research article fractional-order two-port...

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Research ArticleFractional-Order Two-Port Networks

M. E. Fouda,1 A. S. Elwakil,2,3 A. G. Radwan,1,3 and B. J. Maundy4

1Engineering Mathematics and Physics Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt2Department of Electrical and Computer Engineering, University of Sharjah, College of Engineering, P.O. Box 27272, UAE3Nanoelectronics Integrated Systems Center (NISC), Nile University, Giza 12588, Egypt4Department of Electrical and Computer Engineering (ECE), University of Calgary, AB, Canada T2N 1N4

Correspondence should be addressed to A. G. Radwan; [email protected]

Received 25 March 2016; Accepted 17 April 2016

Academic Editor: Riccardo Caponetto

Copyright © 2016 M. E. Fouda et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce the concept of fractional-order two-port networks with particular focus on impedance and admittance parameters.We show how to transform a 2 × 2 impedance matrix with fractional-order impedance elements into an equivalent matrix with allelements represented by integer-order impedances; yet the matrix rose to a fractional-order power. Some examples are given.

1. Introduction

Two-port networks are widely used in linear circuit analysisand design [1, 2]. The system under consideration is repre-sented by a describing matrix which relates its input and out-put variables (voltages and currents). Such a representationenables the treatment of the system as a black box where theinternal details become irrelevant. It also offers an extremelyefficient computational technique which can be used tomodel series, parallel, or cascade interconnects of severalsystems. Standard Network Analyzers can be configuredto measure several types of two-port network parametersincluding impedance, admittance, transmission, and scatter-ing parameters.

Consider, for example, the impedance matrix representa-tion of a system in which case we have

(

𝑉1

𝑉2

) = (

𝑍11

𝑍12

𝑍21

𝑍22

)(

𝐼1

𝐼2

) = (𝑍)(

𝐼1

𝐼2

) , (1)

where 𝑉1,2

(𝐼1,2) are the voltages (currents) at the input

port and output port, respectively, as shown in Figure 1. Allelements in the 2 × 2 impedance matrix are measured in Ω

and if 𝑍11

= 𝑍22

the network is known to be symmetricalwhile if 𝑍

12= 𝑍21

it is known to be reciprocal. How-ever, with the increasing use of fractional-order impedancemodels, particularly in representing supercapacitors [3, 4],energy storage devices [5], oscillators [6], filters [7], and new

electromagnetic charts [8], it is possible that the elementsof (𝑍) are of fractional order. Consider the simple case ofthe grounded impedance 𝑍, shown in Figure 2(a). Treatedas a two-port network, this impedance is described by theimpedance matrix

(

𝑉1

𝑉2

) = (

𝑍 𝑍

𝑍 𝑍)(

𝐼1

𝐼2

) . (2)

Let 𝑍 be a supercapacitor operating in its Warburg mode,where 𝑍 = 1/𝑄√𝑠; 𝑠 = 𝑗𝜔. In this region of operation, themagnitude is proportional to 1/√𝜔, the phase angle is fixedat −45∘, and 𝑄 is the pseudocapacitance of the device [9, 10].As a two-port network, this device would be described as

(

𝑉1

𝑉2

) =1

𝑄(

1

√𝑠

1

√𝑠

1

√𝑠

1

√𝑠

)(

𝐼1

𝐼2

) =1

𝑄(𝑍𝐹) (

𝐼1

𝐼2

) . (3)

Therefore all elements of the (𝑍𝐹) matrix are of fractional

order. However, we can rewrite the above equation in thealternative form

(

𝑉1

𝑉2

) =1

𝑄(

2

𝑠

2

𝑠

2

𝑠

2

𝑠

)

1/2

(

𝐼1

𝐼2

) =1

𝑄(𝑍𝐼)1/2

(

𝐼1

𝐼2

) (4)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 5976301, 5 pageshttp://dx.doi.org/10.1155/2016/5976301

2 Mathematical Problems in Engineering

Two-port network

I1

I1

+

−

V1

I2

I2

+

−

V2

Figure 1: Two-port network variables.

since

(

1

√𝑠

1

√𝑠

1

√𝑠

1

√𝑠

)

2

= (

2

𝑠

2

𝑠

2

𝑠

2

𝑠

) . (5)

It is clear that the elements 𝑍11

→ 𝑍22

of (𝑍𝐼) are all

integer-order impedances, each representing a capacitor of(1/2) Farad whereas the power of the matrix is fractional;that is, √(𝑍

𝐼) = (𝑍

𝐹). In this paper we seek to generalize

this procedure by obtaining the equivalent matrix (𝑍𝐼) and

its fractional exponent such that (𝑍𝐼)𝑛= (𝑍𝐹).The procedure

is not restricted to the impedance matrix and can be applied

to any other type of two-port network parameters. The mainadvantage of this conversion is that an equivalent circuit of(𝑍𝐼) can be easily obtained with integer-order components.

For example, if (𝑍𝐼) is reciprocal, then its equivalent circuit is

that shown in Figure 2(b). However, it is not yet known howto use this equivalent circuit in association with the fractionalexponent of the matrix to construct an overall equivalentmodel of the originally fractional-order two-port network.

2. Power of a Matrix

A matrix 𝑀𝑛 for a nonnegative exponent 𝑛 is defined as the

matrix product of 𝑛 copies of𝑀. However, if 𝑛 is a noninteger,then we need to revert to the Cayley-Hamilton theorem. Inparticular, if 𝑀 is a 2 × 2 matrix and 𝐼 is an identify matrix,then

𝑀𝑛= 𝛼0𝐼 + 𝛼1𝑀, (6)

where

(

𝛼0

𝛼1

) =1

𝜆2− 𝜆1

(

𝜆2𝜆𝑛

1− 𝜆2𝜆𝑛

1

𝜆𝑛

2− 𝜆𝑛

1

) (7)

and 𝜆1,2

are nonrepeated eigenvalues. Hence, for a givenimpedance matrix 𝑍

𝐼, we have

(𝑍𝐼)𝑛

=1

𝜆2− 𝜆1

(

𝜆𝑛

1(𝜆2− 𝑍11) + 𝜆𝑛

2(𝑍11

− 𝜆1) 𝑍

12(𝜆𝑛

2− 𝜆𝑛

1)

𝑍21(𝜆𝑛

2− 𝜆𝑛

1) 𝜆

𝑛

1(𝜆2− 𝑍22) + 𝜆𝑛

2(𝑍22

− 𝜆1)

) , (8)

where 𝜆1,2

= (1/2)(𝑍11+𝑍22±√(𝑍

11− 𝑍22)2+ 4𝑍12𝑍21). For

the case of repeated eigenvalues (𝜆1= 𝜆2= 𝜆) we obtain

(𝑍𝐼)𝑛

=(𝑍11

+ 𝑍22)𝑛−1

2𝑛

⋅ (

𝑍11

+ 𝑍22

+ 𝑛 (𝑍11

− 𝑍22) 2𝑛𝑍

12

2𝑛𝑍21

𝑍11

+ 𝑍22

+ 𝑛 (𝑍11

− 𝑍22)

) .

(9)

If the two-port network is symmetrical, that is,𝑍11

= 𝑍22,

then for nonrepeated eigenvalues

(𝑍𝐼)𝑛

=1

2(

𝑥 √𝑍12

𝑍21

𝑦

√𝑍21

𝑍12

𝑦 𝑥

), (10)

where 𝑥 = (𝑍11

− √𝑍12𝑍21)𝑛+ (𝑍11

+ √𝑍12𝑍21)𝑛 and 𝑦 =

(𝑍11

+ √𝑍12𝑍21)𝑛− (𝑍11

− √𝑍12𝑍21)𝑛 while for repeated

eigenvalues we obtain

(𝑍𝐼)𝑛

= 𝑍𝑛−1

11(

𝑍11

𝑛𝑍12

𝑛𝑍21

𝑍11

) . (11)

If, in addition to being symmetrical, the network is alsoreciprocal (i.e., 𝑍

12= 𝑍21), we then obtain

(𝑍𝐼)𝑛

=1

2(

𝑥 𝑦

𝑦 𝑥

) , (12)

where 𝑥 = (𝑍11+𝑍12)𝑛+ (𝑍11−𝑍12)𝑛 and 𝑦 = (𝑍

11+𝑍12)𝑛−

(𝑍11−𝑍12)𝑛. Noting the semigeneral case of amatrix𝑀 given

by (𝑀) = ( 𝑎 𝑏𝑐 𝑎

), then it can be easily shown that

(𝑀)𝑛=

1

2

⋅(

(𝑎 − √𝑏𝑐)𝑛

+ (𝑎 + √𝑏𝑐)𝑛 (𝑎 − √𝑏𝑐)

𝑛

− (𝑎 + √𝑏𝑐)𝑛

√𝑐/𝑏


− (𝑎 + √𝑏𝑐)𝑛

√𝑏/𝑐


+ (𝑎 + √𝑏𝑐)𝑛

).

(13)

In Section 3 a number of circuit applications are considered.

3. Applications

Case 1. Consider the case of floating impedance, as shownin Figure 3(a), and assume that this impedance represents afractional-order inductor with impedance 𝑍 = 𝐿𝑠

𝛼; 𝑠 = 𝑗𝜔

and 𝐿 is the pseudoinductance.This single impedance cannot

Mathematical Problems in Engineering 3

+

I2I1

I2I1

−

V2V1Z

+

−

(a)

+

−

I2

I2

I1

I1

V1

+

−

V2Z12

Z22 − Z12Z11 − Z12

(b)

Figure 2: (a) Single grounded impedance as a two-port network and (b) general equivalent circuit from a reciprocal impedance matrix.

+

I1

I1

I2

I2

−

+

−

V2V1

Z

(a)

I1

I1

I2

I2

+ +

− −

V2V1

Z

YY

(b)

L/2C

C2

C2

0.5L2

(c)

Figure 3: (a) Single floating impedance as a two-port network, (b) 𝜋-model of a transmission line, and (c) equivalent circuit of (𝑌𝐼) in (21).

be described by an impedancematrix but can be described byan admittance matrix in the form

(𝑌𝐹) = (

𝑌11

𝑌12

𝑌21

𝑌22

) = (

1

𝑍

−1

𝑍

−1

𝑍

1

𝑍

) = (

1

𝐿𝑠𝛼

−1

𝐿𝑠𝛼

−1

𝐿𝑠𝛼

1

𝐿𝑠𝛼

) (14)

which is both symmetrical and reciprocal since𝑌11

= 𝑌22and

𝑌12

= 𝑌21. Using (12) we can write

(𝑌𝐼)𝑛

= 2𝑛−1

(

1

(𝐿𝑠𝛼)𝑛

−1

(𝐿𝑠𝛼)𝑛

−1

(𝐿𝑠𝛼)𝑛

1

(𝐿𝑠𝛼)𝑛

). (15)

Choosing 𝑛 = 1/𝛼 we obatin

(𝑌𝐹) = (𝑌

𝐼)𝛼

= (21/𝛼−1

𝐿1/𝛼

(

1

𝑠−1

𝑠

−1

𝑠

1

𝑠

))

𝛼

, (16)

where the elements inside (𝑌𝐼) represent an integer-order

inductor with inductance (𝐿/2𝛼−1)1/𝛼.

Case 2. Consider the transmission line 𝜋-model shown inFigure 3(b). The admittance matrix for this section is

(𝑌𝐹) = (

𝑌 +1

𝑍−1

𝑍

−1

𝑍𝑌 +

1

𝑍

) (17)

which is both symmetrical and reciprocal. Assume that𝑍 is afractional-order inductor (𝑍 = 𝐿𝑠

𝛼) and that𝑌 is a fractional-order capacitor (𝑌 = 𝐶𝑠

𝛽); then

(𝑌𝐼)𝑛

=1

2

⋅ (

(𝐶𝑠𝛽)𝑛

+ (2

𝐿𝑠𝛼+ 𝐶𝑠𝛽)

𝑛

(𝐶𝑠𝛽)𝑛

− (2


𝑛

(𝐶𝑠𝛽)𝑛

− (2


𝑛

(𝐶𝑠𝛽)𝑛

+ (2


𝑛).

(18)

Selecting 𝛼 = 𝑙/𝑛 and 𝛽 = 𝑘/𝑛 such that 𝑙 + 𝑘 = 𝑛 willguarantee the existence of integer-order elements in (𝑌

𝐼)

which is then given by

(𝑌𝐼)𝑛

=1

2(

𝑠𝑘𝐶𝑛+

1

𝐿𝑛𝑠𝑙(2 + 𝐿𝐶𝑠

(𝑙+𝑘)/𝑛)𝑛

𝑠𝑘𝐶𝑛−

1



𝑠𝑘𝐶𝑛−

1



𝑠𝑘𝐶𝑛+

1



). (19)

4 Mathematical Problems in Engineering

I1 I2

I2I1

+ +

− −

V1 V2

ZZ

Y

(a)

2L/C

0.5C2

L2

L2

(b)

Figure 4: (a) 𝑇-model and (b) equivalent circuit of (𝑌𝐼) in (25).

Note that (𝑌𝐼)𝑛 is symmetrical and recoprical where its

elements can be expanded to𝑌11

= 𝑌22

=

(2𝑛−1

)

𝑠𝑙𝐿𝑛

+ 𝑠𝑘𝐶𝑛+

𝑛−1

∑

𝑗=1

C𝑛𝑗

2𝑛−𝑗−1

𝐶𝑗𝑠𝑗(𝑙+𝑘)/𝑛−𝑙

𝐿𝑛−𝑗

,

𝑌12

= 𝑌21

= −

(2𝑛−1

)

𝑠𝑙𝐿𝑛

−

𝑛−1

∑

𝑗=1

C𝑛𝑗

2𝑛−𝑗−1

𝐶𝑗𝑠𝑗(𝑙+𝑘)/𝑛−𝑙

𝐿𝑛−𝑗

,

(20)

where C𝑛𝑗= 𝑛!/𝑗!(𝑛 − 𝑗)!. If the condition 𝑙 + 𝑘 = 𝑛 is not

satisfied, fractional elementswill still exist inside the two-portnetwork. Now consider, for example, the case 𝛼 = 𝛽 = 0.5;then choosing 𝑙 = 𝑘 = 1 and 𝑛 = 2 yields

(𝑌𝐹) = (𝑌

𝐼)1/2

= (

𝐶2𝑠 +

2𝐶

𝐿+

2

𝐿2𝑠

−2𝐶

𝐿−

2

𝐿2𝑠

−2𝐶

𝐿−

2

𝐿2𝑠

𝐶2𝑠 +

2𝐶

𝐿+

2

𝐿2𝑠

)

1/2

.

(21)

The elements inside (𝑌𝐼) can be represented by the equivalent

circuit in Figure 3(c), all of which are integer-order elements.Alternatively, as an example for nonequal fractional-orderelements, let 𝛼 = 1/3 and 𝛽 = 2/3; then 𝑙 = 1, 𝑘 = 2, and𝑛 = 3; the corresponding fractional matrix is given as follows:

(𝑌𝐹) = (𝑌

𝐼)1/3

= (

4

𝐿3𝑠+ 𝐶3𝑠2+3𝐶2

𝐿𝑠 +

6𝐶

𝐿2

−4

𝐿3𝑠−3𝐶2

𝐿𝑠 −

6𝐶

𝐿

−4

𝐿3𝑠−3𝐶2

𝐿𝑠 −

6𝐶

𝐿

4

𝐿3𝑠+ 𝐶3𝑠2+3𝐶2

𝐿𝑠 +

6𝐶

𝐿2

)

1/3

,

(22)

where all elements of (𝑌𝐼) can also be easily realized. It is

worth noting that the restriction 𝛼 + 𝛽 = 1 imposed abovealso guarantees that the equivalent circuit of (𝑌

𝐼) is a𝜋-model

(see Figure 3(c)). However, lifting this restriction is possible.

Case 3. Consider the transmission-line 𝑇-model shown inFigure 4(a) which has the admittance matrix

(𝑌𝐹) = (

𝑍 +1

𝑌−1

𝑌

−1

𝑌𝑍 +

1

𝑌

) (23)

and assume that 𝑍 is a fractional-order inductor (𝑍 = 𝐿𝑠𝛼)

while 𝑌 is a fractional-order capacitor (𝑌 = 𝐶𝑠𝛽); then in this

case

(𝑌𝐼)𝑛

=1

2

⋅ (

(𝐿𝑠𝛼)𝑛

+ (2

𝐶𝑠𝛽+ 𝐿𝑠𝛼)

𝑛

(2


𝑛

− (𝐿𝑠𝛼)𝑛

(2


𝑛

− (𝐿𝑠𝛼)𝑛

(𝐿𝑠𝛼)𝑛

+ (2


𝑛).

(24)

Following a similar procedure to that of the 𝜋-model for thecase 𝛼 = 𝛽 = 0.5 we can show that

(𝑌𝐹) = (𝑌

𝐼)1/2

= (

𝐿2𝑠 +

2𝐿

𝐶+

2

𝐶2𝑠

2𝐿

𝐶+

2

𝐶2𝑠

2𝐿

𝐶+

2

𝐶2𝑠

𝐿2𝑠 +

2𝐿

𝐶+

2

𝐶2𝑠

)

1/2

(25)

and hence (𝑌𝐼) has the equivalent 𝑇-model given in Fig-

ure 4(b).

4. Conclusion

We attempted to introduce the idea of fractional-order two-port networks and its application to impedance and admit-tance parameters of fractional-order elements. The topic isstill in its early stages [11] and much more work needs to bedone both theoretically and experimentally. In particular, weconsidered the following.

(i) We demonstrated here application to the impedanceand admittance matrices; however there are otherimportant two-port network parameters such asthe transmission and scattering matrices for whichsome of the elements inside the matrix are unit-less and obtaining an equivalent circuit requirestransformation from these types of parameters backto impedance or admittance parameters. Thereforematrix transformations (conversions) need to bestudied in the context ofmatrices raised to nonintegerpower and the table of conversions updated.

(ii) Network interconnects (series, parallel and cascadeinterconnects) with matrices raised to a fractional-order power need to be studied. Such interconnects

Mathematical Problems in Engineering 5

require the addition and multiplication of matrices inthe classical integer-order context.

(iii) The restriction imposed in our analysis (𝑙 + 𝑘 = 𝑛)if not possible to satisfy would imply the existence offractional-order matrix parameters in addition to thefractional-order matrix exponent. A solution to thisproblem is required.

(iv) It should be somehow possible to define a nonintegersquarematrix of dimension 𝛼×𝛼, 𝛼 ≤ 1, equivalent toan 𝑛×𝑛 square matrix raised to the non-integer-order𝛼. Such a definition and relationship require furtherinvestigation.

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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[9] G. Tsirimokou, C. Psychalinos, A. Allagui, and A. S. Elwakil,“Simple non-impedance-based measuring technique for super-capacitors,” Electronics Letters, vol. 51, no. 21, pp. 1699–1701,2015.

[10] L. Said, A. Radwan, A. Madian, and A. Soliman, “Fractionalorder two port network oscillator with equal order,” in Pro-ceedings of the 26th International Conference onMicroelectronics(ICM ’14), pp. 156–159, December 2014.

[11] E. Gonzalez, I. Petras, L. Dorcak, and J. Terpak, “On themathematical properties of generalized fractional-order two-port networks using hybrid parameters,” in Proceedings of the

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