Fractional-order mutual inductance: analysis and design

Transcription

1 INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS Int. J. Circ. Theor. Appl. (2015) Published online in Wiley Online Library (wileyonlinelibrary.com) Fractional-order mutual inductance: analysis and design Ahmed Soltan 1, Ahmed G. Radwan 2,3, *, and Ahmed M. Soliman 4 1 School of Electrical and Electronic Engineering, Newcastle University, UK 2 Department of Engineering Mathematics and Physics, Cairo University, Cairo, Egypt 3 Nanoelectronics Integrated Systems Center (NISC), Nile University, Cairo, Egypt 4 Department of Electronics and Communications Engineering, Cairo University, Cairo, Egypt SUMMARY This paper introduces for the first time the generalized concept of the mutual inductance in the fractional-order domain where the symmetrical and unsymmetrical behaviors of the fractional-order mutual inductance are studied. To use the fractional mutual inductance in circuit design and simulation, an equivalent circuit is presented with its different conditions of operation. Also, simulations for the impedance matrix parameters of the fractional mutual inductance equivalent circuit using Advanced Design System and MATLAB are illustrated. The Advanced Design System and MATLAB simulations of the double-tuned filter based on the fractional mutual inductance are discussed. A great matching between the numerical analysis and the circuit simulation appears, which confirms the reliability of the concept of the fractional mutual inductance. Also, the analysis of the impedance matching using the fractional-order mutual inductance is introduced. Copyright 2015 John Wiley & Sons, Ltd. Received 13 July 2014; Revised 9 October 2014; Accepted 6 January 2015 KEY WORDS: mutual inductance; fractional elements; double-tuned filter; equivalent circuit 1. INTRODUCTION In recent years, fractional calculus has been widely used in modeling the dynamics of many real life phenomena because of the fact that it has higher capability of providing accurate description than integer dynamical systems. This added flexibility is mainly due to the fact that fractional-order systems can be characterized by infinite memory, whereas integer-order systems are characterized by finite memory [1]. Moreover, because of the extra fractional-order parameters, more flexibility is added in the modeling, analysis, and control of many applications such as determining voltage current relationship in a non-ideal capacitor [2, 3], fractal behavior of a metal insulator solution interface [4], electromagnetic waves [5], and recently in electrical circuits such as filters [6 11] and oscillators [12 14]. Furthermore, applications of fractional calculus have been reported in many areas such as physics [15], nonlinear oscillation of earthquakes [16], and mathematical biology [17]. The Caputo definition of the fractional derivative of order α is written as follows [18]: 8 1 f ðmþ ðþ τ >< ad α t fðþ:¼ t Γðm αþ t dτ m 1 < α < m 0 ð ðt τþ αþ1 m Þ >: d m dt m fðþ t α ¼ m (1) *Correspondence to: Ahmed G. Radwan, Department of Engineering Mathematics and Physics, Cairo University, Cairo, Egypt. agradwan@ieee.org Copyright 2015 John Wiley & Sons, Ltd.

2 A. SOLTAN, A. G. RADWAN AND A. M. SOLIMAN where a and t are the initial and the required time of calculation. Applying the Laplace transform to the general fractional derivative of (1) with zero initial conditions yields: L 0 D α t fðþ t ¼ s α Fs ðþ (2) Therefore, it becomes possible to define a general fractance device with impedance proportional to s α [19], where the traditional circuit elements capacitor, resistor, and inductor are special cases of this fractional-order element when the order is 1, 0, and 1, respectively. During the last 10 years, several promising trials have been introduced for the realizations of the fractional element and based on different techniques such as chemical reactions [20], fractal shapes [4], and graphene material [21]. Moreover, many finite circuit approximations were suggested to model fractional-order elements, for example, a finite element approximation of the special case Z = 1/(Cs 0.5 ) was reported in [22]. This finite element approximation relies on the possibility of emulating a fractional-order capacitor via semi-infinite resistor capacitor trees. The technique was later developed by the authors of [23 25] for any order. On the other hand, the conventional mutually coupled circuits (MCCs) have a wide range of applications in instrumentation, communications, control systems, signal processing, and modeling [26 28]. They can also be used for analog filters, particularly for replacing the magnetic transformer in stagger-tuned filters [29, 30]. Using the MCC in the previous blocks has improved the design flexibility and the system performance. The MCC is characterized by primary inductance, secondary inductance, mutual inductance, and the coupling factor. The MCC is characterized by the following impedance matrix [31]: V 1 V 2 ¼ s L 11± M M M L 22 ± M I1 I 2 (3) where L 11, M, and L 22 are the primary inductance, mutual inductance, and the secondary inductance, respectively. From the MCC T-model illustrated in Figure 1, the MCC depends on using the inductors to fulfill the mutual inductance. Although, there are very good inductors or capacitors, they are considered a fractional-order behavior with fractional order α [32, 33]. In addition, this fractional-order value could be less in Radio Frequency (RF) applications [34]. Physical prototypes of the fractional-order capacitors and inductors are presented in [3,35]. Thus, it becomes necessary to propose the analysis of the MCC based on the fractional-order model of the inductors. In this case, a new concept arises, which can be referred to as fractional-order mutual coupled circuits (FMCCs). The FMCC could be used instead of the integer-order mutual inductance because it increases the design degree of freedom because of the increased parameters in the design. This paper is organized as follows: Section 2 discusses the idea of the fractional mutual inductance (FMI). After that, an FMCC is presented in Section 3. Applications based on the FMI and the FMCC like the double-tuned filters and impedance matching are introduced in Section 4. Finally, the conclusion of the paper is presented in Section 5. Figure 1. T-model of the integer-order mutual inductance and T-model of the proposed fractional mutual inductance.

3 FRACTIONAL MUTUAL INDUCTANCE ANALYSIS AND DESIGN 2. PROPOSED FRACTIONAL MUTUAL INDUCTANCE The conventional MCC is a lossless network, which is not practical; thus, fractional parameters can be used to add the losses terms, which were proven to be frequency dependent as mentioned by the Coilcraft report [34,36]. For generalization, assume the primary and secondary inductors are of different fractional orders α, β, respectively. Consequently, in this case the mutual inductance is not symmetric, and the mutual inductance could be considered of the fractional-order γ. Hence, the induced emf at each fractional-order inductor should be given by d α i 1 emf 1 ¼ v 1 ¼ L 11 dt α ± M d γ 12 i2 12 dt γ 12 (4a) emf 2 ¼ v 2 ¼ M 21 d γ 21 i1 dt γ 21 ± L 22 d β i 2 dt β (4b) Then, by taking the Laplace transform of (4), the FMI can be represented by the following impedance matrix equation: V 1 V 2 ¼ sα L 11 s γ 12 M12 s γ 21 M21 s β L 22 I1 I 2 (5) For the traditional case α = β = γ 12 = γ 21 = 1, the matrix of (5) represents the matrix equation of the integer-order MCC presented in (3). The impedance matrix of (5) represents the behavior of the proposed FMI, which is the general case of the traditional mutual inductance. The FMI has unequal phase response for the primary and the secondary inductors. In addition, the coupling between the primary and the secondary inductors is unsymmetrical. The FMI modeled by (5) could be represented by the equivalent T-circuit shown in Figure 1 for only the case of symmetrical coupling γ 12 = γ 21 = γ and M 12 = M 21. Now, while a resistor capacitor ladder can be used to approximate a fractional-order capacitor, this same topology is difficult to realize a fractional-order inductor as it requires many inductors [37].Therefore, the fractional-order capacitor could be used with the general impedance converter circuit (GIC) to implement the grounded fractional-order inductors as shown in Figure 2 [12,38]. On the other hand, to implement a floating fractional-order inductor, two cascaded GICs are used [39, 40]. Hence, the input impedance of the GIC of Figure 2 is given as follows: Figure 2. General impedance converter circuit used to simulate a grounded fractional-order inductor using a fractional-order capacitor.

4 A. SOLTAN, A. G. RADWAN AND A. M. SOLIMAN Z in ¼ s α CR 1R 3 R 5 R 2 (6) From (6), the inductance of the fractional-order inductor of order α is given as L = CR 1 R 3 R 5 /R 2. In addition, the proposed FMI symbol for the special case of symmetrical coupling is depicted in Figure 3. For simulation purposes, the T-model of Figure 1 can be replaced with the circuit of Figure 3. Then, the impedance matrix becomes as follows: V 1 V 2 ¼ sα L 11 þ s γ M s γ M s γ M s β L 22 þ s γ M I1 I 2 (7) Subsequently, for the case of α = β = γ, the model represents a symmetric fractional-order mutual inductance. In this case, the phase of all the parameters of the impedance matrix is απ/2. Then, the phase value depends on the fractional-order α, which increases the design degree of freedom. The circuit simulation for the FMCC of Figure 3 is depicted in Figure 4 at different values of α. The phase response for the traditional case is π/2, but when the inductor elements are replaced with the fractional-order elements, the phase changes depending on the value of α as illustrated in Figure 4. The phase error between the simulated and the ideal phase responses (the dashed lines in Figure 4) is ± 3 o for α = 0.8 and ± 4 o for α = 0.7 during the simulated frequency range. This small error indicates that the FMCC is suitable for a wide bandwidth of applications. Another important case arises when α = β γ, the phase of the impedance matrix parameters in this case can be calculated using the formula of [36] as follows: Figure 3. Proposed symbol of the fractional mutual inductance and equivalent T-model of the fractional mutual inductance. Figure 4. Circuit simulation for the fractional mutual inductance for α = β = γ and L 11 = L 22 = M =10μH.

5 FRACTIONAL MUTUAL INDUCTANCE ANALYSIS AND DESIGN 8 2tan 1 y q 11;22 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x > 0 or y 0 >< x 11;22 þ x 2 11;22 Z 11;22 ¼ þ y2 11;22 π x < 0 and y ¼ 0 >: Undefined x ¼ 0 and y ¼ 0 (8a) where Z 12;21 ¼ γπ 2 (8b) y 11;22 ¼ ω α L 11;22 sinð0:5απþþω γ Msinð0:5γπÞ (9a) x 11;22 ¼ ω α L 11;22 cosð0:5απþþω γ Mcosð0:5γπÞ (9b) where L 11,22 are the primary and secondary inductances of the FMI model. Actually, the phase response of the impedance parameters Z 11 and Z 22 is the same as the phase response of the practical inductor model [41, 42]. This should be expected, because these impedance parameters represent the impedance of the mutual inductance inductors. On the other hand, the terms Z 12 and Z 21 represent the coupling between the two inductors and do not represent a real inductor. From this simple analysis, the impedance matrix of (7) represents the behavior of FMCC even if the fractional orders are different. In addition, the fractional-order model gives the ability to control the parasitic components of the FMCC by changing the value of the fractional orders (α, γ). Also, the phase is a function of the inductance values (L 11,22 and M) and the frequency of operation that increases the design degree of freedom as shown in Figure 5 and for different frequency points. From (8a), the effect of the fractional orders (α, γ) is symmetric as shown in Figure 5. For the special case of α = γ = 0, the elements tend to work as a resistance because the phase in this case equals zero as Figure 5. Phase response for the impedance parameters Z 11 and Z 22 with respect to α and γ at different frequencies at L 11 = L 22 = M =10μH ω o = 1 rad/s and ω o = 1 krad/s.

6 A. SOLTAN, A. G. RADWAN AND A. M. SOLIMAN expected. On the other hand, for α = γ = 1, the elements act as a pure inductor, and the phase equals π/2 as illustrated in Figure 5. Yet, with the fractional-order elements, this is not the only condition to fulfill the pure inductance behavior. From (8a), the condition to satisfy the pure inductance behavior is given as follows: ω α γ 11;22 pure ¼ Mcos ð 0:5γπ Þ L 11;22 cosð0:5απþ (10a) y 11;22pure ¼ ω α 11;22 pure L 11;22 sinð0:5απþþω γ 11;22 pure Msinð0:5γπÞ (10b) From (10a), there is an infinite number of conditions at which the pure inductance response can be obtained. So, for a given value of α and at the required frequency, the value of the fractional-order γ could be calculated to fulfill the condition of (10a) as shown in Figure 6. Then, the value of the pure inductance is calculated using (10b) as depicted in Figure 6. Similarly, the condition of the pure resistance response and its value is obtained by replacing the cosine function with the sine function as follows: ω α γ 11;22 pure ¼ Msin ð 0:5γπ Þ L 11;22 sinð0:5απþ (11a) x 11;22pure ¼ ω α 11;22 pure L 11;22 cosð0:5απþþω γ 11;22 pure Mcosð0:5γπÞ (11b) It is interesting to note here that the phase response is a function of the frequency of operation, which matches with real response of the electrical circuit components [34]. Finally, the more general case of α β γ represents fractional-order mutual inductance, but the phase of the impedance matrix parameters Z 11 and Z 22 is different. The mutual inductance can be considered in this case unsymmetrical, which defines a new concept of the asymmetric mutual inductance. The phase equations of the impedance parameters Z 11, Z 22 are similar to that of (8a) but after using the proper fractional orders α, β and the proper inductance values L 11, L 22, respectively, for each element. Circuit simulations for the fractional-order mutual inductance model of Figure 3 are presented in Figure 7 in the case of different fractional orders. The case of (α, β, γ) = (1, 1, 0.7) is presented in Figure 7 where the phase response of the impedance parameters Z 12,21 is close to 63 o, and the frequency effect on it is negligible as expected by (8b). Yet, the effect of the frequency on the phase response of the impedance parameters and Z 11,22 is larger as discussed before in (8a). On the other hand, the simulation of the general case of different fractional orders (α, β, γ) = (0.5, 1.4, 0.4) is Figure 6. The value of γ that satisfy the condition of (10a) at different frequencies at L 11,22 = M =10μH and pure inductance value versus the fractional-order α at different values of the frequencies at L 11,22 = M =10μH.

7 FRACTIONAL MUTUAL INDUCTANCE ANALYSIS AND DESIGN Figure 7. Phase response of the impedance parameters for fractional-order T-model at different values of the fractional orders and at L 11 = L 22 = M =10μH α = β = 1 and γ = 0.7 and α = 0.5, γ = 0.4, and β = 1.4. depicted in Figure 6. Although, the phase value of the impedance parameters Z 12,21 can be predicted from (8a) and equals 36 o, the phase response of Z 11,22 is frequency dependant. Consequently, the circuit simulations prove the previous discussion, where the output phase of the impedance matrix parameters becomes variable and dependent on the order of the fractional-order element. According to the previous analysis, the fractional-order mutual inductance can be considered the generalization of the traditional mutual inductance. Also, to model the behavior of the real transformers, many resistors and capacitors are added to represent the dependence of the phase on the frequency and the inductors values [41, 42]. Yet in the case of fractional-order mutual inductance, the addition of these extra modeling elements is not necessary. Because the phase response depends on the frequency and the inductance values (as shown in (8)) without adding any extra modeling elements, the FMI is closer to the real transformer behavior than the integer mutual inductance as mentioned before in the analysis. 3. EQUIVALENT CIRCUIT In this section, an equivalent circuit for the FMI based on the differential voltage current-controlled conveyor transconductance amplifier (DVCCCTA) [29] is presented in Figure 8. The FMI is floating in nature. The equivalent circuit consists of two DVCCCTAs and three grounded fractional-order capacitors of orders (α c, γ c, β c ). The port relationships of the DVCCCTA can be written as follows: I Y1 I Y2 V X I Z I O ¼ 1 1 R X g m 0 V Y1 V Y2 I X V Z V O (12) Figure 8. Fractional-order mutual coupled circuit version of the mutually coupled circuit presented in [29]. DVCCCTA, differential voltage current-controlled conveyor transconductance amplifier.

8 A. SOLTAN, A. G. RADWAN AND A. M. SOLIMAN where R X is the intrinsic resistance at the X terminal and g m is the transconductance from the Z terminal to the O terminal of the DVCCCTA. The circuit realization of the DVCCCTA used in this work is presented in [29]. Routine analysis of the FMCC equivalent circuit illustrated in Figure 8 gives the impedance matrix of (13), which is similar to the impedance matrix of the general case of the FMI given in (5). Consequently, the circuit of Figure 8 works as an unsymmetrical coupling fractionalorder mutual inductance. The relations between the parameters of the impedance matrix of (5) and the components of the equivalent circuit of Figure 8 are tabulated in Table I. The circuit of Figure 8 fulfills different behaviors for the fractional-order mutual inductance at different conditions. A summary of these responses and their required conditions are summarized in Table II where the traditional mutual inductance behavior happens at α c = β c = 1 and γ c = 0 and also C 3 = C 1. The phase response for the impedance matrix parameters in this case is π/2 as shown in Figure 9. In addition, the case of symmetrical FMI behavior is fulfilled at α c = β c, γ c = 0 and C 3 = C 1. Phase response of the impedance matrix parameters for the case of symmetrical FMI is depicted in Figure 9. V 1 V 2 2 ¼ 6 4 s α c C 1R X1 g m1 þ s α c γ c C 1 g m1 C 2 s β c γ c C 3 g m2 C 2 s α c γ c C 1 g m1 C 2 s β c C 3R X2 g m2 þ s β c γ c C 3 g m2 C I 1 I 2 (13) Finally, a capacitance effect appears at the coupling points (Z 12, Z 21 ) when the fractional-order α c or β c is less than the fractional-order γ c. This property could be used in the impedance matching circuit to compensate the inductance of the mutual inductance circuit. 4. APPLICATIONS Now, it is important to prove the reliability of the FMCC. So, the goal of this section is to use the FMCC in different applications. Table I. Summary of the relation between the impedance matrix parameters and the circuit elements. Parameter α β γ 12 γ 21 Relation to the equivalent circuit components α c β c α c γ c β c γ c C1RX L 1 11 g m1 C3RX L 2 22 g m2 M 12 C1 g m1 C2 M 21 C3 g m2 C2 Table II. Required conditions to satisfy the proposed fractional mutual inductance. Behavior description Required condition Traditional mutual inductance α c = β c = 1 and γ c = 0 and C 1 = C 3 Symmetrical fractional-order mutual inductance α c = β c and γ c =0 Unsymmetrical fractional-order mutual inductance α c = β c > γ c Fractional mutual inductance with capacitive effect α c or β c < γ c Fractional mutual inductance with resistive effect α c or β c = γ c

9 FRACTIONAL MUTUAL INDUCTANCE ANALYSIS AND DESIGN Figure 9. Circuit simulation of the equivalent circuit of the fractional-order mutual coupled circuit at different cases at C 1 = C 3 =10nF and C 2 = 1/1330 α c = β c = 1 and γ c = 0 and α c = β c = 0.4 and γ c = Double-tuned filter Double-tuned filters are considered one of the applications that are mostly based on the mutual inductance. The conventional double-tuned filter is composed of a series resonance circuit and a parallel resonance circuit. So, the double-tuned filter can achieve a wider bandwidth band-pass filter than the traditional filters. So, the goal of this section is to use the fractional-order mutual inductance to build a fractional-order double-tuned filter. The circuit diagram of the double-tuned filter is illustrated in Figure 10, and its transfer function is given as follows: Ts ðþ¼ V out ¼ V in M RPCPLPCSLS sγ c s α cþ1 þ 1 RPCP sα c þ 1 CPLP 1 (14) s β c þ1 þ 1 sβ c þ 1 RSCS CSLS Although the transfer function in (14) represents a band-pass filter of order α c + β c + 2, the circuit does not give this response. Yet, the circuit works as a two band-pass filters operating at different half power frequencies. So, this circuit is called double-tuned circuit. Then, the transfer function can be represented as follows: Ts ðþ¼ a 1 s γ 1 s α cþ1 þ 1 RPCP sα c þ 1 CPLP a 2 s γ 2 (15) s β cþ1 þ 1 sβ c þ 1 RSCS CSLS Assuming for simplicity that γ 1 = α c, γ 2 = β c, and γ 1 + γ 2 = γ c and also a 1 a 2 ¼ transfer function can be rewritten as follows: M RPCPLPCSLS. Then, the Ts ðþ¼ a 1 s α c s α cþ1 þ 1 RPCP sα c þ 1 CPLP a 2 s β c (16) s β cþ1 þ 1 sβ c þ 1 RSCS CSLS Accordingly, the transfer function of (16) represents two cascaded fractional-order band-pass filters. So, the critical frequency points (cut-off frequency, maximum and minimum frequency points, and the right phase frequency) of this filter can be determined using the algorithm presented in [7 9, 43]. Hence, the critical frequency points become functions of the fractional orders α, β, which increase the design degree of freedom [7 9, 43]. Actually, the frequency response of the double-tuned filter based on the FMI has two resonance frequencies as shown in the numerical analysis of Figure 10. Then, the filter bandwidth is increased as expected. In addition, Figure 10 presents the frequency response of the filter using the same element values but at different fractional orders. It is clear that the filter cut-off frequency changes with the fractional orders as mentioned before.

10 A. SOLTAN, A. G. RADWAN AND A. M. SOLIMAN (c) Figure 10. Double-tuned filter using the fractional-order mutual coupled circuit equivalent circuit, numerical analysis of the double-tuned filter with R P = R s =11kΩ and C s =20nF, C P =10nF, and (c) circuit simulation for the double-tuned filter R P = R s =11kΩ and C s =20nF, C P =10nF. Finally, the circuit simulation of the double-tuned filter based on the FMI is depicted in Figure 10(c) for the same cases of the numerical analysis. There is a great matching between the numerical analysis and the circuit simulations of Figure 10 and (c), respectively. This matching confirms the reliability of the FMI Fractional-order mutual coupled circuit in impedance matching The problem of impedance matching is an important one, which must be addressed in most microwave designs [44, 45, 31]. One of the most common techniques used in the impedance matching is the mutual inductance. Hence, impedance transformation using the FMCC is presented here. The equivalent T-model of the FMCC illustrated in Figure 3 is used for the impedance matching analysis as shown in Figure 11.To simplify the analysis, the case of the symmetrical FMI that has α = β = γ is discussed here. Then, from a simple routine analysis, the load impedance of the circuit can be calculated from (17).

11 FRACTIONAL MUTUAL INDUCTANCE ANALYSIS AND DESIGN Figure 11. Impedance matching circuit using the fractional-order mutual coupled circuit T-model and numerical analysis for Z Load with respect to Z source and α at L 11 =1mH and L 22 = M =10μH. Z Load ¼ ωα cosðαπþðl 11 L 22 þ ML ð 22 þ L 11 ÞÞþZ source cosð0:5απþðl 22 þ MÞ L 11 cosð0:5απþþz source ω α þ Mcosð0:5απÞ (17a) Z Load ¼ ωα sinðαπþðl 11 L 22 þ ML ð 22 þ L 11 ÞÞþZ source sinð0:5απþðl 22 þ MÞ ðl 11 þ MÞsinð0:5απÞ (17b) For the traditional case α = β = γ = 1, the relation between the input impedance and the load impedance is as follows: Z Load ¼ L 22 þ M Z source L 11 þ M (18a) Z Load Z source ¼ ω 2 ðl 11 L 22 þ ML ð 22 þ L 11 ÞÞ (18b) The relation in (18a) is the same as the well-known relation of the integer-order mutual inductance [31], which confirms (17). Yet, the value of the resonance frequency in this case is negative as given in (18b). This means that the system is unstable and requires a compensation capacitor to eliminate the inductance effect. On the other hand, from (17) for the fractional-order mutual inductance, this capacitor can be ignored because as mentioned before the FMI can have self-compensation behavior. Consequently, matching using the fractional-order mutual inductance is simpler and hence cheaper from the circuit implementation point of view because it requires fewer components. From (17a), the load impedance is a function of the fractional-order α besides the circuit components {L 11, L 22, M, Z in }, which increases the design flexibility. The effect of the input impedance on the

12 A. SOLTAN, A. G. RADWAN AND A. M. SOLIMAN output impedance can be minimized at large values of α as shown in Figure 11. Then, the system can be designed for fixed load impedance although the input impedance changes. On the other hand, for small values of α, the effect of the input impedance on the load impedance is very large as illustrated in Figure 11. For the case of equal orders (α = β = γ) and L 11 =1mH, L 22 = M =10μH, the system is stable without using a compensation capacitor in the range demonstrated in Figure 11. Consequently, the system can be designed for matching without using the compensation capacitor but at a specific frequency range. This frequency range could be changed by changing the value of the fractional order and the circuit component values. 5. CONCLUSION Fractional-order mutual inductance analysis is discussed. To use the FMI, an equivalent circuit is presented. It has been found that the phase response of the impedance matrix parameters of the equivalent circuit can be controlled by the value of the fractional orders as shown in the Advanced Design System and MATLAB simulations. Different applications based on the proposed FMCC have been discussed like the double-tuned filter and impedance matching. For the double-tuned filter, a good matching is found between the numerical analysis and the Advanced Design System simulations. In addition, for the impedance matching based on the proposed FMCC, the design equations are derived, and the design degree of freedom is increased because of the increased design variables. REFERENCES 1. Caponetto R, Dongola G, Fortuna L, Petras I. Fractional Order Systems: Modeling and Control Applications, Vol. 72. World Scientific Publishing Company: Singapore, Mondal D, Biswas K. Performance study of fractional order integrator using single-component fractional order element. Circuits, Devices & Systems, IET 2011; 5(4): Coopmans C, Petras I, Chen Y. Analogue fractional-order generalized memristive devices. In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers: California, 2009; 4: Cisse Haba T, Ablart G, Camps T, Olivie F. Influence of the electrical parameters on the input impedance of a fractal structure realised on silicon. Chaos, Solitons & Fractals 2005; 24(2): Radwan AG, Shamim A, Salama KN. Theory of fractional order elements based impedance matching networks. Microwave and Wireless Components Letters, IEEE 2011; 21(3): Radwan AG, Soliman AM, Elwakil AS. First-order filters generalized to the fractional domain. Journal of Circuits, Systems, and Computers 2008; 17(1): Soltan A, Radwan AG, Soliman AM. Fractional order filter with two fractional elements of dependant orders. Microelectronics Journal 2012; 43(11): Soltan A, Radwan AG, Soliman AM. CCII based fractional filters of different orders. Journal of Advanced Research 2014; 5(2): Soltan A, Radwan AG, Soliman AM. CCII based KHN fractional order filter. In Circuits and Systems (MWSCAS), 2013 IEEE 56th International Midwest Symposium on, Aug 2013; Tripathy MC, Mondal D, Biswas K, Sen S. Design and performance study of phase-locked loop using fractional-order loop filter. International Journal of Circuit Theory and Applications Tsirimokou G, Laoudias C, Psychalinos C. 0.5-V fractional-order companding filters. International Journal of Circuit Theory and Applications Radwan AG, Elwakil AS, Soliman AM. Fractional-order sinusoidal oscillators: design procedure and practical examples. Circuits and Systems I: Regular Papers, IEEE Transactions on 2008; 55(7): Radwan AG, Soliman AM, Elwakil AS. Design equations for fractional-order sinusoidal oscillators: four practical circuit examples. International Journal of Circuit Theory and Applications 2008; 36(4): Soltanc A, Radwan AG, Soliman AM. General procedure for two integrator loops fractional order oscillators with controlled phase difference. In Microelectronics (ICM), th International Conference on, Dec 2013; Alikhani R, Bahrami F. Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations. Communications in Nonlinear Science and Numerical Simulation 2013; 18(8): Zhang X, Liu L, Wu Y. The uniqueness of positive solution for a singular fractional differential system involving derivatives. Communications in Nonlinear Science and Numerical Simulation 2013; 18(6): Moaddy K, Radwan AG, Salama KN, Momani S, Hashim I. The fractional-order modeling and synchronization of electrically coupled neuron systems. Computers & Mathematics with Applications 2012; 64(10):

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