Analysis and Passive Synthesis of Immittance for ... ·...

Circuits, Systems, and Signal Processing (2019) 38:3661–3681https://doi.org/10.1007/s00034-019-01035-y

Analysis and Passive Synthesis of Immittancefor Fractional-Order Two-Element-Kind Circuit

Guishu Liang1 · Jiawei Hao1

Received: 14 May 2018 / Revised: 11 January 2019 / Accepted: 14 January 2019 /Published online: 25 January 2019© Springer Science+Business Media, LLC, part of Springer Nature 2019

AbstractFractional circuits have attracted an extensive attention of scholars and researchersfor their superior performance and potential applications. The passive realization ofthe fractional-order immittance function plays an important role in fractional circuitstheory, which is useful in fractional circuits design andmodeling. This paper dealswiththe analysis and passive synthesis of fractional two-element-kind network. Firstly, thetime-domain response of fractional two-element-kind network is analyzed based on itsimmittance function expressions, and the response shows oscillation only in fractionalLβCα circuits. Then, necessary and sufficient conditions to realize the fractional-order immittance functions by a passive network with only two kinds of elements areobtained in view of impedance scaling. A procedure is also proposed to realize suchimmittance functions using two-element-kind network. Finally, three examples aregiven to illustrate the proposed method.

Keywords Fractional circuits · Fractional-order impedance · Network synthesis ·Analysis · Impedance scaling

1 Introduction

In recent years, fractional calculus has been widely used in modeling the dynamicsof many natural phenomena, which is because of its higher capability of provid-ing accurate description than integer dynamical systems. Applications of fractionalcalculus have been reported in many areas [12, 13, 20, 35–39], such as physics, biol-ogy, biomedical engineering, financial market and signal processing. And in electrical

B Jiawei [email protected]

Guishu [email protected]

1 Department of Electric Engineering, North China Electric Power University, No. 619 YongHua NorthStreet, Baoding 071003, China

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3662 Circuits, Systems, and Signal Processing (2019) 38:3661–3681

engineering, the application of fractional calculus is also growing, such as modelingof electrical equipment [4, 14, 30, 32, 59] and wireless power transmission systemdesign [51]. Furthermore, the fractional electrical circuits have been studied fromdiverse aspects, for instance synthesis of filters [44, 55, 56], realization of oscillators[45], stability analysis [40, 46, 50], frequency-domain analysis [6, 22, 34, 41–43, 58],time–frequency-domain analysis [10, 15, 21, 23, 25], energy efficiency [19], sensitivityanalysis [6, 28], fractional-order reconfigurable filters [24], fractional-order parallelresonator [1, 57], fractional-order GIC [3] and passive realization of the fractional-order immittance function [31, 48, 49].

In the study of the fractional-order circuits, for circuits that contain at most twofractional-order elements, the paper [42] presents the generalized impedance, phaseresponse, resonant frequency, quality factor and the sensitivity analyses with respectto the fractional orders of the LβCα circuits; the concept of pure real frequency andpure imaginary frequencywas also proposed. The similarities and differences betweenfractional LβCα circuits and traditional RC/LC circuits are discussed in the case of α+β � 1 in [43]. Thepure imaginary impedance and sensitivity analyses of RCα and RLβ

circuits are studied in thework [34]. For RLβCα circuits which contain three elements,the stability analysis of the series RLβCα circuits is introduced [40], which is basedon the F-plane [46] and S-plane. In [41], the resonance frequencies, stability analysis,quality factor and damping factors of RLαCα circuits are investigated analytically.The resonance in series fractional-order circuits is analyzed in [22]. The papers [6, 58]discussed about the admittance and the resonance conditions of the parallel RLβCα

circuits. In addition, analysis of free oscillations in series RLβCα circuit is describedin [15]. The mathematical model of parallel RLβCα circuit is established and solvedby both Adomain decomposition and Laplace transform methods in [10]. In [25], ageneralized method of solving transient states in RLβCα circuits is introduced bymeans of state equations. The paper [21, 23] present the analysis of transient statesin series and parallel RLβCα circuits for any kind of excitation and α, β ∈ R

+. Atpresent, most of the research on fractional-order circuits analysis focused on circuitswith a small number of components and a simple structure. Transient analysis ofthe two-element-kind network with a number of components has not been reported.There are some issues that require more discussions, such as whether the oscillatoryresponse, which cannot happen in traditional RL/RC circuits, occurs in fractionalRLβ /RCα circuits and the conditions under which the fractional LβCα circuits showa decayed oscillation response.

With the development of numerical calculations about fractional calculus equations[11, 17, 26], it is feasible to simulate and analyze the fractional-order systems usingfractional models. Meanwhile, due to the progresses in fabrication of fractional capac-itors and inductors [2, 8, 52], the fractional-order systems can also be realized directlywith real fractional elements [53]. At present, most of the developed FO impedancesare FO capacitors [2], while the floating or grounded FO inductor can be realizedby active devices like GIC [3], OTA [8] and current-carrying conveyor [52]. Obvi-ously, both the fractional circuits modeling and design motivate the need to study thesynthesis methods for fractional networks. Although several attempts into this goalhave been done, the fractional network synthesis is still in its infancy. In [48, 49], thenecessary and sufficient conditions for the realization of fractional-order impedance

Circuits, Systems, and Signal Processing (2019) 38:3661–3681 3663

function with a passive RLC two-port terminated one fractional element is found.By transforming the immittance function to multivariable positive-real function, asynthesis method for a class of fractional-order immittance function is proposed in[31], where the n-variable positive-real function Z(p1, p2, . . . , pn) contains a high-order variable p1 and n-1 one-order variables p2, p3, . . . , pn , and

∂Z(p1,p2,...,pn)∂pi

� ±(a perfect square), (i � 2, 3, . . . , n) is satisfied. Recently, the positive-real propertyand passivity of the fractional-order circuits were discussed in [27, 29, 33].

However, those proposed synthesis methods for fractional circuits are only appli-cable to several kinds of specific immittance function (matrix) forms. This is mainlydue to the fact that the synthesis for fractional circuits is much more complex thanthat for classical circuits. The fractional one-port circuit composed of two kinds ofelements is a relatively simple network, but its general synthesis method has not beenproposed yet.

In summary, the oscillation conditions and the synthesis methods for two-element-kind network still need more discussions. In conventional circuit theory, the one-portRC, RL and LC have mature synthesis methods and the characters of impedancefunction of such circuits are discussed well [18, 54]. If the fractional-order two-element-kind circuit can be converted into traditional two-element-kind circuit, thesynthesis of fractional one-port circuit composed of two kinds of elements can beaccomplished through the mature synthesis methods. Meanwhile, based on the immit-tance function of fractional two-element-kind network, the zero-state responses canalso be analyzed.

To this end, the aim of the work is to deal with the analysis and passive synthesisof fractional networks using only two kinds of elements, where the order of fractionalelements included canbe any real number between0 and1, and the number of fractionalelements is arbitrary. Specifically, fractional elements with orders not exceeding±1can be considered passive elements [29], even if active devices are inside.

The impulse response of fractional two-element-kind network is analyzed withthe properties of Mittag–Leffler (ML) function. Based on the concept of impedancescaling, the necessary and sufficient conditions to realize fractional two-element-kindnetwork are obtained, and a general synthesis method for such networks is also pro-posed.

The rest of the paper is organized as follows: Some preliminaries are presented inSect. 2. In Sect. 3, the transient analysis of fractional two-element-kind network isgiven. And Sect. 4 elaborates the synthesis method and specific synthesis procedurefor the fractional networks composed of two kinds of elements. In Sect. 5, numericalexamples are presented. The conclusions are drawn in Sect. 6.

2 Preliminaries

This section is devoted to presenting some preliminaries. Fractional calculus is anextension of integral-order calculus. At present, there are three widely used definitionsof fractional differential [5], namely Riemann–Liouville (RL), Grünwald–Letnikov(GL) and Caputo definitions.


Fig. 1 Symbols of fractionalelements

( )a ( )b

( ),Cα α ( ),Lβ β

(a) fractional capacitor (b)fractional inductor

The RL definition of fractional derivative of order α is written as follows:

dα f (t)

dtα≡ 1

Γ (1 − α)

dn

dtn

[∫ t

a(t − τ)n−α−1 f (τ )dτ

](1a)

where Γ (·) is the Euler’sGamma function [5], n � [α] +1, and [α] means the integralpart of α.

The GL definition of fractional derivative of order α is defined as:

dα f (t)

dtα≡ lim

h→0

1

hα

[ t−ah

]∑n�0

(−1)nΓ (α + 1)

n!Γ (α − n + 1)f (t − nh) (1b)

The Caputo definition of fractional derivative of order α is given by:

dα f (t)

dtα≡ 1

Γ (n − α)

∫ t

a(t − τ )−(α+1−n) f (n)(τ )dτ (1c)

The Laplace transforms of Eq. (1a) (1b) and (1c) at zero initial conditions are thesame and given by:

L{0D

αt f (t)

} � sαF(s) (2)

where 0Dαt � dα

dtα denotes the fractional derivative operator of order α.In this paper, we assume that α ∈ [0, 1].The common fractional elements are fractional capacitor and inductor [5].The characteristic equation of a fractional capacitor is:

iα(t) � Cα

dαuα(t)

dtα, (3)

where α ∈ (0, 1] is the order of the passive fractional capacitor, Cα with unit F/s1−α

denoting the pseudo-capacitance of the fractional capacitor; uα(t) and iα(t) are,respectively, the voltage and current of the fractional capacitor. The impedance ofthe fractional capacitor in the Laplace domain equals Z(s) � 1/Cαsα . The circuitsymbol of the fractional capacitor [28] is depicted in Fig. 1a.

The characteristic equation of a fractional inductor is

uβ(t) � Lβ

dβ iβ(t)

dtβ(4)


where β ∈ (0, 1] is the order of the passive fractional inductor, Lβ with unit H/s1−β

denoting the pseudo-inductance of the fractional inductor; uβ(t) and iβ(t) are, respec-tively, the voltage and current of the fractional inductor. The impedanceof the fractionalinductor in the Laplace domain equals Z(s) � Lβsβ . The circuit symbol of fractionalinductor [28] is depicted in Fig. 1b.

Fractional capacitors and fractional inductors are collectively referred to as frac-tional reactance elements (fractances for short). Distinct with classical reactanceelements, fractances are lossy when the orders of the fractional elements are in (0, 1).

Then, the definition and some properties of ML function are described below.Mittag–Leffler function, or one-parameter Mittag–Leffler function [16] is defined

by (5),

Eα(z) �∞∑k�0

zk

Γ (1 + αk), �(α) > 0, z ∈ C (5)

Generalized Mittag–Leffler function with two parameters [16] is defined as (6),

Eα,β(z) �∞∑k�0

zk

Γ (β + αk), �(α),�(β) > 0, z ∈ C (6)

The Laplace transform of (5) and (6) is [16]

L−1[

sα−β

sα − λ

]� tβ−1Eα,β

(λtα

)(7)

as β � 1, we get L−1[sα−1

sα−λ

]� Eα(λtα).

Lemma 1 For any real number λ > 0,

(1) if 0 ≤ α ≤ 1, theML function Eα(−λxα) is completelymonotonic on (0,∞), i.e.,(−1)k Dk Eα(−λxα) ≥ 0 for every x > 0 and every k ∈ N0 [16]. If 1 < α < 2,Eα(−λxα) decays but is not monotonic [7], and there will be more than one zero[9] as α > 1.42.

(2) If 0 ≤ α ≤ 1, β ≥ α, the generalized ML function Eα,β(−λxα) is completelymonotonic [16], and if β ≤ 1 is also satisfied, then tβ−1Eα,β(−λtα) is completelymonotonic.

Lemma 2 [16] Let 0 < α < 1. Then

(1) For β ∈( ∞∪n�0

[−n + α,−n + 1]

)∪ [1,+∞] the function Eα,β(z) has no negative

zero;

(2) For β ∈ ∞∪n�0

(−n,−n + α) the function Eα,β(z) has one negative zero and it is

simple.


Also, the concepts of the reactance function and the commensurate network aregiven as follows.

The function Z(s) is an reactance function of s, if the following conditions aresatisfied [54]:

(1) Z(s) is a rational real function of s.(2) ReZ(s) � 0, when s � jω.(3) The poles of Z(s), which are placed on the imaginary axis of s, are simple, and

their residues are real and nonnegative.

In this paper, the order of a fractional element is referred to as the element order. Thenetwork that the element orders are all the same is called the fractional commensuratenetwork (commensurate network for short). A commensurate network only composedof fractances is called a commensurate reactance network. Differently with the reac-tance network, the commensurate reactance network is lossy, when the orders of thefractional elements are all in (0, 1).

3 Time-Domain Analysis

This section discusses the time-domain response of fractional two-element-kindnetwork. Theoretically, the impulse response of conventional RL/RC circuits mono-tonically decreases, and sustained oscillations happen in conventional LC circuits.Considering that the passive fractional elements are lossy, the oscillation in fractionalcircuit will decay with time. Based on the immittance function of fractional two-element-kind network, the time-domain impulse response of the network is obtained.Then oscillation condition is discussed using the properties of ML function.

3.1 Impedance Function of Fractional Two-Element-Kind Network

The fractional two-element-kind network is classified into five kinds:RLβ ,RCα ,LβCα ,Lβ1Lβ2 , and Cα1Cα2 networks.

For narrative convenience, the impedance of RLC and other fractional elementsis expressed in a unified way, Z(s) � Aγ sγ , where γ ∈ [−1, 1]. For fractionalinductor γ > 0, Aγ � Lγ , for fractional capacitor γ < 0, Aγ � 1/C|γ | and forresistor γ � 0, Aγ � R. Unlike the classical circuit elements, fractional capacitors(fractional inductors) with different element orders are considered as different kindsof elements in this paper. And γ1, γ2 are used to distinguish the value of γ of two kindsof elements (if γ1 � γ2, the two kinds of elements reduce to one kind of elements);for instance, the element of γ1 � 1 (inductor) and the other elements with γ2 � −1(capacitor) are considered as two kinds of elements.

Without loss of generality, we assume that γ1 > γ2 in the two-element-kind net-work, and the two-element-kind network can cover the RL , RC , and LC networkswhen γ1, γ2 ∈ {−1, 0, 1}. In the case of γ1 � −γ2 � γ , the network is a com-mensurate reactance network, in which the element orders of fractional inductors andfractional capacitors are all γ .


The impedance of the two-element-kind network can be written as follows (seeAppendix A):

Z(s) � sv0cMsMv + cM−1s(M−1)v + · · · + c0dN sNv + dN−1s(N−1)v + · · · + d0

, (8)

where v ∈ (0, 2], |M − N |≤ 1, the coefficients c0, . . . , cM , d0, . . . , dN > 0, and sv0

is a fractional-order multiplier and v0 ∈ [−1, 1].

3.2 Transient Response

From (A-1), the impedance of two-element-kind network is Z(s) � sγ1+γ2

2 ZB(s). AsZB(s) is an reactance function, Z(s), as well as Y (s) that can be obtained by replacingγ1, γ2 with −γ2,−γ1, is expressible as a partial expansion,

Z(s) � kγ1sγ1 + kγ2s

γ2 +n∑

i�1

ki sγ1

sγ1−γ2 + bi(9)

Y (s) � kγ1s−γ1 + kγ2s

−γ2 +n∑

i�1

ki s−γ2

sγ1−γ2 + b′i

(10)

where kγ1 , kγ2 , ki , bi , b′i > 0.

With the inverse Laplace transform given by (7), we can rewrite the terms of (9)and (10) as follows:

L−1[s−γi] � tγi−1E1,γi (0) � tγi−1

Γ (γi ), i � 1, 2 (11)

L−1[

ki sγ1

sγ1−γ2 + bi

]� ki t

−γ2−1Eγ1−γ2,−γ2

(−bi tγ1−γ2

)(12)

where Reα > 0,Reβ > 0.Then, for fractional Lγ1Lγ2 circuit, the time-domain impulse response is,

Z(t) � dγ1

dtγ1δ(t) +

dγ2

dtγ2δ(t) +

n∑i�1

ki t−γ2−1Eγ1−γ2,−γ2

(−bi tγ1−γ2

)(13)

Y (t) � tγ1−1

Γ (γ1)+

tγ2−1

Γ (γ2)+

n∑i�1

ki tγ1−1Eγ1−γ2,γ1

(−b′i t

γ1−γ2)

(14)

for fractional C|γ1|C|γ2| circuit,

Z(t) � t−γ1−1

Γ (−γ1)+

t−γ2−1

Γ (−γ2)+

n∑i�1


(−bi tγ1−γ2

)(15)


Y (t) � d−γ1

dt−γ1δ(t) +

d−γ2

dt−γ2δ(t) +

n∑i�1


(−b′i t

γ1−γ2)

(16)

for fractional Lγ1C|γ2| circuit,

Z(t) � dγ1

dtγ1δ(t) +

t−γ2−1

Γ (−γ2)+

n∑i�1


(−bi tγ1−γ2

)(17)

Y (t) � tγ1−1

Γ (γ1)+

d−γ2

dt−γ2δ(t) +

n∑i�1


(−b′i t

γ1−γ2)

(18)

For the sake of brevity, we assume that ki � 1. As t > 0, the first two termsof the above expressions either monotonically decrease or equal to zero, oscillatory

response depends on the following items Zn(t) �n∑

i�1t−γ2−1Eγ1−γ2,−γ2

(−bi tγ1−γ2)

or Yn(t) �n∑

i�1tγ1−1Eγ1−γ2,γ1

(−b′i t

γ1−γ2). The impulse response will not oscillate, if

Zn(t) and Yn(t) are monotonic.For the admittance of RLγ1 and Lγ1Lγ2 circuits, where γ2 ≥ 0, according to

Lemma 1, Yn(t) is completely monotonic. Hence, Y (t) of RLγ1 and Lγ1Lγ2 circuitmonotonically decrease.

For the impedance of RC|γ2| and C|γ1|C|γ2| circuits, as γ1 ≤ 0, with Lemma 1, Zn

(t) is completelymonotonic. Then, Z(t) of RC|γ2| andC|γ1|C|γ2| circuit monotonicallydecrease.

As for Lγ1C|γ2| circuits, the impedance function is discussed as an example. Thederivative of t−γ2−1Eγ1−γ2,−γ2

(−bi tγ1−γ2)is t−γ2−2Eγ1−γ2,−γ2−1

(−btγ1−γ2). When

γ1 − γ2 ∈ (0, 1), the condition −1 < −γ2 − 1 < γ1 − γ2 − 1 < 0 is satisfied, withLemma 2; t−γ2−2Eγ1−γ2,−γ2−1

(−btγ1−γ2)has only one simple zero in t > 0. There

are no oscillation components in Z(t), since the oscillation occurs when the derivativeof the function has at least two zeros.

When γ1−γ2 ∈ (1, 2), the oscillation condition is discussed based on the propertiesof one-parameter ML function. Let γ2 � −1 and γ1 > 0.42, then Zn(t) shows adecayed oscillation. This is because, with Lemma 1, Eγ1+1,1

(−btγ1+1)decayed but

not monotonic and has at least two zeroes in the case of γ1 > 0.42. In a similar way,if γ1 � 1 and γ2 ∈ (−1,−0.42), Yn(t) decayed with oscillation.

In summary, the impulse response of RLβ , Lβ1Lβ2 , RCα and Cα1Cα2 circuitsdecreases monotonically. For LβCα circuits, there are two sufficient conditions here,

(1) when α + β ∈ (0, 1], there are no oscillation components in impulse response.(2) when α + β ∈ (1, 2), in the case that (a) α � 1, β ∈ (0.42, 1) and (b) β � 1, α ∈

(0.42, 1), the impulse response contains decayed oscillation components. Whilein the case that (c) α � 1, β ∈ (0, 0.41) and (d) β � 1, α ∈ (0, 0.41), there areno oscillation components in impulse response.

Figure 2a shows that the impulse response u(t) of parallel RCα andCα1Cα2 circuitsdecreases monotonically when R � 0.1 Ω , Cα � 1 F/s1−a ; Cα1 � 0.1 F/s1−a1 ,


(a) (b)

Fig. 2 Impulse response u(t) of a parallel RCα and Cα1Cα2 circuits when R � 0.1 Ω,Cα �1 F/s1−a ;Cα1 � 0.1 F/s1−a1 ,Cα2 � 1 F/s1−a2 , and b parallel LβCα circuit when Lβ �0.1 H/s1−β ,Cα � 1 F/s1−α for α + β ∈ (0, 1]

Cα2 � 1 F/s1−a2 . Figure 2b shows that the impulse response u(t) of parallel LβCα

circuit decays without oscillation when Lβ � 0.1 H/s1−β,Cα � 1 F/s1−α , for α+β ∈(0, 1]. The impulse response of parallel LβCα circuit when Lβ � 0.1 H/s1−β,Cα �1 F/s1−α for α + β > 1 is as shown in Fig. 3.

Then, we discuss the effect of element value. Changing the element value onlyaffects the coefficients ki and bi in Eq. (9). Therefore, the changes in element valuehave no effect on the oscillation of the impulse response when the circuits satisfy theabove conditions.

For the component of the impulse response fi (t) � ki t−γ2−1Eγ1−γ2,−γ2(−bi tγ1−γ2), the increase of ki can be seen as an amplitude scaling of fi (t), and

the change in bi will make both amplitude and time scaling on fi (t). For example, ifwe set fa(t) � k0t−0.5E1, 0.5(−b0t), τ � b0t and fb(t) � t−0.5E1,0.5(−t), then,

fa(t) � k0b0.50 τ−0.5E1, 0.5(−τ) � k0b

0.50 fb(τ ).

In particular, when −γ2 � 1, the component

fi (t) � ki t−γ2−1Eγ1−γ2,−γ2

(−bi tγ1−γ2

) � ki Eγ1−γ2,−γ2

(−bi tγ1−γ2

),

changing bi will only compressed or expanded fi (t).Figure 4 shows the impulse response of parallel L0.4C0.4 and L0.8C circuits. It can

be seen that changing the fractional-order element values has the effect of scale on theamplitude and time of the impulse response of two-element-kind network, but doesnot affect its oscillation.


(a) (b)

Fig. 3 Impulse response a u(t) of LβCα circuits when α � 1, β ∈ (0, 1), and b i(t) of LβCα circuits when

β � 1, α ∈ (0, 1), where Lβ � 0.1 H/s1−β ,Cα � 1 F/s1−α

Fig. 4 Impulse response of L0.8C circuits for different element values

4 The Synthesis of Fractional Two-Element-Kind Network

In this section, we take the realization of impedance functions as an example; therealization of the admittance functions can be treated similarly.


4.1 Necessary and Sufficient Conditions

Theorem The impedance function Z(s) is realizable by a two-element-kind networkif and only if the following conditions are simultaneously satisfied:

(1) Z(s) is in the form of (8), and γ1, γ2 ∈ [−1, 1].The γ1, γ2 can be obtained from the following two cases,Case 1 γ1 � v0 and γ2 � v0 − v, if M � N and c0

cM> d0

dM, or N � M + 1.

Case 2 γ1 � v0 + v and γ2 � v0, if M � N and c0cM

< d0dM

, or N � M − 1.

(2) ZB(s) � s− γ1+γ22 Z(s), the impedance function of the network after impedance

scaling is an reactance function with respect to sγ1−γ2

2 .

The proof of Theorem is shown in Appendix B.

Remark 1 Whether ZB(s) � s− γ1+γ22 Z(s) is an reactance function with respect to

sγ1−γ2

2 can be identified based on Routh criterion.

From Eq. (8), if we assume that p � sγ1−γ2

2 , ZB(s) will be a function of p, denotedas Z(p). According to [54], Z(p) is an reactance function if and only if H(p), the sumof the numerator and denominator of Z(p), is a strictly Hurwitz polynomial. StrictlyHurwitz polynomial has no zeros placed in the right half closed plane (the imaginaryaxis is included), and this can be ensured if the values in the first column of the Routhtable are all positive [47]. Therefore, if H(p) satisfies the Routh criterion [47], Z(p)is an reactance function. For instance, consider the following impedance function:

Z(s) � 2s0.8 + 8s0.4 + 6

s0.4 + 2,

where N � M − 1, and v0 � 0, v � 0.4. From Theorem, it is deduced that γ2 �v0 � 0, γ1 � v0 + v � 0.4. The sum of the numerator and denominator of Z(s)/s0.2

is H(p) � 2p4 + p3 + 8p2 + 2p + 6, where p � s0.2. The Routh table of H(p) is asfollows,

p4 2 8 6

p3 1 2 0

p2 4 6

p1 0.5 0

p0 6

As the values in the first column of the Routh table of H(p) are all positive, we

can conclude that Z(s)/s0.2 is an reactance function with respect to sγ1−γ2

2 . As for theimpedance function

Z(s) � 2s1.6 + 4s0.8 + 1

s1.7 + 2s0.9 + s0.1� s−0.1 · 2s

1.6 + 4s0.8 + 1

s1.6 + 2s0.8 + 1,


where N � M and c0/cM � 0.5, d0/dM � 1, v0 � −0.1, v � 0.8. Also according toTheorem, we have γ2 � v0 � −0.1, γ1 � v0 +v � 0.7. And the sum of the numeratorand denominator of Z(s)/s0.3 is H(p) � p5+2p4+2p3+4p2 + p+1, where p � s0.4.The Routh table of H(p) is as follows,

p5 1 2 1

p4 2 4 1

p3 δ+ ≈ 0 0.5 0

p2 4 − 1/δ+ ≈ −1/δ+ 1

p1 0.5 0

p0 1

As the signs of the values changed twice in the first column of Routh table, H(p)has two zeros in the open right half-plane, which means H(p) is not a strictly Hurwitz

polynomial. Hence, Z(s)/sγ1+γ2

2 is not an reactance function with respect to sγ1−γ2

2 .As for the remaining case where the values in one row of Routh table of H(p) are all

zero, H(p) has zeros that are symmetry to the original point, and clearly Z(s)/sγ1+γ2

2

is not an reactance function with respect to sγ1−γ2

2 either.

Remark 2 For the impedance function Z(s) of a two-element-kind network, with scal-ing parameter s−γ1 , ZB(s) � s−γ1 Z(s) will be an RC impedance function of sγ1−γ2 .Besides, ZB(s) � s−γ2 Z(s) will be an RL impedance function of sγ1−γ2 with scalingparameter s−γ2 . Therefore, it is also available to realize Z(s) by synthesis methodsused in RC networks or RL networks.

Anyway, a two-element-kind network can always be translated to a commensuratenetwork by suitable impedance scaling.

4.2 Synthesis procedure

The following procedure can finish the realization of the fractional impedance functionin Eq. (8) by using two-element-kind network.

Step 1 Check the realizability. Calculate γ1,γ2 from the impedance function Z(s) and

determine that γ1, γ2 ∈ [−1, 1]. Identify whether ZB(s) � s− γ1+γ22 Z(s) is an

reactance function with respect to sγ1−γ2

2

Step 2 Realize Z(s)/sγ1+γ2

2 as a passive commensurate reactance network using tra-ditional LC network synthesis methods

Step 3 Realize Z(s) by impedance scaling. Scaling the impedance levels of the ele-ments in the commensurate reactance network obtained in step 2, with the

scaling parameter sγ1+γ2

2 , the realization of Z(s) is then obtained.

It should be noted that as a fractional inductor is changed to fractional capacitor,and vice versa, the value of the element should take the reciprocal; to the value of afractional inductor is Lγ � Aγ , while the value of a fractional capacitor is C|γ | �


1/Aγ . For instance, with the scaling parameter s−1, the impedance scalingwill modifythe impedance of a fractional inductor of order 0.5 and L0.5 � 2H/s0.5 from 2s0.5 to2 1s0.5

. Obviously, the element becomes a fractional capacitor of order 0.5 and C0.5

equals 0.5F/s0.5, which exactly the reciprocal of L0.5.

5 Application

In this section, three numerical examples are presented to show the usefulness ofthe results of the previous section in implementation of fractional-order impedancefunctions.

(1) Consider the impedance function

Z(s) � s0.7 + 4s0.3

2s0.8 + 20s0.4 + 18� s0.3 · s0.4 + 4

2s0.8 + 20s0.4 + 18(19)

It can be obtained from this impedance function that N � M + 1, γ0 � 0.3 andγ � 0.4. Then, from Theorem we have γ1 � γ0 � 0.3, γ2 � γ0 − γ � −0.1.

The sum of the numerator and denominator of Z(s)/s0.1 is:

H(p) � 2p4 + p3 + 20p2 + 4p + 18,

where p � s0.2. The Routh table of H(p) is as follows,

p4 2 20 18

p3 1 4 0

p2 12 18

p1 2.5 0

p0 18

As there is no sign change in the first column of the array, Z(s)/s0.1 is an reactancefunction of s0.2. Hence, by Theorem it is deduced that the impedance function (19) isrealizable by using fractional inductors of order 0.3 and fractional capacitors of order0.1.

According to Cauer’s synthesis method [54], the continued fraction expansion ofthe impedance function (19) at s → ∞ is as follows,

Z (s) � s0.1 · s0.6 + 4s0.2

2s0.8 + 20s0.4 + 18� s0.1 · 1

2s0.2 + 1112 s

0.2+ 1245 s0.2+ 1

536 s

0.2

� 1

2s0.1 + 1112 s

0.3+ 1245 s0.1+ 1

536 s

0.3


Fig. 5 Ladder form of theimpedance function (19) inExample 1

Fig. 6 Foster form of theimpedance function (19) inExample 1

Then, the impedance function (19) can be realized as a one port with fractionalinductors of order 0.3 and fractional capacitors of order 0.1. The corresponding realizednetwork is shown in Fig. 5. It is a L0.3C0.1 network.

Also, it is available to realize the impedance function (19) by Foster’s synthesismethod [54] used in RC networks. From Remark 2, with the scaling parameter s−0.3,Z(s)/s0.3 will be an RC impedance function with respect to s0.4, namely that the polesof Z(s)/s0.3 are placed on the negative real axis of s0.4, simple and with nonnegativeresidues.

Rewrite Z(s)/s0.3 in the form of partial-fraction expansion with respect to s0.4, wehave:

Z(s)/s0.3 � 5

16

1

s0.4 + 9+

3

16

1

s0.4 + 1,

and then,

Z(s) � 5

16

1

s0.1 + 9/s0.3+

3

16

1

s0.1 + 1/s0.3.

Consequently, the impedance function (19) can be realized as Foster form, leadingto the network shown in Fig. 6.

(2) Given the impedance function

Z(s) � 2s0.8 + 8s0.4 + 6

s0.4 + 2(20)

We can obtain from impedance function (20) that N � M−1, v0 � 0 and v � 0.4.Then, according to Theorem, we have γ2 � v0 � 0, γ1 � v0 + v � 0.4. The sum ofthe numerator and denominator of Z(s)/s0.2 is:


Fig. 7 Ladder network of theimpedance function (20) inExample 2

H(p) � 2p4 + p3 + 8p2 + 2p + 6,


p4 2 8 6

p3 1 2 0

p2 4 6

p1 0.5 0

p0 6

There are all positive values in the first column of Routh table, indicating thatZ(s)/s0.2 is an reactance function with respect to s0.2. According to Theorem, theimpedance function (20) can be realized with fractional inductors of order 0.4 andresistors, which is corresponding to an RL0.4 network.

A continued fraction expansion of the impedance function (20) about infinity yields:

Z(s) � 2s0.4 +1

14 + 1

8s0.4+12

,

which corresponds to the ladder network shown in Fig. 7.

(3) Consider the admittance function

Y (s) � s + 4s0.6

2s0.8 + 20s0.4 + 18� s0.6 · s0.4 + 4

2s0.8 + 20s0.4 + 18(21)

It can be obtained from this admittance function that N � M − 1, γ0 � −0.6 andγ � 0.4. Consequently, γ2 � γ0 � −0.6, γ1 � γ0 + γ � −0.2.

The sum of the numerator and denominator of YB(s) � 1ZB (s) � Y (s)/s0.4 is:

H(p) � 2p4 + p3 + 20p2 + 4p + 18,



Fig. 8 Ladder network of theadmittance function (21) inExample 3

p4 2 20 18

p3 1 4 0

p2 12 18

p1 5/2 0

p0 18

The values in the first column of Routh table are all positive, which indicates that Y(s)/s0.4 is an reactance function of s0.2. Also from Theorem, the admittance function(21) can be realized as a C0.2C0.6 network.

Carrying out a continued fraction expansion of the impedance function (21) ats → ∞, we obtain

Y (s) � s0.4 · s0.6 + 4s0.2

2s0.8 + 20s0.4 + 18� 1

2s−0.2 + 1112 s

0.6+ 1245 s−0.2+ 1

536 s

0.6

The above expression is now realized directly to give the network in Fig. 8.

6 Conclusions

Analysis and synthesis of the fractional two-element-kind network are discussed inthis paper. The fractional two-element-kind network is of more kinds than that of theinteger-order two-element-kind network, which not only includes the commensuratenetworks, such as RLβ , RCα and LβCβ networks, but also include the noncommen-surate networks, such as LβCα , Lβ1Lβ2 and Cα1Cα2 networks. In time domain, theimpulse response of RLβ , Lβ1Lβ2 , RCα andCα1Cα2 circuits decreasesmonotonically.For LβCα circuits, there are two sufficient conditions here, (1) when α + β ∈ (0, 1],there are no oscillation components in impulse response. (2) When α + β ∈ (1, 2),in the case that (a) α � 1, β ∈ (0.42, 1) and (b) β � 1, α ∈ (0.42, 1), the impulseresponse contains decayed oscillation components. While in the case that (c) α � 1,β ∈ (0, 0.41) and (d) β � 1, α ∈ (0, 0.41) there are no oscillation components inimpulse response. And if these conditions are met, the changes in element value willscale the amplitude and time of the impulse response of two-element-kind network,but does not affect its oscillation. Then, based on the fact that a fractional two-element-kind network can always be transformed into a commensurate network by impedance


scaling, the necessary and sufficient conditions were derived for the realizability ofimmittance functions to be realized by two-element-kind network. Also, a generalsynthesis method was proposed for such networks. The investigation in this paperfurther enriched the synthesis methods of the fractional electrical networks.

Acknowledgements This research was supported in part by Natural Science Foundation of BeijingMunic-ipality under Grant No. 3192039 and the Natural Science Foundation of Hebei Province under Grant No.E2018502121.

Appendix A: The Impedance Function Expression of PassiveTwo-Element-Kind Network

By scaling the impedance levels of the elements with parameter s− γ1+γ22 , a network

containing two kinds of elements will be transformed to a commensurate reactancenetwork. This is because that, after the impedance scaling, the elements whose valueof γ is γ1 become fractional inductors of order (γ1 − γ2)/2, and the elements whosevalue of γ is γ2 become fractional capacitors of order (γ1 − γ2)/2; both the order of

fractional inductors and fractional capacitors are (γ1 − γ2)/2. If we set p � sγ1−γ2

2 ,the impedance function of the network after impedance scaling ZB(p) will be anreactance function with respect to p, and (A-1) holds,

ZB(p) � s− γ1+γ22 Z(s) (A-1)

where p � sγ1−γ2

2 .The general form of Z(s) is given below by studying the form of ZB(p).Based on the existence of zero or pole at p � 0, the expression of ZB(p) has two

forms as (A-2) and (A-4).

(1) If ZB(p) has a zero at p � 0.

If ZB(p) has a zero at p � 0, we have

ZB(p) � k · p

M∏m�0

(p2 + am

)N∏

n�0

(p2 + bn

) (A-2)

where ai , bi (i � 0, 1, 2, . . .),k are positive real, and [54] bi < ai < bi+1N − 1 ≤M ≤ N .

From (A-1) and (A-2), (A-3) is obtained,

Z(s) � k · sγ1M∏

m�0

(sγ1−γ2 + am

)N∏

n�0

(sγ1−γ2 + bn

) � sγ1cMsM(γ1−γ2) + cM−1s

(M−1)(γ1−γ2) + · · · + c0dN sN (γ1−γ2) + dN−1s(N−1)(γ1−γ2) + · · · + d0

,

(A-3)


and the coefficients c0, . . . , cM , d0, . . . , dN > 0, which is due to ai , bi(i � 0, 1, 2, . . .),k are positive real. In the case that M � N , since

∏Mm�0 am >∏N

n�0 bn , we havec0cM

> d0dM

. And sγ1 can be viewed as a fractional-order multiplier.

(2) If ZB(p) has a pole at p � 0.

If ZB(p) has a pole at p � 0, we have

ZB(p) � k · 1p

M∏m�0

(p2 + a′

m

)N∏

n�0

(p2 + b′

n

) (A-4)

where a′i , b

′i (i � 0, 1, 2, . . .), k are positive real, and [54] a′

i < b′i < a′

i+1, M − 1 ≤N ≤ M .

From (A-1) and (A-4), (A-5) is obtained,

Z(s) � k · sγ2

M∏m�0

(sγ1−γ2 + a′

m

)N∏

n�0

(sγ1−γ2 + b′

n

) � sγ2cMsM(γ1−γ2) + cM−1s(M−1)(γ1−γ2) + · · · + c0dN sN (γ1−γ2) + dN−1s(N−1)(γ1−γ2) + · · · + d0

(A-5)

and the coefficients c0, . . . , cM , d0, · · · , dN > 0, which is due to a′i , b

′i

(i � 0, 1, 2, . . .), k, are positive real. In the case M � N , since∏M

m�0 a′m <

∏Nn�0 b

′n ,

we have c0cM

< d0dM

. And the fractional-order multiplier is sγ2 .For convenience, we assume that Z(s) → c0

d0sv0 as s → 0, and the difference of

the degrees between two adjacent items is v, and v � γ1 − γ2. As a result of γ1, γ2 ∈[−1, 1], we have v0 ∈ [−1, 1] and v ∈ (0, 2]. With the assumptions above, we canarrive at (A-6) from (A-3) and (A-5).

For a passive two-element-kind network, the expression of the impedance functionZ(s) is as follows:

Z(s) � sv0cMsMv + cM−1s(M−1)v + · · · + c0dN sNv + dN−1s(N−1)v + · · · + d0

(A-6)

where M, N is, respectively, the number of items in numerator and denominator and|M−N |≤ 1, v � γ1−γ2 and v ∈ (0, 2], the coefficients c0, · · · , cM , d0, · · · , dN > 0.sv0 is a fractional-order multiplier and v0 ∈ [−1, 1], and we can conclude that (1)γ1 � v0 if M � N and c0

cM> d0

dM, or N � M + 1; (2) γ1 � v0 + v, if M � N and

c0cM

< d0dM

, or N � M − 1.In the dual way, the expression (A-6) is true for admittance function. QED


Appendix B: Proof of Theorem

NecessityAs the twokinds of elements in a network are both passive, the correspondingtwo-element-kind network must be passive. The value of Aγ of a passive element willbe positive and the values of γ of the element will be in [−1, 1], which meansγ1, γ2 ∈ [−1, 1]. Scaling the impedance levels of the elements in this network with

parameter s− γ1+γ22 , the network becomes a passive commensurate reactance network,

where the value of Aγ keeps the same, the values of γ of elements changed fromγ1γ2 to

γ1−γ22

γ2−γ12 , and γ1−γ2

2 ∈ (0, 1] γ2−γ12 ∈ [−1, 0). Thus, ZB(s), the impedance

function of the network after impedance scaling, should be an reactance function of

sγ1−γ2

2 , and ZB(s) � s− γ1+γ22 Z(s).

Consequently, the conditions (1) and (2) are necessary for passive realization of Z(s) by a passive two-element-kind network.

Sufficiency Assume that ZB(s) � s− γ1+γ22 Z(s) is an reactance function of s

γ1−γ22 ,

and γ1, γ2 ∈ [−1, 1]. To synthesis ZB(s) which is an reactance function of sγ1−γ2

2 ,there are two classical methods to realize the reactance function in synthesis theory,which are the Foster’s synthesis based on partial expansion and the Cauer’s synthesisbased on continued fraction expansion. Thus, it is always feasible to realize ZB(s) bya commensurate network, where the orders of elements are γ1−γ2

2 and the values ofAγ of elements are positive.

Then, by scaling the impedance levels of the elements in the networkwith parameter

sγ1+γ2

2 , the commensurate network corresponding to ZB(s) becomes a two-element-kind network corresponding to Z(s), where the values of Aγ of elements are positive,and the values of γ of elements become, respectively, γ1, γ2. As γ1, γ2 ∈ [−1, 1], theelements in the two-element-kind network corresponding to Z(s) are passive. Thus,the conditions (1) and (2) are also sufficient conditions for realization of Z(s) by apassive two-element-kind networks QED

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