Fractional-Order Two-Port Networks.

1. Introduction

Two-port networks are widely used in linear circuit analysis and design [1, 2]. The system under consideration is represented by a describing matrix which relates its input and output variables (voltages and currents). Such a representation enables the treatment of the system as a black box where the internal details become irrelevant. It also offers an extremely efficient computational technique which can be used to model series, parallel, or cascade interconnects of several systems. Standard Network Analyzers can be configured to measure several types of two-port network parameters including impedance, admittance, transmission, and scattering parameters.

Consider, for example, the impedance matrix representation of a system in which case we have

[mathematical expression not reproducible], (1)

where [V.sub.1,2] ([I.sub.1,2]) are the voltages (currents) at the input port and output port, respectively, as shown in Figure 1. All elements in the 2 x 2 impedance matrix are measured in [OMEGA] and if [Z.sub.11] = [Z.sub.22] the network is known to be symmetrical while if [Z.sub.12] = [Z.sub.21] it is known to be reciprocal. However, with the increasing use of fractional-order impedance models, particularly in representing supercapacitors [3, 4], energy storage devices , oscillators , filters , and new electromagnetic charts , it is possible that the elements of (Z) are of fractional order. Consider the simple case of the grounded impedance Z, shown in Figure 2(a). Treated as a two-port network, this impedance is described by the impedance matrix

[mathematical expression not reproducible]. (2)

Let Z be a supercapacitor operating in its Warburg mode, where Z = 1/Q[square root of s]; s = j[omega]. In this region of operation, the magnitude is proportional to 1/[square root of [omega]], the phase angle is fixed at -45[degrees], and Q is the pseudocapacitance of the device [9, 10]. As a two-port network, this device would be described as

[mathematical expression not reproducible]. (3)

Therefore all elements of the ([Z.sub.F]) matrix are of fractional order. However, we can rewrite the above equation in the alternative form

[mathematical expression not reproducible] (4)

since

[mathematical expression not reproducible]. (5)

It is clear that the elements [Z.sub.11] [right arrow] [Z.sub.22] of ([Z.sub.I]) are all integer-order impedances, each representing a capacitor of (1/2) Farad whereas the power of the matrix is fractional; that is, [square root of ([Z.sub.I])] = ([Z.sub.F]). In this paper we seek to generalize this procedure by obtaining the equivalent matrix ([Z.sub.I]) and its fractional exponent such that [([Z.sub.I]).sup.n] = ([Z.sub.F]). The procedure is not restricted to the impedance matrix and can be applied to any other type of two-port network parameters. The main advantage of this conversion is that an equivalent circuit of ([Z.sub.I]) can be easily obtained with integer-order components. For example, if ([Z.sub.I]) is reciprocal, then its equivalent circuit is that shown in Figure 2(b). However, it is not yet known how to use this equivalent circuit in association with the fractional exponent of the matrix to construct an overall equivalent model of the originally fractional-order two-port network.

2. Power of a Matrix

A matrix [M.sup.n] for a nonnegative exponent n is defined as the matrix product of n copies of M. However, if n is a noninteger, then we need to revert to the Cayley-Hamilton theorem. In particular, if M is a 2 x 2 matrix and I is an identify matrix, then

[M.sup.n] = [[alpha].sub.0]I + [[alpha].sub.1]M, (6)

where

[mathematical expression not reproducible] (7)

and [[lambda].sub.1,2] are nonrepeated eigenvalues. Hence, for a given impedance matrix [Z.sub.I], we have

[mathematical expression not reproducible], (8)

where [mathematical expression not reproducible]. For the case of repeated eigenvalues ([[lambda].sub.1] = [[lambda].sub.2] = [lambda]) we obtain

[mathematical expression not reproducible]. (9)

If the two-port networkis symmetrical, that is, [Z.sub.11] = [Z.sub.22], then for nonrepeated eigenvalues

[mathematical expression not reproducible], (10)

where [mathematical expression not reproducible] and [mathematical expression not reproducible] while for repeated eigenvalues we obtain

[mathematical expression not reproducible]. (11)

If, in addition to being symmetrical, the network is also reciprocal (i.e., [Z.sub.12] = [Z.sub.21]), we then obtain

[mathematical expression not reproducible], (12)

where x = [([Z.sub.11] + [Z.sub.12]).sup.n] + [([Z.sub.11] - [Z.sub.12]).sup.n] and y = [([Z.sub.11] + [Z.sub.12]).sup.n] - [([Z.sub.11] - [Z.sub.12]).sup.n]. Noting the semigeneral case of a matrix M given by [mathematical expression not reproducible], then it can be easily shown that

[mathematical expression not reproducible]. (13)

In Section 3 a number of circuit applications are considered.

3. Applications

Case 1. Consider the case of floating impedance, as shown in Figure 3(a), and assume that this impedance represents a fractional-order inductor with impedance Z = L[s.sup.[alpha]]; s = j[omega] and L is the pseudoinductance. This single impedance cannot be described by an impedance matrix but can be described by an admittance matrix in the form

[mathematical expression not reproducible] (14)

which is both symmetrical and reciprocal since [Y.sub.11] = [Y.sub.22] and [Y.sub.12] = [Y.sub.21]. Using (12) we can write

[mathematical expression not reproducible]. (15)

Choosing n = 1/[alpha] we obatin

[mathematical expression not reproducible], (16)

where the elements inside ([Y.sub.I]) represent an integer-order inductor with inductance [(L/[2.sup.[alpha]-1]).sup.1/[alpha]].

Case 2. Consider the transmission line [pi]-model shown in Figure 3(b). The admittance matrix for this section is

[mathematical expression not reproducible] (17)

which is both symmetrical and reciprocal. Assume that Z is a fractional-order inductor (Z = L[s.sup.[alpha]]) and that Y is a fractional-order capacitor (Y = C[s.sup.[beta]]); then

[mathematical expression not reproducible]. (18)

Selecting [alpha] = l/n and [beta] = k/n such that l + k = n will guarantee the existence of integer-order elements in ([Y.sub.I]) which is then given by

[mathematical expression not reproducible]. (19)

Note that [([Y.sub.I]).sup.n] is symmetrical and recoprical where its elements can be expanded to

[mathematical expression not reproducible], (20)

where [C.sup.n.sub.j] = n!/j!(n - j)!. If the condition l + k = n is not satisfied, fractional elements will still exist inside the two-port network. Now consider, for example, the case [alpha] = [beta] = 0.5; then choosing l = k = 1 and n = 2 yields

[mathematical expression not reproducible]. (21)

The elements inside ([Y.sub.I]) 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-order elements, let [alpha] = 1/3 and [beta] = 2/3; then l = 1, k = 2, and n = 3; the corresponding fractional matrix is given as follows:

[mathematical expression not reproducible], (22)

where all elements of ([Y.sub.I]) can also be easily realized. It is worth noting that the restriction [alpha] + [beta] = 1 imposed above also guarantees that the equivalent circuit of ([Y.sub.I]) is a [pi]-model (see Figure 3(c)). However, lifting this restriction is possible.

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

[mathematical expression not reproducible] (23)

and assume that Z is a fractional-order inductor (Z = L[s.sup.[alpha]]) while Y is a fractional-order capacitor (Y = C[s.sup.[beta]]); then in this case

[mathematical expression not reproducible]. (24)

Following a similar procedure to that of the [pi]-model for the case [alpha] = [beta] = 0.5 we can show that

[mathematical expression not reproducible] (25)

and hence ([Y.sub.I]) has the equivalent T-model given in Figure 4(b).

4. Conclusion

We attempted to introduce the idea of fractional-order two-port networks and its application to impedance and admittance parameters of fractional-order elements. The topic is still in its early stages  and much more work needs to be done both theoretically and experimentally. In particular, we considered the following.

(i) We demonstrated here application to the impedance and admittance matrices; however there are other important two-port network parameters such as the transmission and scattering matrices for which some of the elements inside the matrix are unit-less and obtaining an equivalent circuit requires transformation from these types of parameters back to impedance or admittance parameters. Therefore matrix transformations (conversions) need to be studied in the context of matrices raised to noninteger power and the table of conversions updated.

(ii) Network interconnects (series, parallel and cascade interconnects) with matrices raised to a fractional-order power need to be studied. Such interconnects require the addition and multiplication of matrices in the classical integer-order context.

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

(iv) It should be somehow possible to define a noninteger square matrix of dimension [alpha] x [alpha], [alpha] [less than or equal to] 1, equivalent to an n x n square matrix raised to the non-integer-order [alpha]. Such a definition and relationship require further investigation.

http://dx.doi.org/10.1155/2016/5976301

Competing Interests

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

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M. E. Fouda, (1) A. S. Elwakil, (2,3) A. G. Radwan, (1,3) and B. J. Maundy (4)

(1) Engineering Mathematics and Physics Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt

(2) Department of Electrical and Computer Engineering, University of Sharjah, College of Engineering, P.O. Box 27272, UAE

(3) Nanoelectronics Integrated Systems Center (NISC), Nile University, Giza 12588, Egypt

(4) Department of Electrical and Computer Engineering (ECE), University of Calgary, AB, Canada T2N 1N4

Received 25 March 2016; Accepted 17 April 2016

Caption: Figure 1: Two-port network variables.

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

Caption: Figure 3: (a) Single floating impedance as a two-port network, (b) [pi]-model of a transmission line, and (c) equivalent circuit of ([Y.sub.I]) in (21).

Caption: Figure 4: (a) T-model and (b) equivalent circuit of ([Y.sub.I]) in (25).