𝐶𝑝 for 𝛼𝛾 and for𝛼𝛽 are
𝑅𝑝𝐶𝑠, the time constant of the system. For (34) and
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Advances in Mathematical Physics 9
Nyquist diagram
0
20
40
60
80
100
120
Imag
inar
y pa
rt
100 150 200 250 300 35050Real part
𝛾 = 1
𝛾 = 0.999
𝛾 = 0.998
𝛾 = 0.997
(a)
Bode diagram
35
40
45
50
55
Mag
nitu
de (d
B)
−30
−20
−10
0
Phas
e (de
g)
101 102 103 104 105100
Frequency (rad/s)
101 102 103 104 105100
Frequency (rad/s)
𝛾 = 1
𝛾 = 0.999
𝛾 = 0.998
𝛾 = 0.997
(b)
Nyquist diagram
100 150 200 250 300 35050Real part
0
20
40
60
80
100
120
Imag
inar
y pa
rt
𝛾 = 1
𝛾 = 0.999
𝛾 = 0.998
𝛾 = 0.997
(c)
Bode diagram
35
40
45
50
Mag
nitu
de (d
B)
−30
−20
−10
0
Phas
e (de
g)
101 102 103 104 105100
Frequency (rad/s)
101 102 103 104 105100
Frequency (rad/s)
𝛾 = 1
𝛾 = 0.999
𝛾 = 0.998
𝛾 = 0.997
(d)
Figure 6: Nyquist and Bode diagram for the heterogeneous
reaction model, Caputo-Fabrizio derivative approach, in (a) and
(b): 𝑅1= 50Ω,
𝑅2= 100Ω, 𝑅
3= 200Ω, 𝐶
1= 1−3 F, and 𝐶
2= 1−2 F for 𝛾 = 1, 𝛾 = 0.999, 𝛾 = 0.998, and 𝛾 = 0.997; for (c)
and (d), 𝑅
1= 50Ω, 𝑅
2= 100Ω,
𝑅3= 200Ω, 𝐶
1= 1−3 F, and 𝐶
2= 3−3 F for 𝛾 = 1, 𝛾 = 0.999, 𝛾 = 0.998, and 𝛾 = 0.997. The
capacitor 𝐶
2is considered ideal, 𝛽 = 1.
(35) when 𝛾 = 𝛽 = 1, we recover the Markovian nature of
themodel.
The capacitor 𝐶 and Warburg element 𝑊 shown inFigure 8 are
replaced by two fractional capacitors 𝐶
𝑝and
𝐶𝑠, respectively. The capacitor 𝐶
𝑠is assigned an exponent
𝛽 = 1/2, causing replacement of pure capacitance andcausing thus
an impedance according to an infinite diffusionlayer or Warburg
impedance. This element arises from one-dimensional diffusion of an
ionic species to the electrode [18].The capacitance of fractional
order 𝐶
𝑝can be modeled in
the range 0 < 𝛾 ≤ 1, causing that this pure
capacitancerepresents a CPE or imperfect capacitance. Equations
(34)and (35) presented here preserve the dimensionality of
thestudied system for any value of the exponent of the
fractionalderivative. The equivalent circuit shown in Figure 8 has
beenused to model phenomena of corrosion stability in the
phys-iological environment such as the simulated body fluid; 𝑅
𝑠
represents the electrolyte resistance,𝑅𝑝represents the
coating
pore resistance, and the capacitances represent the
frequency-dependent electrochemical phenomena, such as the
coating
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10 Advances in Mathematical Physics
Nyquist diagram
0
10
20
30
40
50
60
Imag
inar
y pa
rt
100 150 200 250 300 35050Real part
(a)
Bode diagram
35
40
45
50
Mag
nitu
de (d
B)
−20
−15
−10
−5
0
Phas
e (de
g)
101 102 103 104 105 106100
Frequency (rad/s)
101 102 103 104 105 106100
Frequency (rad/s)
(b)
Nyquist diagram
0
10
20
30
40
50
60
70
80
90
100
110
Imag
inar
y pa
rt
150 200 250 300 350100Real part
(c)
Bode diagram
424446485052
Mag
nitu
de (d
B)
−20
−10
0
Phas
e (de
g)
101 102 103 104 105100
Frequency (rad/s)
101 102 103 104 105100
Frequency (rad/s)
(d)
Figure 7: Nyquist and Bode diagram for the heterogeneous
reaction model, Caputo approach in (a) and (b) and Caputo-Fabrizio
approachin (c) and (d). For (a) and (b), 𝑅
1= 50Ω, R
2= 100Ω, 𝑅
3= 200Ω, 𝐶
1= 1−3 F, 𝐶
2= 1−2 F (solid line), 𝐶
2= 3−3 F (dash line), 𝐶
2= 3−3 F (dot
line), and 𝐶2= 3−4 F (dash-dot line). For the capacitor 𝐶
1, 𝛾 = 1/2; for the capacitor 𝐶
2, 𝛽 = 1/2. For (c) and (d), 𝑅
1= 50Ω, 𝑅
2= 100Ω,
𝑅3= 200Ω, 𝐶
1= 1−3 F, 𝐶
2= 1−2 F (solid line), 𝐶
2= 3−3 F (dash line), 𝐶
2= 3−3 F (dot line), and 𝐶
2= 3−4 F (dash-dot line). For the capacitor
𝐶1, 𝛾 = 1/2; for the capacitor 𝐶
2, 𝛽 = 1/2.
WCPE
+ i i
−
+
−
Rs
Rp
iR iC
Cp
Rs
Rp
iR iC
Cs
Figure 8: Equivalent electrical circuit for the limited
diffusionmodel.
impedance and passive oxide film capacitance. CPE is usedin
these models to compensate the nonhomogeneity in thesystem
[53].
Consider that the values of the parameters of the circuitshown
in Figure 8 correspond to 𝑅𝑠 = 100Ω, 𝑅𝑝 = 5000Ω,𝐶𝑠 = 3
−6 F, 𝐶𝑠 = 3−7 F, and 𝐶𝑝 = 1
−3 F. Figures 9(a), 9(b),9(c), and 9(d) show the Nyquist and
Bode plots for (34).
Now consider that the values of the parameters of thecircuit
shown in Figure 8 correspond to 𝑅𝑠 = 100Ω, 𝑅𝑝 =5000Ω, 𝐶
𝑠= 3−6 F, 𝐶
𝑠= 3−7 F, and 𝐶
𝑝= 1−3 F. Figures 10(a),
10(b), 10(c), and 10(d) show the Nyquist and Bode plots
for(35).
Replacing capacitor 𝐶 by a CPE introduces a pseudoca-pacitive
constant at circuit and causes the attenuation of theimaginary part
of the impedance causing a flattening of thesemicircles in the
Nyquist diagrams; see Figure 9(a) for theCaputo approach and Figure
10(a) for the Caputo-Fabrizio
-
Advances in Mathematical Physics 11
Nyquist diagram
2000 4000 6000 8000 100000Real part
0
1000
2000
3000
4000
5000
6000
Imag
inar
y pa
rt
𝛾 = 1
𝛾 = 0.95
𝛾 = 0.9
𝛾 = 0.85
(a)
Bode diagram
0
50
100
Mag
nitu
de (d
B)
−100
−50
0
Phas
e (de
g)
𝛾 = 1
𝛾 = 0.95
𝛾 = 0.9
𝛾 = 0.85
101 102 103 104 105 106100
Frequency (rad/s)
100 102 103 104 105 106101
Frequency (rad/s)
(b)
Nyquist diagram
2000 4000 6000 8000 100000Real part
0
1000
2000
3000
4000
5000
6000
Imag
inar
y pa
rt
𝛾 = 1
𝛾 = 0.95
𝛾 = 0.9
𝛾 = 0.85
(c)
Bode diagram
50
60
70
80
90
Mag
nitu
de (d
B)
−80
−60
−40
−20
0
Phas
e (de
g)
𝛾 = 1
𝛾 = 0.95
𝛾 = 0.9
𝛾 = 0.85
101 102 103 104 105 106100
Frequency (rad/s)
101 102 103 104 105 106100
Frequency (rad/s)
(d)
Figure 9: Nyquist and Bode diagram for the limited diffusion
model, Caputo derivative approach, in (a) and (b): 𝑅𝑠= 100Ω, 𝑅
𝑝= 5000Ω,
𝐶𝑠= 3−6 F, and 𝐶
𝑝= 1−3 F for 𝛽 = 1/2 corresponding to 𝐶
𝑠, 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85; for (c) and (d),
𝑅
𝑝= 5000Ω, 𝐶
𝑠= 3−7 F,
and 𝐶𝑝= 1−3 F for 𝛽 = 0.4 corresponding to 𝐶
𝑠, 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85.
approach. Increasing the value of the pseudocapacitive con-stant
implies an increase in the contribution of the imaginarypart of the
impedance and this changes the system’s timeconstants [32],
creating fractal structures which correspondto particular values of
𝛾. In Figures 9(b), 9(d), 10(b), and 10(d)for the Caputo and
Caputo-Fabrizio approach, respectively,the Bode plots show
variations in the cutoff frequency, hencethe phase shift and the
decrease of the magnitude.
If 0 < 𝛽 < 0.5, then we have a model that
geometricallydescribes the activation surface inhomogeneities or
devia-tions from linear diffusion processes. In our example 𝛽 =
0.4.
This occurs naturally when the diffusion occurs in a
dilutesolution or in the case that the spread does not obey the
lawsof Fick; see Figures 9(c) and 10(c).
4. Conclusions
An impedance spectrum which is obtained in response tothe small
amplitude signal excitation is often interpreted interms of an
equivalent electrical circuit. This one is basedon a physical model
that may represent and characterize
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12 Advances in Mathematical Physics
Nyquist diagram
0
500
1000
1500
2000
2500
3000
3500
4000
Imag
inar
y pa
rt
1000 2000 3000 4000 5000 60000Real part
𝛾 = 1
𝛾 = 0.999
𝛾 = 0.998
𝛾 = 0.997
(a)
Bode diagram
40
60
80
100
Mag
nitu
de (d
B)
−100
−50
0
Phas
e (de
g)
101 102 103 104 105 106100
Frequency (rad/s)
101 102 103 104 105 106100
Frequency (rad/s)
𝛾 = 1
𝛾 = 0.999
𝛾 = 0.998
𝛾 = 0.997
(b)
Nyquist diagram
𝛾 = 1
𝛾 = 0.999
𝛾 = 0.998
𝛾 = 0.997
1000 2000 3000 4000 5000 60000Real part
0
500
1000
1500
2000
2500
3000
3500
4000
Imag
inar
y pa
rt
(c)
Bode diagram
5060708090
100
Mag
nitu
de (d
B)
−80−60−40−20
0
Phas
e (de
g)
100 104 106102
Frequency (rad/s)
102 104 106100
Frequency (rad/s)
𝛾 = 1
𝛾 = 0.999
𝛾 = 0.998
𝛾 = 0.997
(d)
Figure 10: Nyquist and Bode diagram for the limited diffusion
model, Caputo-Fabrizio derivative approach, in (a) and (b): 𝑅𝑠=
100Ω,
𝑅𝑝= 5000Ω, 𝐶
𝑠= 3−6 F, and 𝐶
𝑝= 1−3 F for 𝛽 = 1/2 corresponding to𝐶
𝑠, 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85; for (c) and (d),
𝑅
𝑝= 5000Ω,
𝐶𝑠= 3−7 F, and 𝐶
𝑝= 1−3 F for 𝛽 = 0.4 corresponding to 𝐶
𝑠, 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85.
elements whose electrochemical properties and structuralfeatures
or the physicochemical processes are taking place inthe studied
being. The resulting spectrum described in awide frequency range
might present one, two, or moretime constants, depending on the
monolayer or multilayerstructure, porosity, or diffusive
limitations caused by thecharge transfer process. The existence of
a time constantindicates a homogeneous layer structure; if two
differentconstants are presented, thismay indicate the existence of
twosublayers.
In the field of biomaterials, the electrochemical impe-dance
spectroscopy technique proves to be useful in char-acterizing
roughness or heterogeneous surfaces, in coatingsanalysis, as cell
suspensions, in studying the adsorption ofprotein, and in
characterizing the performance of materials,mainly with regard to
electrocatalysis and corrosion. In thiscontext, FC allows the
investigation of the nonlocal responseof electrochemical systems.
FC has been used successfullyto modify many existing models of
physical processes;the representation of equivalent models in
integer-order
-
Advances in Mathematical Physics 13
derivatives provided a good approximation of the
electro-chemical response of the model.
On the basis of Cole’s proposal to add an extra degree offreedom
in order to solve the RC circuits for characterizationpurposes and
improve the correlation in the adjustment toexperimental data, we
have developed analytical argumentsto derive this result based on
integration in weighted indi-vidual relaxation processes. However,
the distributions ofrelaxation times involve complex functions and
are diffi-cult to measure. This study has shown a pattern in
whichthe Cole type behavior appears as a result of
competitionbetween a capacitive and resistive behavior within the
sam-ple, characterized by the fractional order derivative of
theapplied voltage. This combination of stored and dissipatedenergy
is conveniently based on the representation of linearviscoelastic
behavior; this dissipation is known as internalfriction. In the
literature it is common to characterize basedon least-squares fit
of equivalent electrical circuit modelson experimental data,
including Cole models. From thedescription of the fractional
differential equation models itcan be noted that the representation
of Colemodels is derivedas a particular solution to the RC circuit
under FC.
Since some authors replace the derivative of a fractionalentire
order in a purely mathematical context, the physicalparameters
involved in the differential equation do not havethe dimensionality
obtained in the laboratory and from thephysical point of view of
engineering this is not entirelycorrect. In this representation an
auxiliary parameter 𝛼 isintroduced; this parameter characterizes
the existence ofthe fractional temporal components and related the
timeconstant of the system; the solutions presented here
preservethe dimensionality of the studied system for any value of
theexponent of the fractional derivative. The advantage of
thisalternative representation inwhen comparisonwith themod-els
presented in the literature is the physical compatibility ofthe
solutions.
TheCaputo representation has the disadvantage that theirkernel
had singularity; this kernel includes memory effectsand therefore
this definition cannot accurately describe thefull effect of the
memory. Due to this inconvenience, Caputoand Fabrizio in [15]
present a new definition of fractionalderivative without singular
kernel, the Caputo-Fabrizio frac-tional derivative.The two
definitions of fractional derivativesmust apply conveniently
depending on the nature of thesystem and the choice of the
fractional derivative dependsupon the problem studied and on the
phenomenologicalbehavior of the system.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
The authors would like to thank Mayra Mart́ınez for
theinteresting discussions. J. F.Gómez-Aguilar acknowledges
thesupport provided by CONACYT: catedras CONACYT parajovenes
investigadores 2014.
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