Comparative Study of Discrete Component Realizations of Fractional-Order Capacitor and Inductor Active Emulators

Due to the absence of commercially available fractional-order capacitors and inductors, their implementation can be performed using fractional-order di®erentiators and integrators, respectively, combined with a voltage-to-current conversion stage. The transfer function of fractional-order di®erentiators and integrators can be approximated through the utilization of appropriate integer-order transfer functions. In order to achieve that, the Continued Fraction Expansion as well as the Oustaloup's approximations can be utilized. The accuracy, in terms of magnitude and phase response, of transfer functions of di®erentiators/integrators derived through the employment of the aforementioned approximations, is very important factor for achieving high performance approximation of the fractional-order elements. A comparative study of the accuracy o®ered by the Continued Fraction Expansion and the Oustaloup's approximation is performed in this paper. As a next step, the corresponding implementations of the emulators of the fractional-order elements, derived using fundamental active cells such as operational ampli ̄ers, operational transconductance ampli ̄ers, current conveyors, and current feedback operational ampli ̄ers realized in commercially available discrete-component IC form,

The impedance of fractional-order capacitor, is described by the expression in (1), where 0 < < 1 is the order of the element and C is the pseudo-capacitance in units of Farad/s ð1ÀÞ . 16 The value of the conventional capacitance (C) in Farad equivalent in its impedance at speci¯c frequency (!) to the fractional-order capacitor can be obtained by the formula in (2).
The impedance of a fractional-order inductor is described by the expression in (3), where 0 < < 1 is the order of the inductor and L is the pseudo-inductance in Henry/s ð1ÀÞ .
The value of the conventional inductance (L) in Henry equivalent in its impedance at speci¯c frequency (!) to the fractional-order inductor can be obtained by the formula in (4).
The straightforward procedure for implementing these types of circuits is the substitution of the integer-order elements by fractional-order ones.Unfortunately, these elements are not yet commercially available [17][18][19] and, therefore, they have to be approximated by appropriate integer-order networks.A possible solution for approximating fractional-order capacitors is the employment of an RC network, [20][21][22] but this su®ers from the requirement for re-designing the network in order to change the characteristics of the fractional-order element.The behavior of fractional-order inductors is approximated through the combination of a fractional-order capacitor emulator and a Generalized Impedance Converter (GIC). 23,6Another solution, offering more design °exibility than that o®ered by the previous one, is the employment of fractional-order integrator/di®erentiation stage and, also, an appropriate voltage-to-current (V =I) converter. 24,25The fractional-order integrator/di®erentiator is approximated by an appropriate integer-order transfer function and the achieved accuracy depends on the order of approximation which re°ects into the circuit complexity required for implementing the corresponding transfer function.Therefore, the aim of this paper is to provide a useful comparative performance study o®ered by various orders of approximations of these stages.Hence, the performance of the 2nd to 5th-order approximations of the Continued Fraction Expansion (CFE) 26 are evaluated and compared with those o®ered by the 3rd and 5th-order Oustaloup's approximation. 27The paper is organized as follows: in Sec. 2, the concept for emulating fractional-order capacitors and inductors is brie°y presented and the expressions of approximations of the integro-di®erential operator in the Laplace domain are presented and evaluated in terms of magnitude and phase response.The implementation of the most suitable approximation is presented in Sec. 3, using discrete-component IC forms of operational ampli¯ers (op-amps), operational transconductance ampli¯ers (OTAs), second-generation current conveyors (CCIIs), and current feedback operational ampli¯ers (CFOAs).The comparison results, obtained using the OrCAD PSpice software, are presented in Sec. 4.

Functional block diagram (FBD)
The emulation of a fractional-order capacitor/inductor is performed according to the concept demonstrated in Fig. 1.In the case that r ¼ , a fractional-order capacitor is emulated with equivalent impedance, de¯ned as Z eq =i, given by Eq. ( 5).

Comparative Study of Discrete Component Realizations of CPE/FI Active Emulators
A fractional-order inductor is emulated in the case that r ¼ À, with equivalent impedance given by Eq. ( 6) Therefore, a fractional-order di®erentiator is utilized in the case of fractional-order capacitor emulation, with unity-gain frequency ð! 0 Þ given by the formula: !0 ¼ 1=.
In the case of fractional-order inductor, a fractional-order integrator with the same unity-gain frequency is employed.Inspecting Eqs. ( 5)-( 6), it is derived that the transconductance g m;VI of the V =I converter determines the impedance of the emulated element at the center frequency !0 , i.e., Zð! 0 Þ ¼ 1=g m;VI .Using Eqs. ( 1) and ( 3), the pseudo-capacitance and the pseudo-inductance design equations are given by Eq. ( 7) De-normalizing at the unity-gain frequency !0 , the capacitance and the inductance take the forms of ( 8) Selecting !¼ !0 , in (8) yields the values of the de-normalized capacitance and inductance as

Approximation of the Laplace operator
From the discussion in Sec.2.1, it is obvious that the level of accuracy of the fractional integro-di®erential operator in the Laplace domain (s r ) is very critical for the emulation of the corresponding fractional-order element.The nth-order approximation of the operator around a center frequency !0 ¼ 1 rad/s, is expressed by a rational function de¯ned by the quotient of two polynomials of the variable s: where in Eq. (10) a i (i ¼ 0 . . .n) are positive real coe±cients.In the case of r ¼ with 0 < < 1 the expression represents a fractional-order di®erentiator with unitygain frequency !0 ¼ 1 rad/s, while for r ¼ À, with 0 < < 1, a fractional-order integrator (with the same unity-gain frequency).The values of the coe±cients a i , for n ¼ 2 . . .5, are summarized in Tables 1 and 2. 26 Employing the Oustaloup's approximation method, the corresponding expression, for geometrically distributed frequencies over the band [! b ; !h ], is the following: Table 2. Coe±cients values of the CFE approximation (for n ¼ 5).

Comparative Study of Discrete Component Realizations of CPE/FI Active Emulators
Here, the variables != k , ! k , and C in Eq. ( 11) are de¯ned by Eq. ( 12): Owing to the geometrical distribution of frequencies, the unity-gain frequency (! 0 ) is calculated according to the formula: . It must be also mentioned that the order of the transfer function is n ¼ 2N þ 1 and, therefore, only odd-order approximations are possible through the Oustaloup's method.
Using the FOMCON Toolbox of MATLAB, 28 the obtained magnitude and phase responses in frequency domain of the 2nd and 5th-order approximations of the variable s 0:5 using CFE method are provided in the plots in Fig. 2.

Comparative Study of Discrete Component Realizations of CPE/FI Active Emulators
summarize the performance comparison, the corresponding error plots for CFE approximation are given in Fig. 4. The corresponding plots for the Oustaloup's approximation are provided in Fig. 5.In addition, the performance of the approximations in terms of accuracy is summarized in Table 3.
According to the provided results, the conclusions are the following: (i) Comparing the accuracy of the 3rd-order CFE and Oustaloup's approximations, it is obvious that CFE is more e±cient.In addition, CFE approximation o®ers also 2nd and 4th-order approximations.This is very important, because using the CFE relatively simple circuits are possible to design approximating fractional-order transfer function. 3(ii) In the case of 5th-order of approximation, the Oustaloup's method o®ers more accuracy in magnitude than that o®ered by the CFE method.On the other hand, the CFE is more e±cient in terms of phase accuracy.

Realizations of Fractional-Order Capacitor and Inductor Emulators
As Eqs. ( 10)-( 11) have the same general form, they can be implemented by the same topology.Thus, the Follow-the-Leader Feedback (FLF) and Inverse-Follow-the-Leader-Feedback (IFLF) structures, 29 represented by the FBDs in Figs.6(a) and 6(b), respectively, can be utilized for this purpose.The transfer function realized by both schemes is given by Eq. ( 13) The calculation of the time-constants i (i ¼ 1 . . .n) and gain-factors G j (j ¼ 0 . . .n) is performed by equating the corresponding coe±cients of the polynomials in Eq. ( 10) or (11) with those in Eq. ( 13).It should be mentioned at this point that both integrator and di®erentiator can be implemented by the same structure, just by appropriately selecting the values of time-constants and gain factors.Consequently, both types of fractional-order elements can be emulated and this is very attractive from the design °exibility point of view.
The implementation of the FBD in Fig. 1, using op-amps as active elements for realizing the lossless integration and summation stages, 29 in the case of a 5th-order approximation, is depicted in Fig. 7.Note that an input bu®er is employed in order to implement the required high input impedance of the integration/di®erentiation stage.In addition, the employed OTA, with transconductance g m;VI , performs the required V =I conversion.The realized time-constants and gain-factors are given by the expressions in Eqs. ( 14) and (15), respectively.
The corresponding implementation using CCIIs as active elements 30 is demonstrated in Fig. 8, where the V =I conversion is performed by a CCII-conveyor with a resistor R VI ¼ 1=g m;VI connected at its low-impedance input (X) terminal.The expressions of time-constants and gain-factors are still given by Eqs. ( 14) and ( 15), respectively.Using CFOAs as active elements, 31 the resulted implementation is shown in Fig. 9.The V =I conversion is performed through a CFOA with a resistor R VI ¼ 1=g m;VI associated with its X terminal.The design equations are the same as in previous cases.
Inspecting Fig. 6, it is concluded that the IFLF form is suitable only for elements with inherent di®erential operation, such as OTAs. 32Thus, the OTA-C implementation of the FBD in Fig. 1 is demonstrated in Fig. 10.The realized time-constants, as well as the gain-factors, are expressed by Eqs. ( 16) and (17), respectively.
Considering that the active elements are available as commercially discretecomponent ICs, Table 4 compares the circuit complexity of the corresponding realizations of the FBDs in Fig. 1, in the case of a 5th-order approximation.According to the provided results, the best option in terms of active component count is the FLF structure implemented using CFOAs as active elements.

Simulation and Comparison Results
In order to verify the results obtained in Sec.2.2, fractional-order capacitors with capacitance C ¼ 6:3 F at center frequency f 0 ¼ 100 Hz and orders ¼ f0:3; 0:5; 0:7g will be emulated using the topology in Fig. 9. Using Eq. ( 2), the corresponding values of the pseudo-capacitance were: 0:57 F/s 0:7 , 0:16 F/s 0:5 , and 0:043 F/s 0:3 , respectively.According to Eq. ( 9), the transconductance of the V =I converter must be: g m;VI ¼ 10 S or, equivalently, the resistance of R VI will be equal to 100 k.This will For controlling the transconductance of OTAs.

Comparative Study of Discrete Component Realizations of CPE/FI Active Emulators
be also the impedance of the element at 100 Hz.Therefore, fractional-order di®erentiators with unity-gain frequency f 0 ¼ 100 Hz and orders ¼ f0:3; 0:5; 0:7g will be realized.
As a second design example, fractional-order inductors with inductance L ¼ 159:15 H at center frequency f 0 ¼ 100 Hz and orders ¼ f0:3; 0:5; 0:7g will be emulated using the topology in Fig. 9. Using Eq. ( 4), the derived values of the pseudo-inductances are: 14.4 kH/s 0:7 , 3.99 kH/s 0:5 , and 1.1 kH/s 0:3 , respectively.Owing to the fact that the impedance at unity-gain frequency will be equal to 100 k, then g m;VI ¼ 10 S and R VI ¼ 1=g m;VI ¼ 100 k.Fractional-order integrators with unity-gain frequency f 0 ¼ 100 Hz and orders ¼ f0:3; 0:5; 0:7g will be employed for emulating the fractional-order inductors.Following a similar procedure as in the case of the fractional-order capacitor emulator, the derived values of the passive elements of the topology in Fig. 9 are provided in Tables 7 and 8, respectively.Using the AD844 33 discrete-component ICs as CFOAs and the OrCAD PSpice simulator, the obtained magnitude and phase responses in the cases of the emulation of fractional-order capacitors and inductors using the CFE approximation, along with the plots that correspond to the ideal (i.e.without approximation) cases, are given in Fig. 11.The corresponding plots, derived using the Oustaloup's approximation are provided in Fig. 12.The accuracy evaluation of the frequency responses from Figs. 11 and 12 is summarized in Tables 9 and 10, respectively, where it is  veri¯ed the superiority of the CFE approximation in terms of phase accuracy and of the Oustaloup's method in terms of magnitude accuracy.The time-domain behavior of the realized emulators has been evaluated through the stimulation of the circuit in Fig. 9, with a sinusoidal voltage with amplitude 100 mV and frequency 100 Hz.The derived current waveforms, obtained using the  CFE approximation for ¼ ¼ 0:5, are demonstrated in Fig. 13 and con¯rm the correct operation of the circuit.The sensitivity performance of the implemented emulators has been evaluated through the utilization of the Monte-Carlo method o®ered by the Advanced Analysis Tool of the PSpice and assuming 5% random variations of the passive elements.The  respectively.In addition, the sensitivity performance characteristics of the corresponding implementations performed using op-amps (Fig. 7), CCIIs (Fig. 8), and OTAs (Fig. 10) as active elements are summarized in Table 11.According to the provided results, it is derived that the implementation with CFOAs has increased sensitivity to the components values variations and this is the price paid for the achieved reduction of the active component count.

Conclusions
The performance of typical approximations for implementing fractional-order differentiators and integrators has been evaluated in this paper.Thus, the important conclusion is that the CFE approximation is more e±cient in terms of phase accuracy, while the Oustaloup's approximation o®ers superiority in the magnitude approximation.With respect to the discrete-component implementation using commercially available active elements, the employment of CFOAs as active elements o®ers the bene¯ts of reduced active component count.The veri¯cation of these derivations has been performed simulation results.

Fig. 2 .
Fig. 2. Frequency responses of the variable s 0:5 using the CFE approximation (a) magnitude and (b) phase.

Fig. 11 .
Fig.11.Fractional-order capacitor and inductor impedance frequency responses, obtained from the circuit in Fig.9using the CFE approximation, (a) magnitude and (b) phase.

Table 3 .
Performance comparison results for CFE and Oustaloup's approximations.Comparative Study of Discrete Component Realizations of CPE/FI Active Emulators

Table 4 .
Performance comparison results for implementations of FBDs in Fig. 1.Notes: Table notes.a Two CCII+ ICs are required for implementing a CCII−. 31b

Table 5 .
Values of passive elements for emulating fractional-order capacitors using CFE approximation.

Table 6 .
Values of passive elements for emulating fractional-order capacitors using the Oustaloup's approximation.

Table 7 .
Values of passive elements for emulating fractional-order inductors using the CFE approximation.Comparative Study of Discrete Component Realizations of CPE/FI Active Emulators

Table 8 .
Values of passive elements for emulating fractional-order inductors using the Oustaloup's approximation.

Table 9 .
Accuracy performance of the CFE approximation for emulating fractional-order capacitors and inductors.