Fractance with Tunable Fractor’s Order for Microwave Circuit Applications

Featured Application: The paper intends to illustrate a methodology to design and implement fractance with tunable fractor’s order, suitable for implementing fractional order circuits in a more ﬂexible way. Abstract: There is an increasing number of studies in the literature to implement fractional order components by means of equivalent circuits based on integer order components. Such implementations aim to realize laboratory equipment that can exhibit a fractional behavior in a certain range of frequencies. One of the main limitations of the existing implementation is the ﬁxed fractor’s order. In practice, every time the experimenter wants to change fractor’s order, it is necessary to calculate again the equivalent circuit and implement it again. In order to overcome this limitation, in this paper we propose a new implementation of a fractional order component that enables to tune the fractor’s order. This is achieved by means of variable resistors and a proper control methodology. The methodology can be applied in microwave circuits, for instance for the implementation of ﬁlters: a low-pass one is discussed in this paper.


Introduction
Fractional calculus is a branch of mathematics dealing with the extension of the concept of derivative and integral. The name is due to fact that the order of derivative is not just an integer value but may be fractional or even real. Namely, for a function f (t), in fractional calculus it is defined as the expression: Fractional derivative concept was discussed for the first time about three centuries ago, but only in the last fifty years it has been studied in deep. Formally, the operator a D α t f (t) denotes the fractional derivative of a function f (t) defined in the interval [a, b]. Several representations of the fractional derivative have been proposed in the literature. The most recognized ones are the so-called Caputo derivative [1] and the Caputo-Weyl derivative [2][3][4].
The Caputo definition is more commonly adopted in time domain, while the Caputo-Weyl definition is usually adopted in frequency domain, where the Fourier transform of the fractional derivative operator exhibits the following property: being F(ω) the Fourier transform of f (t) and W −∞ D α t f (t) the Caputo-Weyl derivative of f (t). This is the fractional extension of the derivative property of the Fourier transform.
In electrical engineering is used for the modeling some kind of supercapacitors, transformers and non-linear circuits [10][11][12]. Additionally, it has been theorized the existence of dual port components with fractional order characteristic. This kind of components, usually called Fractances [13][14][15], exhibit an impedance that can be expressed as: Fω α e jαπ/2 (3) where ω is the angular frequency, α a real parameter ranging from -1 to 1, and F is a constant value termed as fractance and its unit is [ s −α ]. Fractances can be considered as a generalization of the traditional components, representing a resistor for α = 0, a traditional inductor for α = −1, and a traditional capacitor for α = 1. However, higher interest towards these components arises for non-integer values of α, where the components produce nontraditional behaviors.
It is worth to note that, according to (3), fractances have a magnitude with a constant slope of −α20 dB/dec and a phase not depending by the frequency and equal to −απ/2. For this reason, the fractances are classified as Constant Phase Components (CPCs).
By a theoretical point of view, fractances find several applications in the design of fractional order filters [16][17][18][19] and in chaotic circuits and systems [20][21][22]. Anyway, one of the most practical limitations in the study of fractances is the difficulty to implement such a component in reality. Several studies in the literature try to implement fractances in different ways. Among the proposed solution, it has been found that the fractor's order is always a pre-defined value of the implementation. This requires changing the implementation any time it is necessary to change the fractor's order. In this paper, we propose a new methodology to implement a fractance with tunable fractor's order, that can be adopted in experimental studies without the need to re-implement the circuit when the fractor's order is changed.
The paper is organized as follow: apart from this Introduction, in Section 2 the state of art of fractances' implementation is discussed. In Section 3, the new methodology for implementing fractances with tunable fractor's order is discussed and applied in a practical example. Finally, the Conclusions are presented.

State of Art of Fractances' Implementation
One of the main problems related to the study and the consequent application of fractances in reality is the absence of physical components exhibiting a fractional behavior. In fact, no ideal circuit is able to implement it so far. Therefore, several research studies nowadays focus on the physical creation of fractances [23]. Although the final challenge is to realize physical components with fractional order behavior, at the current stage this goal seems far from being achieved. Therefore, most of the efforts focus on the realization of equivalent circuits that can behave as fractances in such ranges of frequencies. In the following sections, the state of art of fractances' implementation is briefly discussed.

Equivalent Passive Circuits
One of the main solutions to realize a fractance in experimental set-up is the implementation of a passive equivalent circuit, composed of traditional integer order components, exhibiting an equivalent fractional impedance in a certain frequency range. These approximate representations generally reconstruct the fractional operator by means of a rational function, whose circuital representation can be easily realized [24][25][26][27][28]. Relevant methods adopted in the literature are the ones based on the Continued Fraction Expansions, the Matsuda's method, the Carlson's method, the Oustaloup's method, and the Chareff's method.

Equivalent Active Circuits
An alternative solution to the equivalent passive circuits is the synthesis of fractances by means of networks involving active components. Most of them are based on operational amplifiers [29]. Some implementations also try to use the non-linear behavior of transistors characteristic in order to reproduce a fractional behavior. However, so far these solutions are not as popular as the ones based on passive networks.

Single Componet
In the literature, some studies are focusing on the possibility to construct the first dual port components that exhibit themselves a fractional order behavior, without the need to use equivalent circuits. This research aims to define processes for the production of commercial fractances. If achieved, this result would really boost the experimental research on fractances and the consequent increase of potential applications.
So far, the research studies focus on the use of particular materials that can exhibit a fractional behavior. In particular, implementations have been tested employing ionic gel-Cu electrode based, porous polymers, graphene-polymer, ionic polymer-metal composite, or CNT-polymer composite [30][31][32]. Although these research studies are promising, a commercial solution is not yet available. Therefore, most of the current studies mainly focus on efficient implementations of equivalent circuits.

Fractances with Tunable Fractor's Order
By the analysis of the experimental studies of fractances published in the last year, it emerges that the representations by means of equivalent circuits with passive integer order components are by far the most adopted ones. Ladder networks nested networks and tree networks are some of the most popular implementations.
One of the main limitations of these implementations is the fixed characteristic of the equivalent circuit. In an experimental set-up, especially at a research stage, it would be highly desirable to easily change the characteristic of the fractance. Instead, the equivalent circuits have of course a fixed amplitude and a fixed fractor's order. If the experimenter wants to change fractor's order, he has to calculate again the equivalent circuit, and physically implement it again. In case of a ladder network with even 50 components, this may be considerably frustrating.
Recently, some studies have been proposed in order to realize equivalent circuits of fractances with tunable amplitude [33]. However, these implementations again do not allow to change the fractor's order, that is the main characterizing parameter of the element.
In this paper, we propose for the first time an implementation that allows changing in some ranges the fractor's order, without the need of substituting the implemented network. This is a relevant improvement by the experimental point of view since it allows realizing just one implementation in order to test circuits with different fractor's order. As example of practical implementation, a microwave filter is considered. Anyway, since the innovative contribution of the paper is the tunable fractance, the efficiency of the proposed method is discussed. The performance of the fractional order filter is not addressed since it has been already discussed in the literature and the tunable fractance does not imply an improvement of the performances.

Equivalent Circuit through Rational Approximation
First of all, the methodology for representing a fractance in terms of equivalent integer order circuit is illustrated. Several methodologies have been proposed in the literature, also corresponding to different kind of circuits' topologies [34].
Let us consider the expression of the fractance Z(ω) = 1/F 1/(jω) α and let us try to find a rational approximation G(ω) as: where A, a i , and b i are all positive numbers. Such an expression consists of n first-order damp elements and n first-derivative elements. Each couple of damp and derivative elements, separately considered, represent a piecewise linear function. So, if we consider the operator Z(ω), we can define a tolerance threshold 2δ and require that the piecewise linear is included in the upper and lower boundaries, as shown in Figure 1.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 9 where A, ai, and bi are all positive numbers. Such an expression consists of n first-order damp elements and n first-derivative elements. Each couple of damp and derivative elements, separately considered, represent a piecewise linear function. So, if we consider the operator Z(ω), we can define a tolerance threshold 2δ and require that the piecewise linear is included in the upper and lower boundaries, as shown in Figure 1. The approximation of the fractional characteristic will be acceptable in a certain bandwidth, depending by the accepted tolerance δ and by the order n of the rational function G(ω). It is worth to define as ω0 the central frequency of the approximation and as Z0 the magnitude of the impedance for ω = ω0. Taking n as odd value and starting from the central point , , it is possible to graphically calculate the coefficients ai and bi as: = 10 / (6) = 10 / (7) Once the parameters of (4) have been identified, it is possible to expand the rational approximation as: where the parameters A0 … An can be calculated as: The single first order rational functions composing (9) correspond, in terms of circuits, to a series of R-C parallels where: The approximation of the fractional characteristic will be acceptable in a certain bandwidth, depending by the accepted tolerance δ and by the order n of the rational function G(ω). It is worth to define as ω 0 the central frequency of the approximation and as Z 0 the magnitude of the impedance for ω = ω 0 . Taking n as odd value and starting from the central point (ω 0 , Z 0 ), it is possible to graphically calculate the coefficients a i and b i as: A = Z 0 (1/a 1 ) α 10 δ/20 (8) Once the parameters of (4) have been identified, it is possible to expand the rational approximation as: where the parameters A 0 . . . A n can be calculated as: The single first order rational functions composing (9) correspond, in terms of circuits, to a series of R-C parallels where: In Figure 2, the obtained equivalent circuit is represented, corresponding to a low-pass filter with constant phase.
In Figure 2, the obtained equivalent circuit is represented, corresponding to a lowpass filter with constant phase. In order to make a numerical example, let us try to represent a fractance with fractor's order α = 0.5 and for the representation we choose n = 7, f0 = 100 Hz, Z0 = 50 Ω, and δ = 2 dB. The parameters for the equivalent circuit are obtained as show in Table 1. In Figures 3 and 4, the amplitude and the angle of impedance are represented. It is worth nothing that while the amplitude well approximates the fractional impedance in about 5 decades, the phase exhibits a faster variation. In order to make a numerical example, let us try to represent a fractance with fractor's order α = 0.5 and for the representation we choose n = 7, f 0 = 100 Hz, Z 0 = 50 Ω, and δ = 2 dB. The parameters for the equivalent circuit are obtained as show in Table 1. In Figures 3 and 4, the amplitude and the angle of impedance are represented. It is worth nothing that while the amplitude well approximates the fractional impedance in about 5 decades, the phase exhibits a faster variation.
In Figure 2, the obtained equivalent circuit is represented, corresponding to a lowpass filter with constant phase. In order to make a numerical example, let us try to represent a fractance with fractor's order α = 0.5 and for the representation we choose n = 7, f0 = 100 Hz, Z0 = 50 Ω, and δ = 2 dB. The parameters for the equivalent circuit are obtained as show in Table 1. In Figures 3 and 4, the amplitude and the angle of impedance are represented. It is worth nothing that while the amplitude well approximates the fractional impedance in about 5 decades, the phase exhibits a faster variation.

Tuning of the Fractor's Order
In order to calculate a good approximation of the fractional impedance, every time the fractor's order α changes, the whole representation should be re-calculated. However, representing the resistances in a logarithmic scale, all the values are on a line that rotates Phase Figure 4. Phase of the impedance Z(ω) and the rational approximation function G(ω).

Tuning of the Fractor's Order
In order to calculate a good approximation of the fractional impedance, every time the fractor's order α changes, the whole representation should be re-calculated. However, representing the resistances in a logarithmic scale, all the values are on a line that rotates clockwise to the variation of α around the (n + 3)/2 value. For the values of n, ω 0 , Z 0 , and δ the variation is show in Figure 5. So, once a set of resistances is found for a given value of α 0 , is possible to suppose to find a rule for the variation of R i as function of α.

Tuning of the Fractor's Order
In order to calculate a good approximation of the fractional impedance, every time the fractor's order α changes, the whole representation should be re-calculated. However, representing the resistances in a logarithmic scale, all the values are on a line that rotates clockwise to the variation of α around the + 3 /2 value. For the values of n, ω0, Z0, and δ the variation is show in Figure 5. So, once a set of resistances is found for a given value of α0, is possible to suppose to find a rule for the variation of Ri as function of α. The family of straight lines can be expressed as: Once the equivalent network is calculated for a reference fractor's order α0, it is possible to easily calculate . The, it is found that applying the empirical law: Additionally, practically it found that the resistances changes with α much more than the capacitances. So, it is worth to consider a network where the resistances vary with α according to (14) and (15) and capacitors are fixed. Such a network is a very good approx- The family of straight lines can be expressed as: Once the equivalent network is calculated for a reference fractor's order α 0 , it is possible to easily calculate m(α 0 ). The, it is found that applying the empirical law: Additionally, practically it found that the resistances changes with α much more than the capacitances. So, it is worth to consider a network where the resistances vary with α according to (14) and (15) and capacitors are fixed. Such a network is a very good approximation of the fractional impedance for the value of α 0 adopted to compute the capacitances. However, it will be able to approximate fractional impedances for different values of α, in a bandwidth always as smaller as the fractor's order differs from the starting value of α 0 .
Such a kind of circuit can be easily implemented by means of the same configuration in Figure 2 and adopting traditional capacitors and variable resistors. For instance, by using digital potentiometers connected to a controller, it is possible to program the synchronous variation of all the potentiometers according to the programmed rules, in order to obtain the wished set of resistances and so in the end the wished fractional impedance.
In order to provide a numerical example, we can consider once again the previously values of n, ω 0 , Z 0 , and δ. If we consider as reference fractor's order α 0 = 0.5, then we obtain m(α 0 ) = −0.402. This allows to compute the coefficients m(α) and then new values of R i for different values of α.
In Figures 6 and 7, we show the amplitude and the phase of the rational approximation G(ω) for different values of α computed with this procedure. The reference amplitudes and phases of Z(ω) are not shown to make the pictures more readable. Anyway, the amplitude comparison would have been neglectable, as for the special case of Figure 3, while the phase of G(ω) has to be compared with a constant value of course.
In order to provide a numerical example, we can consider once again the previously values of n, ω0, Z0, and δ. If we consider as reference fractor's order = 0.5, then we obtain = −0.402. This allows to compute the coefficients and then new values of for different values of α. In Figures 6 and 7, we show the amplitude and the phase of the rational approximation G(ω) for different values of α computed with this procedure. The reference amplitudes and phases of Z(ω) are not shown to make the pictures more readable. Anyway, the amplitude comparison would have been neglectable, as for the special case of Figure 3, while the phase of G(ω) has to be compared with a constant value of course.  Again, the amplitude plot represents the fractional impedance in a wider bandwidth than the phase plot. In order to provide a numerical example, we can consider once again the previously values of n, ω0, Z0, and δ. If we consider as reference fractor's order = 0.5, then we obtain = −0.402. This allows to compute the coefficients and then new values of for different values of α. In Figures 6 and 7, we show the amplitude and the phase of the rational approximation G(ω) for different values of α computed with this procedure. The reference amplitudes and phases of Z(ω) are not shown to make the pictures more readable. Anyway, the amplitude comparison would have been neglectable, as for the special case of Figure 3, while the phase of G(ω) has to be compared with a constant value of course.  Again, the amplitude plot represents the fractional impedance in a wider bandwidth than the phase plot. Again, the amplitude plot represents the fractional impedance in a wider bandwidth than the phase plot.
Moreover, as expected, the quality of the approximation, especially in the phase plot, decreases for higher values of α.

Conclusions
In this paper, a methodology for designing a fractional order component with tunable fractor's order has been proposed. It represents a step forward in the research of equivalent circuits for the practical realization of fractances, since the proposed method allows to realize a unique implementation able to cover a range of fractor's orders, not requiring any more a new circuit implementation any time the fractor's order has to be changed. The procedure has been presented in detail and applied to the case of a low-pass microwave filter. Being the procedure generic, it can be easily applied in order to physically implement the circuit on a small board. With a simple control of the digital potentiometers, it is possible to change fractor's order in a certain range, depending by the number of elements in the ladder and by the accepted tolerance.
A more detailed investigation on the control methodology may allow extending the range of tuning. For this reason, we expect to introduce in the near future an optimization algorithm instead of the empirical procedure in order to improve the method's performance. In addition, comparison between the designed model and the real implementation will allow to better understand the real effectiveness of the design process.