This paper introduces the fundamentals of the conventional
LC
lowpass filter circuit in the fractional domain. First, we study the new fundamentals of fractionalorder
LC
lowpass filter circuit including the pure real angular frequency, the pure imaginary angular frequency and the short circuit angular frequency. Moreover, sensitivity analysis of the impedance characteristics and phase characteristics of the
LC
lowpass filter circuit with respect to the system variables is studied in detail, which shows the greater flexibility of the fractionalorder filter circuit in designs. Furthermore, from the filtering property perspective, we systematically investigate the effects of the system variables (
LC
, frequency
f
and fractional orders) on the amplitudefrequency characteristics and phasefrequency characteristics. In addition, the detailed analyses of the cutoff frequency and filter factor are presented. Numerical experimental results are presented to verify the theoretical results introduced in this paper.
1. Introduction
There has been more than four centuries since the advent of fractional calculus. However, the study of the fractional calculus has been made dramatic advances in recent four decades
[1]
. In view of the extra fractionalorder variables, the application of fractional calculus has many advantages superior to that of traditional integer calculus, such as more flexibility, freedom, best fit, and optimization techniques and so on. Therefore, many scientists have attempted to broaden the scope of fundamentals and theorems from integer order systems into fractional ones in biomedical applications
[2
,
3]
, chaotic systems
[4
,
5]
, signal processing
[6]
, control design
[7]
, and more. More specifically, the applications of fractional calculus on the filter design have yielded much recent progress in theory
[8

10]
, noise analysis
[11]
and stability analysis
[12]
. However, fewer researchers have concentrated on the
LC
lowpass filter circuit, which is an important basic filter circuit.
Fractionalorder mathematical models developed for inductors and capacitors could describe the electrical characteristics more accurately. In other words, the actual inductors and capacitors are fractionalorder in nature
[13

16]
. More specifically, the fractionalorder capacitors are obtained by different designing ways
[17

20]
. Also, Machado and Galhano
[21]
pointed that the discretional fractional orders of inductors could be designed based on the skin effect. Motivated by the above analysis, circuit designers have to face new challenges on the new phenomenon and laws owing to applications of the fractionalorder components.
Recently, some researchers have published some papers on the filter circuit
[22

26]
. While few researchers have concentrated on the fractionalorder filter circuit
[27
,
28]
. From these considerations, the fundamentals of fractionalorder
LC
filter circuit are mainly studied. Because the fractionalorder capacitors and inductors are introduced into the filter circuit, there are four variables:
L, C
plus two fractional orders
α
and
β
. Therefore, there should be new fundamentals and laws which cannot be obtained in the conventional filter circuit.
From the above analyses, the following advanced research contents can make our research attractive. First, the new fundamentals of the impedance characteristics and phase characteristics have been systematically investigated, which lays the groundwork for the design of the fractionalorder filter circuit. Second, from the filter perspective, the amplitudefrequency characteristics, phasefrequency characteristics and cutoff frequency are studied in detail with respect to
LC
and fractional orders to show the better filter properties of the fractionalorder
LC
filter circuit.
This paper is organized as follows: Section 2 simply analyzes the fundamentals of the conventional
LC
filter circuit. In Section 3, basic definitions of fractional capacitors and inductors are presented, and the new fundamentals of the fractionalorder filter circuit are discussed, including the pure real angular frequency, the pure imaginary angular frequency, the short circuit angular frequency and the sensitivity analysis of the impedance and phase characteristics with respect to the system variables. In Section 4, as for the filtering characteristics, amplitudefrequency characteristics, phasefrequency characteristics and cutoff frequency are studied in detail. Section 5 concludes the paper.
2. Integer OrderLCFilter Circuit
The diagram of integer order
LC
filter circuit is shown in
Fig. 1()
. Thus, we can get its impedance as
Diagram of the LC filter circuit.
Hence, its magnitude of the impedance can be expressed by
From Eq. (1), the impedance is pure imaginary, which means the
LC
filter circuit is power lossless. And there is a critical operating frequency
which is called as resonance frequency. Therefore, when
ω
>
ω_{c}
, the impedance will be inductive. Otherwise, it will be capacitive.
Fig. 2
shows the relation between the impedance magnitude versus
ω
and
LC
. The large magnitude response will be obtained at very low frequency, or very high frequency and larger
LC
. Also, it’s clear from the figure that the minimal magnitude of the impedance exists at
ω_{c}
, which is zero.
The graphic model of the magnitude of the impedance versus LC and ω of the conventional LC filter circuit.
As for phase, it is a fixed value
with the inductive impedance, or
with capacitive impedance.
3. Fractional OrderLβCαFilter Circuit
 3.1 Basic definitions of fractional capacitor and inductor
At present, there are three frequentlyused definitions of the fractional derivative, which are the GrunwaldLetnikov, the RiemannLiouville and the Caputo definition
[29]
. Here, we use the Caputo definition of a fractional derivative over other approaches because the initial conditions for this definition take the same form as the more familiar integerorder differential equations. The fractional derivative by Caputo is denoted as
where Γ(
x
) is the wellknown Euler’s Gamma function and
n
−1 ≤
α
≤
n
.
Under zero initial conditions, we apply the Laplace transform to the Caputo definition (3). Thus, one gets
Then, we use the relationship between the voltage and current of the fractional order capacitor and fractional order inductor. Consequently, the following expressions are given as
and
Similarly, under zero initial conditions, by applying the Laplace transform to Eq. (5) and Eq. (6), the impedances of the fractional order capacitor and inductor can be given as
and
, respectively
The diagram of fractionalorder
L_{β}C_{α}
filter circuit is shown in
Fig. 1()
. So, we get the impedance as
Furthermore, one gets
The equivalent impedance of the fractionalorder
L_{β}C_{α}
filter circuit is presented by Eq. (7). Note that there are both the real and imaginary parts in the impedance. For the integer order LC filter circuit, the previous cases cannot be satisfied, which validates the deficiency of the integer order filter circuit. Additionally, both the real and imaginary parts in the impedance vary with the four parameters, fractional orders
α
and
β
,
LC
and
ω
.
Therefore, we will systematically analyze the new fundamentals of the fractionalorder
L_{β}C_{α}
filter circuit in the following subsections.
 3.2 Pure real angular frequency
The equivalent impedance of the fractionalorder
L_{β}C_{α}
filter circuit can be pure real only at a certain frequency which we named pure real angular frequency
[27]
. In this case, the energy is converted into loss in the fractionalorder filter circuit. From Eq. (7), if the imaginary part is zero, the pure real angular frequency can be given by
where
α
,
β
≠2.
Furthermore, the pure real impedance with the pure real angular frequency can be expressed by
Fig. 3
illustrates the effects of the fractional orders with different
LC
values on the pure real angular frequency. From Eq. (9), if
α
=
β
, the pure real angular frequency can be simplified as
and the corresponding pure real impedance can be rewritten as
which can also explain the results in
Fig. 4
. Specifically, the results in
Fig. 3
agree with the expected outcome, in that
ω_{pri}
is a fixed value 1 when
LC
=1. When
LC
<1,
ω_{pri}
decreases with the increase of fractional orders
α
=
β
, and it also decreases as
LC
increases with fixed fractional orders
α
=
β
. In addition,
ω_{pri}
has a very wide range varying with the fractional orders. However, when
LC
>1, it’s quite opposite as shown in
Fig. 3()
.
The relation between ω_{pri} versus α=β for different LC values. (a) LC<1. (b) LC≥1.
The relation between the pure real impedance versus α = β for different L and C values.
From
Fig. 4
, we can get that when
α
=
β
=1, the pure real impedance is zero, which is the integer order case. Interestingly, if
α
=
β
>1, we can get a negative resistor.
 3.3 Pure imaginary angular frequency
Similarly, the pure imaginary impedance can also be obtained at a certain frequency which we named as pure imaginary angular frequency
[25]
. In this case, all the power is stored in the filter circuit without energy loss. From Eq. (7), the pure imaginary angular frequency can be given by
where
α
,
β
≠ 1.
From Eq. (11), the pure imaginary impedance can be derived as
From Eq. (11), if
α
=
β
, pure imaginary angular frequency
ω_{pii}
can be simplified as
Since
ω_{pii}
must be positive, the acceptable results of
ω_{pii}
can be obtained if and only if
If
α
≠
β
, the acceptable results of
ω_{pii}
can be obtained in the following two cases. First,
Y
is positive when
α
∈(0, 1) and
β
∈(1, 2) or
α
∈(1, 2) and
β
∈(0, 1). Second case is
without considering the positive or negative
Y
. As a result, we can get acceptable results of
ω_{pii}
only at some critical values of the fractional orders.
As for the pure imaginary impedance shown in Eq. (12), it must be pure imaginary. If
α
=
β
, the corresponding pure imaginary impedance can be simplified as
However, there is no acceptable result because of the existence of ( −1)
^{α}
and
which is illustrated in
Fig. 5
. Specifically, the imaginary part of
X
is always negative. On the other hand, if
α
≠
β
, there are also two cases shown in
Fig. 6
. First, from
Fig. 6
, some acceptable results of the pure imaginary impedance exist when the imaginary part of
X
is 0 and the real part of
X
is not zero. In this case, the pure imaginary impedance could be inductive or capacitive. Second, acceptable results of the pure imaginary impedance can be obtained when
. However, it is impossible for 0<
α
,
β
<2.
The imaginary of X versus 1/4α when α=β with different LC values.
The graphic model of the imaginary and real part of X versus the fractional orders with LC = 10^{−3} : (a) Imaginary part of X when α ∈(0, 1) and β ∈(1, 2); (b) Real part of X when α ∈(0, 1) and β ∈(1, 2); (c) Imaginary part of X when α ∈(1, 2) and β ∈(0, 1); (d) Real part of X when α ∈(1, 2) and β ∈(0, 1).
 3.4 Short circuit angular frequency
If Eq. (10) is zero, the real and imaginary part of the impedance of the fractionalorder filter circuit will be zero, which means there are no power loss and storing energy. In this case, the fractionalorder filter circuit can oscillate freely without any source, which is impossible in the integer order filter circuit. Therefore, the frequency of oscillation (also called as short circuit angular frequency)
[27]
can be obtained as
ω_{sci}
=
ω_{pri}
, and the condition of oscillation can be expressed by
With 0<
α
,
β
< 2,
α
+
β
= 2 can be derived, where
α
≠
β
≠ 1 .
 3.5 Impedance characteristics
 3.5.1 Effect of LC
From Eq. (8), the magnitude of the impedance can be amplified by
because the values of
LC
are very small in practice. And the magnitude of the impedance will increase rapidly with the increase of
LC
.
Also, the critical magnitude of the impedance can be obtained when
Therefore, the critical
LC
can be expressed by
where 1<
α
+
β
<3.
Hence, the critical magnitude of the impedance is derived as
Fig. 7()
shows the values of (
LC
)
_{cr}
versus
α
+
β
for different
ω
values. Clearly, the critical
LC
decreases as
α
+
β
increases. In addition, the critical
LC
also decreases with the increase of
ω
with fixed fractional orders. The relation between the critical magnitude of the impedance at critical
LC
versus
α
for different
β
values is presented in
Fig. 7()
, where some local minimum exist when
α
+
β
=2.
(a) The relation between the critical LC versus α+β for different ω values. (b) The critical magnitude of the impedance at critical LC versus α for different β values when ω=100.
 3.5.2 Effect of fractional orders
From Eq. (8), the effects of fractional orders on the magnitude of the impedance are systematically discussed in the following contents.
It’s clear from the
Fig. 8()
that the critical maximum values exist at different fractional orders and its location changes as
ω
changes. The critical minimum values will be discussed in detail later in
Fig. 10
. The graphic model of the magnitude of the impedance versus fractional orders of the fractionalorder filter circuit is presented in
Fig. 8()
, where the larger effects of the fractional orders on the magnitude of the impedance can be obtained with the larger
β
or smaller
α
.
(a) The relation between magnitude of the impedance versus the fractional orders for different ω values when LC = 10^{−3} . (b) The graphic model of the magnitude of the impedance versus fractional orders of the fractionalorder filter circuit when ω=100 and LC = 10^{−3} .
(a)The critical value of α=β versus ω for different LC values. (b) The critical value of magnitude of impedance versus ω for different LC values.
The critical magnitude of the impedance versus
α
and
β
can be easily obtained when
and
which are presented in Eqs. (16) and (17), respectively.
Fig. 9
shows the relations between the critical
α
(
β
) and the critical value of magnitude of impedance versus the operating frequency for different
β
(
α
) values. From
Fig. 9()
, the critical
α
gradually decreases with the increase of frequency, and the critical
α
also decreases as
β
increases. However, it’s quite opposite for the critical value of magnitude of impedance shown in
Fig. 9()
. It’s clear from the
Fig. 9()
that the local maximal critical β exists at different frequency for different
α
. Note that there is more than one critical
β
in the high frequency when
α
<1. With the increase of frequency in
Fig. 9()
, the critical magnitude of impedance rapidly decreases, which means the critical magnitude of impedance can become zero at very high frequency.
(a) The critical value of α versus ω for different β when LC = 10^{−3}. (b) The critical value of magnitude of impedance versus ω for different β when LC = 10^{−3} . (c) The critical value of β versus ω for different α when LC = 10^{−3} . (d) The critical value of magnitude of impedance versus ω for different α when LC = 10^{−3} .
Also, the critical magnitude of the impedance versus
α
and
β
can be obtained when
and
α
=
β
, which is given by
The relations between the critical value of
α
=
β
and magnitude of impedance versus
ω
for different
LC
values are presented in
Fig. 10()
. As expected, the critical fractional orders decrease with the increase of frequency, which can also explain the phenomena happened in
Fig. 8()
. From
Fig. 10
, the three minimal points can be obtained in
Fig. 8()
, which are (0.77, 533.8), (0.71, 817.3) and (0.68, 1018), respectively.
 3.6 Phase characteristics
From Eq. (7), the phase response of fractional order
L
_{β}
C
_{α}
filter circuit is given by
 3.6.1 Effect of LC
From Eq. (19), the relation between the phases of the impedance of the fractionalorder
LC
filter circuit versus
LC
for different fractional orders is presented in
Fig. 11
.
The phase of the fractionalorder filter circuit versus LC for different fractional orders when ω=100.
Comparing the figures in
Fig. 11
, when
α
=
β
<1, with the increase of
LC
, the phase is gradually increases. In addition, the rate of increase is becoming larger with the increase of fractional orders. However, it’s quite opposite when
α
=
β
>1.
When
α
=
β
=1, phase could be infinite because the real part of the impedance is 0 in the conventional case.
 3.6.2 Effect of the fractional orders
From Eq. (19), the relation between the phase of the impedance of the fractionalorder filter circuit versus
α
=
β
is illustrated in
Fig. 12
.
Phase of the impedance versus the fractional orders α = β when LC=0.001. (a) 0<α =β<1. (b) 1<α =β<2.
Fig. 12
shows the relation between the phases of the impedance versus the fractional orders for the different
ω
values. Clearly, from
Fig. 12()
, the phase decreases as the fractional orders increase at the low frequency, while there is a critical phase at the high frequency, which will be illustrated in detail. In addition, the effect of the frequency on the phase is minimal when the fractional orders are very small. However, there is no critical phase at the high frequency as shown in
Fig. 12()
, and the minimal effect of the phase on impedance can be obtained with the large fractional orders.
Based on above analyses, the critical phase of the impedance versus
α
=
β
can be easily obtained when
Therefore, the relations between the critical fractional orders and phase of the impedance can be presented in
Fig. 13
.
(a) The critical fractional orders versus the frequency with different LC values. (b) The critical phase of the impedance versus the frequency with different LC values.
With increase of the frequency and
LC
in
Fig. 13()
, the critical fractional orders decrease. However, the interrupted phenomena happens for all curves when
α
=
β
=1, which is the conventional case. From
Fig. 13()
, more than one critical phase of the impedance exists at the low frequency. Furthermore, the minimal phase of the impedance happened in
Fig. 12()
for
ω
=100 can be proved by
Fig. 13
. Specifically,
α_{cp}
=
β_{cp}
=0.56 and corresponding phase of the impedance is −0.85.
Fig. 14
shows the effects of fractional orders on the magnitude of phase. The results of the magnitude of the phase agree with the expected outcome, in that the larger magnitude of the phase can be obtained when
α
,
β
→1 shown in
Fig. 14()
,
()
and
()
. In addition, note that the local critical magnitudes of the phase exist in
Fig. 14()
.
The graphic model of the phase of the impedance versus fractional orders of the fractionalorder filter circuit when ω=100 and LC=10^{−3}. (a) α ∈(0,1) , β ∈(0,1) . (b) α ∈(0,1) , β ∈(1,2) . (c) α ∈(1,2) , β ∈(0,1) . (d) α ∈(1,2) , β ∈(1,2) .
Accordingly, new fundamentals of the fractionalorder
LC
filter circuit can be obtained, including the critical factional orders,
LC
, the impedance and phase. Conequently, we can get the suitable electrical characteristics of the filter circuit by adjusting the two extra parameters and
LC
, which shows the greater flexibility of the fractionalorder filter circuit on circuit design.
4. Filtering Characteristics
 4.1 Amplitudefrequency characteristics
From
Fig. 1 ()
, the voltage gain function is presented as
Therefore, the expression of frequency characteristics is easily given according to the relationship between the frequency characteristics and the voltage gain function as
Furthermore, the amplitude of Eq. (21) can be expressed as
The amplitudefrequency characteristic is an important reflection of the filtering properties. In this subsection, we will systematically analyze the amplitudefrequency characteristics of the fractionalorder
L_{β}C_{α}
filter circuit varying with the fractional orders,
α
+
β
and
LC
.
 4.1.1 Effect of fractional orders
Fig. 15
illustrates the relation between the magnitude responses of Eq. (22) versus
f
for different fractional orders
α
+
β
values. From
Fig. 15
, the following new amplitudefrequency characteristics of the fractionalorder
L_{β}C_{α}
filter circuit can be obtained.
Amplitudefrequency characteristics of the fractional order L_{β}C_{α} filter circuit versus different fractional orders α+β values when LC=10^{−3}.
Ripple output: the ripple output of the fractional order
L_{β}C_{α}
filter circuit is becoming larger with the increase of the fractional orders
α
+
β
.
Passband gain: the fractional order
L_{β}C_{α}
filter circuit will get larger passband gain as
α
+
β
increases in the passing band of the fractionalorder filter circuit, which means the preferable effect of the passband gain can be acquired with smaller fractional orders.
Cutoff frequency: the cutoff frequency of the fractional order
L_{β}C_{α}
filter circuit decreases with the increase of the fractional orders just when
LC
=10
^{−3}
, which will be proved later
Bandwidth: the bandwidth of the fractionalorder filter circuit attenuations as
α
+
β
increase, which is also illustrated in
Table 1
.
The relation between the filter factor versus the fractional orders whenLC=10−3.
The relation between the filter factor versus the fractional orders when LC=10^{−3}.
Damping coefficient and Quality factor: power loss of the inductance is less at a fixed frequency with the increase of
α
+
β
due to the smaller damping coefficient. However, it’s quite opposite for the quality factor because of the reciprocal relation between damping coefficient and quality factor.
With the increase of the fractional orders shown in
Table 1
, the filter factor is decreasing, which indicate that the filter circuit can get higher frequency resolution and selectivity.
Cutoff frequency, alias half power frequency, is also an important reflection of the filtering properties. In order to study the effects of cutoff frequency on the filtering properties better,
Fig. 16
is presented.
Cutoff frequency of the fractional order L_{β}C_{α} filter circuit versus the fractional orders α+β for different LC values.
As shown in
Fig. 16
, as expected, these results show a general trend that as
α
+
β
increase, the cutoff frequency rapidly decreases.
 4.1.2 Effect of LC
Fig. 17
shows the effect of
LC
on the magnitude response with fixed fractional orders. Also, the new amplitudefrequency characteristics can be described as follows.
Amplitudefrequency characteristics of the fractional order L_{β}C_{α} filter circuit with different LC values when α+β =1.5.
Ripple output: the ripple output of the fractional order
L_{β}C_{α}
filter circuit is unaffected by the change of
LC
with fixed
α
+
β
.
Passband gain: the passband gain is unaffected by
LC
when
α
+
β
is fixed.
Cutoff frequency: the cutoff frequency of the fractionalorder filter circuit decreases with the increase of
LC
with fixed
α
+
β
, which will be proved in
Fig. 18
.
Cutoff frequency of the fractional order L_{β}C_{α} filter circuit versus LC for different fractional orders.
Bandwidth: the bandwidth of the fractionalorder filter circuit attenuations as
LC
increases, which is also illustrated in
Table 2
.
The relation between the filter factor versusLCwhenα+β=1.5.
The relation between the filter factor versus LC when α+β =1.5.
Damping coefficient and Quality factor: power loss of the inductance remains unchanged at fixed
α
+
β
with the increase of
LC
due to the unchanged damping coefficient. In other words,
LC
doesn’t affect the damping coefficient. Also, the quality factor is unchanged.
Also, the relation between the filter factor versus the
LC
is presented in
Table 2
.
From
Table 2
, the effect of
LC
on the filter factor is minimal.
To describe the effects of
LC
on the cutoff frequency in detail,
Fig. 18
is presented below.
Fig. 18
shows the relation between the cutoff frequency versus
LC
for different fractional orders. Clearly, the cutoff frequency rapidly decreases as
LC
increases. And it’s worth noting that the cutoff frequency may not exist for the fixed
LC
when
α
+
β
=2, which validates the deficiency of the integerorder filter circuit in designs.
 4.2 Phasefrequency characteristics
The phasefrequency characteristic is also an important indicator of the filtering properties, and the analysis of it is the main part of the subsection. Furthermore, the relationships between the phasefrequency characteristics of the fractionalorder filter circuit and fractional orders
α
+
β
,
LC
are analyzed in the following contents.
 4.2.1 Effect of LC
From Eq. (21), the effects of
LC
on the phase response are presented in
Fig. 19
.
(a) The phase response of the fractionalorder filter circuit versus frequency for different LC when α + β =1.5; (b) The phase response of the fractionalorder filter circuit versus frequency for different LC when α + β=2; (c) The graphic model of the phase response of the fractionalorder filter circuit versus different LC when f =100 and α + β =1.5; (d ) The graphic model of the phase response of the fractionalorder filter circuit versus different LC when f =100 and α + β =2.
From
Fig. 19()
, with increase of operating frequency, the lagging phase angle of the fractionalorder filter circuit decreases to a fixed value with fixed fractional order
α
+
β
. Moreover, the filter circuit can obtain the same minimal and maximal lagging phase angle in the high frequency and very low frequency for different values. However, it’s quite different in the conventional case as shown in
Fig. 19()
, in that the phase angle is zero at very low frequency, and it’s
π
at high frequency. Such results can be explained when the real part of Eq. (21) tends to zero at very low frequency, while it’s negative at high frequency. From
Fig. 19()
and
()
, we see that
L
and
C
values have the same effects on the phase response, which agrees with theoretical analysis.
 4.2.2 Effect ofα+β
Also, the effects of
α
+
β
on the phase response are presented in
Fig. 20
.
(a) The phase response of the fractionalorder filter circuit versus frequency for different α +β when LC=10^{−3}. (b) The graphic model of the phase response of the fractionalorder filter circuit versus fractional orders when f =100 and LC=10^{−3}.
From
Fig. 20()
, as for the fractionalorder filter circuit, the lagging phase angle decreases in the high frequency as fractional orders increase, while it’s a fixed value
π
for the conventional case. And the fractional order
β
plays a leading role in the phasefrequency characteristics as shown in
Fig. 20()
.
5. Conclusions and Discussions
This paper systematically studies the fundamentals of using fractionalorder capacitors and fractionalorder inductors instead of integerorder elements, and creatively advances the
LC
filter circuit. From the point of view of the circuit, the new fundamentals of the fractionalorder filter circuit are discussed in detail, including the pure real angular frequency, the pure imaginary angular frequency, the short circuit angular frequency and the sensitivity analysis of the impedance and phase characteristics, where many interesting phenomena are presented, and it shows a broad view of the fractionalorder filter circuit since two extra parameters are introduced. Furthermore, we systematically study the amplitudefrequency characteristics and phasefrequency characteristics of the fractionalorder
LC
filter circuit to show the greater flexibility of the fractionalorder filter circuit.
In this paper, we make an important theoretical contribution to study the fractionalorder
LC
lowpass filter circuit. Many new fundamentals are obtained by the results of the rigorous mathematical analyses and numerical experiments. In the future, we will devote ourselves to the physical experiments of the fractionalorder filter circuit with the new fundamentals in the paper. Additionally, we will also try our best to the applications of the fractionalorder
LC
lowpass filter circuit.
Acknowledgements
This work was supported by the scientific research foundation of National Natural Science Foundation (51479173), the Fundamental Research Funds for the Central Universities of Ministry of Education of China (201304030577), Northwest A&F University Foundation, China (2013BSJJ095), the scientific research foundation on water engineering of Shaanxi Province (2013slkj12).
BIO
Rui Zhou received the B.Sc. degree in electrical engineering and automation from Northwest A&F University, China in 2013. Now, he is a postgraduate student in Northwest A&F University. His research interests include the fractional circuit and nonlinear circuit networks.
Runfan Zhang received the B.E. degree in electrical engineering and its automation from Northwest A&F University, he is studying for the M. E. degree in Hydropower Engineering in Northwest A&F University, in 2013, and 2016 respectively. His research interests involve nonlinear circuit.
Diyi Chen received the B.Sc. degree in electrical engineering and automation from China University of Mining and Technology, the M. Sc. degree in theory and new technology of electrical engineering from Shandong University and the Ph.D. degree in electrical engineering and automation from Northwest A&F University, in 2005, 2008 and 2013, respectively. In 2008, he joined the Department of Electrical Engineering, Northwest A&F University as a Lecturer. He is currently an Associate Professor at the same school. His research interests include nonlinear dynamics, fractional circuit, nonlinear circuit networks and stability of hydropower and power system. He has published over 30 papers in these areas. He served as a Visiting Scholar at School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ, USA from Jul. 27, 2012 to Jul. 29, 2013.
Sabatier J.
,
Agrawal O. P.
,
Tenreiro Machado J. A.
2007
“Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering,”
SpringerVerlag New York Inc.
Berlin
Doehring T. C.
,
Freed A. H.
,
Carew E. O.
,
Vesely I.
2005
“Fractional order viscoelasticity of the aortic valve cusp: An alternative to quasilinear viscoelasticity,”
J. Biomech. Eng.Trans. ASME
127
(4)
700 
708
DOI : 10.1115/1.1933900
Radwan A. G.,
,
Soliman A. M.
,
Elwakil A. S.
,
Sedeek A.
2009
“On the stability of linear systems with fractionalorder elements,”
Chaos. Solition. Fract.
40
(5)
2317 
2328
DOI : 10.1016/j.chaos.2007.10.033
Zhang W. B.
,
Fang J. A.
,
Tang Y.
2012
“Robust stability for genetic regulatory networks with linear fractional uncertainties,”
Commun. Nonl. Sci. Numer. Simul.
17
(4)
1753 
1765
DOI : 10.1016/j.cnsns.2011.09.026
Tang Y.
,
Gao H. J.
,
Kurths J.
2014
“Distributed robust synchronization of dynamical networks with stochastic coupling,”
IEEE T. CASI
61
(5)
1508 
1519
Tan L.
,
Jiang J.
,
Wang L. M.
2014
“Multirate Processing Technique for Obtaining Integer and FractionalOrder Derivatives of LowFrequency Signals,”
IEEE Trans. Instrum. Meas.
63
(4)
904 
912
DOI : 10.1109/TIM.2013.2289578
Li H.
,
Luo Y.
,
Chen Y. Q.
2010
“A fractionalorder proportional and derivative (FOPD) motion controller:tuning rule and experiments,”
IEEE Trans. Control Syst. Technol.
18
(2)
516 
520
DOI : 10.1109/TCST.2009.2019120
Radwan A. G.
,
Elwakil A. S.
,
Soliman A. M.
2008
“Fractionalorder sinusoidal oscillators: Design procedure and practical examples,”
IEEE T. Circ. Syst.
55
(7)
2051 
2063
DOI : 10.1109/TCSI.2008.918196
Seo U. J.
,
Chun Y. D.
,
Choi J. H.
,
Chung S. U.
,
Han P. W.
2013
“General characteristic of fractional slot double layer concentrated winding synchronous machine,”
J. Electr. Eng. Techn.
8
(3)
282 
287
DOI : 10.5370/JEET.2013.8.2.282
Ahmadi P.
,
Maundy B.
,
Elwakil A. S.
,
Belostostski L.
2012
“Highquality factor asymmetricslope bandpass filters: a fractionalorder capacitor approach,”
IET Circ. Dev. Syst.
6
(3)
187 
197
DOI : 10.1049/ietcds.2011.0239
Shah S. M.
,
Samar R.
,
Raja M. A. Z.
,
Chambers J. A.
2014
“Fractional normalised filterederror least mean squares algorithm for application in active noise control systems,”
Electron. Lett.
50
(14)
973 
U9874
Tabatabaei M.
,
Haeri M.
2012
“Sensitivity analysis of CRA based controllers in fractional order systems,”
Signal Process.
92
(9)
2040 
2055
DOI : 10.1016/j.sigpro.2012.01.014
Bak G. W.
,
Jonscher A. K.
1999
“Lowfrequency dispersion in hopping electronic systems,”
Journal of Materials Science
34
(22)
5505 
5508
DOI : 10.1023/A:1004712512052
Westerlund S.
,
Ekstam L.
1994
“Capacitor theory,”
IEEE T. Dielectr. Electr. Insulat.
1
(5)
826 
839
DOI : 10.1109/94.326654
Westerlund S.
2002
“Dead Matter Has Memory,”
Causal Consulting
Kalmar, Sweden
Jesus I. S.
,
Tenreiro Machado J. A.
2012
“Development of fractional order capacitors based on electrolyte processes,”
Nonlinear Dyn.
68
(12)
107 
115
DOI : 10.1007/s110710110207z
Sivarama Krishna M.
,
Das S.
,
Biswas K.
,
Goswami B.
2011
“Fabrication of a fractional order capacitor with desired specifications: A study on process identification and characterization,”
IEEE Trans. Electron. Devices
58
(11)
4067 
4073
DOI : 10.1109/TED.2011.2166763
Haba T.
,
Ablart G.
,
Camps T.
,
Olivie F.
2005
“Influence of the electrical parameters on the input impedance of a fractal structure realised on silicon,”
Chaos, Solitons Fractals
24
(2)
479 
490
DOI : 10.1016/j.chaos.2003.12.095
Elshurafa A. M.
,
Almadhoun M. N.
,
Salama K. N.
,
Alshareef H. N.
2013
“Microscale electrostatic fractional capacitors using reduced graphene oxide percolated polymer composites,”
Appl. Phys. Lett.
102
(23)
Machado J.A.T.
,
Galhano A.M.S.F.
2012
“Fractional order inductive phenomena based on the skin effect,”
Nonlinear Dyn.
68
(12)
107 
115
DOI : 10.1007/s110710110207z
Benchouia M. T.
,
Golea A.
2013
“Harmonic Current Compensation based on Threephase Threelevel Shunt Active Filter using Fuzzy Logic BandstoptoBandpass Switchable Filter,”
IEEE Trans. Microw. Theory Tech.
61
(3)
1114 
1123
DOI : 10.1109/TMTT.2012.2237036
Park J. H.
,
Cheon S. J.
,
Park J. Y.
2013
“A Compact LTCC DualBand WLAN Filter using Two Notch Resonators,”
J. Electr. Eng. Technol.
8
(1)
168 
175
DOI : 10.5370/JEET.2013.8.1.168
Kahng S.
,
Lee B.
,
Park T.
2013
“Compact UHF 9thorder bandpass filter with sharp skirt by cascadedtriplet CRLHZor,”
J. Electr. Eng. Technol.
8
(5)
1152 
1156
DOI : 10.5370/JEET.2013.8.5.1152
Ruiz J. D.
,
Hinojosa J.
2014
“Shunt series LC circuit for compact coplanar waveguide notch filter design,”
IET Microw Antenna P.
8
(2)
125 
129
DOI : 10.1049/ietmap.2013.0401
Cheon S. J.
,
Park J. H.
,
Park J. Y.
2012
“Highly Miniaturized and Performed UWB Bandpass Filter Embedded into PCB with SrTiO3 Composite Layer,”
J. Electr. Eng. Technol.
7
(4)
528 
588
Radwan A. G.
,
Fouda M. E.
2013
“Optimization of FractionalOrder RLC Filters,”
Circuits Syst. Signal Process.
32
(5)
2097 
2118
DOI : 10.1007/s0003401395809
Ali A. S.
,
Radwan A. G.
,
Soliman A. M.
2013
“Fractional Order Butterworth Filter: Active and Passive Realizations,”
IEEE Journal on emerging and selected topics in circuits and systems
3
(3)
346 
354
DOI : 10.1109/JETCAS.2013.2266753
Podlubny I.
1999
“Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of their Applications,”
Academic Press
San Diego, Calif, USA