Fractional-Order Sinusoidal Oscillators: Design ... - Semantic Scholar
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 7, AUGUST 2008
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Fractional-Order Sinusoidal Oscillators: Design Procedure and Practical Examples Ahmed Gomaa Radwan, Ahmed S. Elwakil, Senior Member, IEEE, and Ahmed M. Soliman, Senior Member, IEEE
Abstract—Sinusoidal oscillators are known to be realized using dynamical systems of second-order or higher. Here we derive the Barhkausen condition for a linear noninteger-order (fractional-order) dynamical system to oscillate. We show that the oscillation condition and oscillation frequency of some famous integer-order sinusoidal oscillators can be obtained as special cases from general equations governing their fractional-order counterparts. Examples including fractional-order Wien oscillators, Colpitts oscillator, phase-shift oscillator and LC tank resonator are given supported by numerical and PSpice simulations. Index Terms—Fractional-order circuits, noninteger order systems, oscillators.
I. INTRODUCTION HE classical linear circuit theory upon which electronic circuits are designed today is based on integer-order differential equations which reflect the behavior of the three wellknown elements: the resistor, the capacitor and the inductor in the time domain. Via Laplace transform, integer-order algebraic equations in the complex frequency -domain are also used to describe linear dynamical systems. Accordingly, electronic circuits are traditionally classified as first-order, second-order or th-order circuits where is an integer number. The circuit order is directly proportional to the number of energy storage elements in the circuit. From a fractional calculus mathematical point of view, differential equations are not necessarily of integer-order [1], [2]. The Riemann-Liouville definition of a fractional derivative [3], [4] is given by
T
(1) . A more physical interpretation of a fractional where derivative is the Grünwald approximation given by
(2)
Manuscript received March 18, 2007; revised October 10, 2007. First published February 8, 2008; last published August 13, 2008 (projected). This paper was recommended by Associate Editor T. B. Tarim. A. G. Radwan is with the Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Cairo 12613, Egypt. A. S. Elwakil is with the Department of Electrical and Computer Engineering, University of Sharjah, Emirates (e-mail: [email protected]). A. M. Soliman is with the Department of Electronics and Communications, Faculty of Engineering, Cairo University, Cairo 12613, Egypt (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSI.2008.918196
where
is the integration step and . Applying the Laplace transform to (1), assuming zero initial conditions, yields (3) A fractance device is one whose impedance is proportional to is arbitrary. In such a device, the phase difference between the voltage across its two terminals and the current en. The special cases tering these terminals is correspond respectively to the resistor, inductor and capacitor. Four decades ago, some researchers investigated realizing a fractional-order capacitor [5], [6]. A finite element approximawas reported in [7]. This tion of the special case finite element approximation relies on the possibility of emulating a fractional-order capacitor via semi infinite self-similar RC trees. The technique was later developed by the authors of [8]–[10] for any . Finite element approximations offer a valuable tool by which the effect of a fractance device can be simulated using a standard circuit simulator, or studied experimentally. However, they do not offer a simple practical two-terminal device. Therefore, investigations of fractional-order circuits remained limited and confined mostly to simulations of special case circuits due to the non existence of a real fractance device and hence the lack of practical motivation [11], [12]. Most recently, the authors of [13], [14] reported the fabrication of a real two-terminal fractance device. The fabricated probe described in [13] and [14] is based on a metal-insulatorliquid interface and was used in [14] to realize a fractional-order differentiator circuit. Although this capacitive probe is bulky and relies on a necessary liquid interface, with more research and increasing application motivation, a better device will surely become commercially available in the near future. Should this happen, circuit designers will be faced with the challenge as to how to make use of a fractance device in constructing their application circuits particularly as the available design equations need to be generalized from the narrow integer-order subset to the more general fractional-order domain. In this paper, we focus on sinusoidal oscillators which are key building blocks. We derive the Barkhausen oscillation condition and the oscillation frequency for any linear system with 2, 3 or fractance devices. It is well-known that pure linear systems, whether integer-order or fractional-order, cannot maintain sustained oscillations. An accurate oscillator model requires the modeling differential equations to be necessarily nonlinear. It is also known that the Barkhausen oscillation condition is a necessary but insufficient condition for oscillation. For example, an oscillator might actually latch-up and never oscillate even if
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the Barkhausen condition is satisfied [15]. However, circuit designers are still accustomed to applying the Barkhausen condition to a linearized (with respect to the equilibrium point at the origin) model of their oscillator in order to derive the oscillation condition and oscillation frequency. We therefore opt to derive the general fractional-order Barkhausen oscillation conditions using stability analysis of fractional order systems [16]–[19]. A number of practical examples including fractional-order RC Wien oscillators, LC tank resonator, phase-shift oscillator and Colpitts oscillator are given in this paper. We show that these famous oscillators can still be designed to oscillate if constructed with fractance devices. We also show that fractionalorder oscillators have an advantage which may be exploited. In particular, the oscillation frequency does not only depend on the values of the reactive elements and/or but also on their fractional-order , which adds an extra degree of design freedom. Numerical and PSpice simulation results are shown.1 It is worth noting that some special case fractional-order oscillators were studied in [20] and [21]. It is also worth noting that experimental results of some fractional-order oscillators using the capacitive probe of [14] were very recently reported in [22].
Let us assume that are two roots of this equation . Using Euler’s identity and ( is a real number and solving separately for the real and imaginary parts yields
(7a) (7b) i) Assuming the system is oscillatory then must be two roots of the characteristic equation. Substiand yields the necessary condition tuting for oscillation (5a) and (5b). ii) Assuming (5a) and (5b) are satisfied, then comparing with and . Hence, it is sufthe (7) yields ficient to satisfy (5) for the system to admit sinusoidal oscillations. and can be calThe phase difference between culated as
II. OSCILLATORS WITH TWO FRACTANCE DEVICES
(8)
In this section, we consider oscillators with two fractance devices of fractional orders and . A. Theorem 1 A linear fractional-order system of the form
(4)
can admit sinusoidal oscillations if and only if there exists a value which satisfies simultaneously the two equations
(5a)
(9) where sgn . The following special cases are important ones. then (5b) can be solved to yield the oscillation fre1) If quency . Substituting this in (5a) yields the condition for oscillaand the phase differtion as ence is found to be . For the special case (secondorder system), the frequency of oscillation is then and the oscillation condition is which are the famous and well-known expressions for any lin. The phase earized second-order oscillator difference then reduces to . then (5b) can be solved2 for yielding 2) If
(5b) is the determinant of the system where coefficient matrix. Proof: Transforming (4) into the -domain, the characteristic equation of the system is obtained as
(10) the oscillation condition can then be obtained [by substituting for in (5a)] as
(6)
(11)
1Numerical simulations are carried out using a backward difference integration algorithm based on the Grünwald approximation of (2) with step size , where is the period of the sinusoid. PSpice simulations are based on the finite element approximations of [8]–[10].
2Equation
(5b) in this case has the form which can be solved using the identity .
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Fig. 1. Oscillations obtained from numerical simulations for examples (1) and (2).
and the phase difference is given by . the special case that are, respectively, given by
In , and
and . This case corresponds to a sinusoidal oscillator of order 1.5 . 3) If then , the oscillation condition and . This case corresponds to a is quadrature oscillator which is clearly not possible to realize . unless the oscillator is second-order then 4) If , the oscillation condition is and . The oscillation condition, frequency and phase difference and in this case. Similarly, are all independent of the same expressions are obtained replacing if with and interchanging with and vice versa.
5) If oscillator.
or
it is impossible to obtain an
B. Examples Example (1): Consider an oscillator with and which belongs to the above case 3. The characteristic equation of this oscillator is . Fig. 1(a) and (b) shows numerical simulaand respectively. Note tion results for for both figures while and , that respectively. Example (2): Consider an oscillator with and which belongs to the above case 4. The characteristic equation of this system is . Table I shows the calculated value of necessary to satisfy the oscillation condition and the corresponding for different values of and . Note from Table I that while this system (second-order oscillator); it can cannot oscillate for oscillate if either of them is fractional. Fig. 1(c) shows numerical
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TABLE I NECESSARY VALUE OF TO SATISFY THE OSCILLATION CONDITION FOR EXAMPLE (2) AND THE CORRESPONDING OSCILLATION FREQUENCY FOR DIFFERENT VALUES OF AND
TABLE II SUMMARY OF STABILITY CONDITIONS FOR
(12b) Hence
(13) Equation (13) contains five variables (then , (13) then becomes met) and define simulation results for where the character. istic equation in this case becomes Using the technique developed in [23], it can be shown that this characteristic equation has the 6 equivalent complex conjugate roots . A root which is visible in the -plane is one with phase rad. For these 6 pairs, is respectively and with hence the only visible pair is the last one . This pair is pure imaginary and the system is therefore oscillatory, as confirmed by Fig. 1(c). Table II summarizes the stability conditions and roots of the . characteristic equation
and
. Let must be
(14) and using (12b)
(15) (16) (17)
It is clear that the relationships between and the other variand are nonlinear and hence it is difficult ables to design for a required and . All seven variables can be diand vided into two groups; group (1) contains the set . The following progroup (2) contains the set cedure can be used if two variables of the first group are given and only one variable from the second group, most likely is known. The procedure helps create a design look-up table which and the values of simplifies the design process. The sign of and can be arbitrary chosen. it can be shown that From (5a) and (5b), isolating
The following two steps can now be followed. Step(1):- Related to group (1) variables: and are known, then (14) can be solved for 1) If . Table III is a look-up table for in the case that . A similar table can be constructed when . Knowing is found from (17). 2) If and are known, can be found from (16). Using look-up Table IV is then obtained. 3) If and are known, can be found from (17) and then is found from Table III. Step(2):- related to group (2) variables: Any known parameter from this group enables the rest to be found using the relations
(12a)
(18)
C. Simplified Design Procedure
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TABLE III LOOK-UP TABLE FOR WHEN
TABLE IV LOOK-UP TABLE FOR WHEN
Note that the special case in Table III.
yields
which is clear
1) RC Oscillators: Consider the Wien oscillator shown in Fig. 2(c) modified to include two fractional capacitors and hence described by
D. Circuit Design Examples In what follows, and for the purpose of performing PSpice simulations, we shall use the circuit in Fig. 2(a), proposed in [8], to simulate a fractional capacitor of order 0.5 while the circuit in Fig. 2(b), proposed in [10], shall be used to simulate a fractional capacitor of ar. To realize for exbitrary order branches with and ample, we need [10]. A number of classical second-order oscillators, modified to contain two fractance devices of orders and , where , are presented below.
(19) where, is the saturation voltage of the operational amplifier . and the gain
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Fig. 2. Circuit realizations of a (a) fractional capacitor of order [8], (b) fractional capacitor of any order [10], (c) fractional-order Wien oscillator, (d) fractional-order negative resistor RC oscillator and (e) fractional-order LC oscillator. TABLE V DESIGN EQUATIONS FOR THE WIEN OSCILLATOR
Fig. 3. PSpice simulation of Wien oscillator with and (a) F F F . (b)
4) With
Design is based in the linear region of operation where it is necessary to find the value of needed to start oscillations. If the values of and are all known then can be easily obtained as follows: and 1) substituting for in (13) the value of is found. can be obtained and hence 2) substituting with in (12a), is found. and oscillation freIf it is required to design for a given quency , the simplified procedure explained above can then and hence be followed noting from (19) that and Tables III and IV can be used. Choosing any value for the remaining steps are as follows. 1) From Tables III and IV find the values of and . is calculated from (15) 2) Knowing and the desired is found. and hence known, (12b) can be solved for 3) With hence is found.
F
k , and
and
known, (12a) can be solved for hence the last element is found. Steps 3 and 4 can be considerably simplified by noting that ; i.e., and can be found directly knowing and . Table V summarizes design equations for some important special cases of this oscillator. Note that the case in row 2 is the one reported in [20]. Fig. 3 shows two different cases simulated in PSpice using a generic TL082 opamp. Note from the second row in Table V that the oscillation frequency depends not only on and but also on and can be made since . much higher than the classical Next consider the famous RC oscillator shown in Fig. 2(d). In the opamp linear region, this oscillator is described by
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(20)
RADWAN et al.: FRACTIONAL-ORDER SINUSOIDAL OSCILLATORS: DESIGN PROCEDURE AND PRACTICAL EXAMPLES
Fig. 5. PSpice simulation for the LC tank oscillator with k and F k
Fig. 4. PSpice simulation for the oscillator in Fig. 2(d) with k (a) F (b) F F .
F
and
Following a procedure similar to the one described for the Wien oscillator, we may also design this fractional-order osciland that if lator. Note here that otherwise . Table V can also be used to design this oscillator applying the minor transformations and noting that is positive . Fig. 4 shows two different cases of circuit simulation using PSpice. 2) LC Oscillator: An active LC oscillator with fractionalorder inductor and capacitor is shown in Fig. 2(e). The opamp and associated resistors form a negative resistor and the oscillator is described by
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.
In the linear region of operation, it is seen that and hence the condition for stability is . It can be shown in the spethat the condition for oscillation is cial case , which reduces to the famous condition when . The oscillation frequency is , which also reduces to the at well-known formula . The phase difference between and is , which becomes frequency independent and tends to as . In the general case where , and where the values of and are given, simplified design steps for a desired oscillation frequency can be summarized as follows: calculate and hence . 1) let 2) let ; then find by solving the equation . Finding , the value of directly results. 3) the condition for oscillation is then . PSpice simulation results are shown in Fig. 5, where the fractional inductor, which has impedance was implemented using a standard gyrator [24] and a fractional capacitor.
III. OSCILLATORS WITH THREE FRACTANCE DEVICES The state space presentation of a linear system with three fracand is given by: tance devices of orders
(21) where are as given by (19) with is the inductor internal parasitic resistance.
and
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(22)
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A. Theorem 2 The above system can sustain sinusoidal oscillations if there exists a value for to satisfy the following equation:
the two conditions sary. Substituting for oscillation as
and are necesin (26a) yields the condition for
(28) (23) 2) If and its counterpart equation obtained by replacing every term in (23) with a term and removing the last term. Here, is the determinant of the 3 3 matrix, and . Proof: transforming (23) into the -domain, the characteristic equation of the system is found to be
(24) To achieve oscillatory behavior must satisfy the characteristic equation. Using Euler’s identity and solving separately for the real and imaginary parts yields (23) and its counterpart. The transfer function is given by
(third-order system), (27) reduces to and (28) reduces to .
B. Examples Example (1): Consider a system described by (22) with and . It can be easily shown that and . Hence, applying the equations above, and the frequency the oscillation condition is is . Fig. 6(a) and (b) shows numerical simulation results for two different sets of fractional orders with . Example (2): Consider a system described by (22) with and which results in and . The oscillation condition and frequency, respectively, are (29a)
(25) from which the phase difference can be easily found. Similar relations for and can be derived. The following special cases are important ones. 1) If (23) and its counterpart reduce to
(29b) Fig. 6(c) and (d) show numerical simulations for the two difand . ferent cases C. Circuit Design Examples
(26a)
(26b) where . Solving (26b) for
and yields
(27) to be real, the condition must hold. In addition, for a positive , the conditions if and if are necessary. For ,
For
Few oscillators with three energy storage elements are practically known. These include the famous RC phase-shift and Twin-T oscillators and the famous LC Colpitts and Hartley oscillators. Two of these oscillators modified to include fractance devices are discussed below, while the other two were studied in [22]. 1) Phase-Shift Oscillator: The phase shift oscillator circuit is shown in Fig. 7(a) and can be described by
where , matrix the bottom of the next page, and replacing with It is easy to show that
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is as given by (30), shown at are as given by (19) after .
.
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Fig. 6. Numerical simulation results for examples (1) and (2) of a system with three fractance devices.
Also let
, and . Using (23) and its counterpart equation yields
(31) which has a solution only if all terms are positive and . Given and , (31) can be solved for the oscillation frequency . Substituting with the obtained in (23) yields the oscillation condition. The following are important special cases.
(30)
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Fig. 7. Oscillators with three fractance devices. (a) phase-shift oscillator, (b) Colpitts oscillator, and its (c) small-signal equivalent model.
1)
: equation (31) may be solved3 to give (32), shown at the bottom of the page. For to be real and positive, two conditions must hold: (i) and (ii) . Substituting with oscillation condition
in (23) yields the
(33) and : the oscillation frequency and oscillation condition are, respectively, as shown in (34a) and (34b), at the bottom of the page. 3) and : this case corresponds to the classical third-order phase-shift oscillator whose oscillation frequency and con2)
3using
the identity
.
Fig.
8. PSpice F
simulations F
for the phase-shift oscillator k and (a) F , and (b) F F
with F .
dition are well-known to be and , as confrimed by (34). Fig. 8 shows PSpice simulations of the phase-shift oscillator for two different sets of fractional orders.
(32)
(34a)
(34b)
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RADWAN et al.: FRACTIONAL-ORDER SINUSOIDAL OSCILLATORS: DESIGN PROCEDURE AND PRACTICAL EXAMPLES
2) Colpitts Oscillator: Fig. 7(b) shows a fractional-order Colpitts oscillator. Under small-signal conditions, the equivalent circuit in Fig. 7(c) can be drawn and is described by the following equations:
(35)
where and
is the bipolar transistor small-signal transconductance . Hence, it is seen that and . Applying (23), the condition for oscillation is found to be that of (36), shown at the bottom of the page, and the oscillation frequency is then obtained by solving for the equation
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IV. OSCILLATORS WITH
FRACTANCE DEVICES
The general state space representation of a linear system with fractance devices is
.. .
.. .
.. .
.. .
.. .
.. .
.. .
(38)
A. Theorem 3 The above system can sustain sinusoidal oscillations if there exists a value for to satisfy (39), shown at the bottom of the page, and its counterpart equation obtained by replacing every term with a term and removing the last term. In the special case that , (39) simplifies to
(37) The classical third-order Colpitts oscillator corresponds to setting which results in the famous oscillaand oscillation frequency tion condition where as confirmed by (36) and (37).
(40)
(36)
(39)
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The only application where an oscillator with fractance devices may be needed is that of an -phase oscillator.
V. CONCLUSION In this work, we have generalized the classical marginal stability sinusoidal oscillator design equations to the case where fractional-order elements are used. Several design examples were given. PSpice simulations were based on the finite element fractional capacitor approximation methods proposed in [8] and [10]. We believe the work presented here will prove valuable to circuit designers once a simple to use fractance device becomes commercially available [14]. We emphasize, in particular, the possibility of obtaining very high oscillation frequencies via adjusting the fractional order independent of the values of or .We have also generalized recently the classical first-order filters to the fractional-order domain [25].
REFERENCES [1] K. B. Oldham and J. Spanier, Fractional Calculus. New York: Academic, 1974. [2] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Application. New York: Gordon & Breach, 1987. [3] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley, 1993. [4] T. T. Hartley and C. F. Lorenzo, “Initialization, conceptualization, and application in the generalized fractional calculus,” National Aeronautics and Space Administration (NASA/TP-1998-208415) 1998. [5] G. Carlson and C. Halijak, “Approximation of fractional capacitors by a regular Newton process,” IEEE Trans. Circuits Syst., vol. CAS-11, no. 2, pp. 210–213, Mar. 1964. [6] S. Roy, “On the realization of a constant-argument immitance or fractional operator,” IEEE Trans. Circuits Syst., vol. CAS-14, no. 2, pp. 264–274, Mar. 1967. ,” IEEE Trans. [7] K. Steiglitz, “An RC impedance approximation to Circuits Syst., vol. 11, pp. 160–161, 1964. [8] M. Nakagawa and K. Sorimachi, “Basic characteristics of a fractance device,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci., vol. E75, no. 12, pp. 1814–1819, 1992. [9] K. Saito and M. Sugi, “Simulation of power-law relaxations by analog circuits: Fractal distribution of relaxation times and noninteger exponents,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci., vol. E76, no. 2, pp. 205–209, 1993. [10] M. Sugi, Y. Hirano, Y. F. Miura, and K. Saito, “Simulation of fractal immittance by analog circuits: An approach to the optimized circuits,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci., vol. E82, no. 8, pp. 1627–1634, 1999. [11] A. Abbisso, R. Caponetto, L. Fortuna, and D. Porto, “Noninteger-order integration by using neural networks,” in Proc. Int. Symp. Circuits Syst. ISCAS’01, Sydney, Australia, 2001, vol. 3, pp. 688–691. [12] A. Arena, R. Caponetto, L. Fortuna, and D. Porto, “Nonlinear noninteger order circuits and systems,” World Scientific Ser. Nonlinear Sci., ser. A, vol. 38, 2002. [13] K. Biswas, S. Sen, and P. Dutta, “Modelling of a capacitive probe in a polarizable medium,” Sens. Actuat. Phys., vol. 120, pp. 115–122, 2005. [14] K. Biswas, S. Sen, and P. Dutta, “Realization of a constant phase element and its performance study in a differentiator circuits,” IEEE Circuits Syst. II, Exp. Briefs, vol. 53, no. 8, pp. 802–806, Aug. 2006. [15] A. S. Elwakil and W. Ahmed W., “On the necessary and sufficient conditions for latch-up in sinusoidal oscillators,” Int. J. Electron., vol. 89, pp. 197–206, 2002.
[16] D. Matignon, “Stability results in fractional differential equations with applications to control processing,” in Proc. Multiconf. Comput. Engi. Syst. Appl. IMICS, IEEE-SMC, Nice, France, 1996, vol. 2, pp. 963–968. [17] C. Bonnet and J. Partington, “Coprime factorizations and stability of fractional differential equations,” Syst. Contr. Lett., vol. 41, pp. 167–174, 2000. [18] K. Diethelm and N. Ford, “Analysis of fractional differential equations,” J. Math. Anal. Appl., vol. 265, pp. 229–248, 2002. [19] R. El-Khazali and S. Al-Momani, “Stability analysis of composite fractional systems,” Int. J. Appl. Math., vol. 12, pp. 73–85, 2003. [20] W. Ahmad, R. El Khazali, and A. S. Elwakil, “Fractional-order Wienbridge oscillator,” Electron. Lett., vol. 37, no. 18, pp. 1110–1112, 2001. [21] R. Meilanov, “Features of the phase trajectory of a fractal oscillator,” Tech. Phys. Lett., vol. 28, pp. 30–32, 2002. [22] A. G. Radwan, A. M. Soliman, and A. S. Elwakil, “Design equations for fractional-order sinusoidal oscillators: Four practical design examples,” Int. J. Circuit Theory Appl., vol. 36, pp. 473–492, Jun. 2008. [23] A. G. Radwan, A. M. Soliman, A. S. Elwakil, and A. Sedeek, “On the stability of linear systems with fractional order elements,” Chaos Solitons Fractals, (in press) (available on line doi:10.1016/j.chaos.2007.10. 033). [24] A. Budak, Passive and Active Network Analysis and Synthesis. Chicago, IL: Waveland , 1991. [25] A. G. Radwan, A. M. Soliman, and A. S. Elwakil, “First-order filters generalized to the fractional domain,” J. Circuits Syst. Comput., vol. 17, pp. 55–66, Feb. 2008.
Ahmed Gomaa Radwan received the B.Sc. degree (with honours) in electronics and communications, the M.Sc. degree (for his thesis entitled “New MOS realizations of some chaotic equations using mathematical transformations”), and the Ph.D. degree (for his research on “Fractional calculus; stability, fractional-oscillators and fractional-order filters”), from Cairo University, Cairo, Egypt, in 1997, 2002, and 2006, respectively. After graduation, he joined the Department of Engineering Mathematics and Physics at Cairo University where he is now an Assistant Professor. He is the author and coauthor of 13 research papers in different scientific journals and his main interest is in chaotic systems, chaos generation, fractional calculus and applications to circuit design as well as implementation of the circuit blocks for hearing-aid systems. He received the Best Thesis Award from Cairo University for his M.Sc. thesis.
Ahmed S. Elwakil (SM’03) was born in Cairo, Egypt, in 1972. He received the B.Sc and M.Sc. degrees in electrical and electronic engineering from the Department of Electronics and Communications, Cairo University, Cairo, Egypt, and the Ph.D. degree in electrical and electronic engineering from the National University of Ireland (University College Dublin), Dublin, Ireland. He has acted as an Instructor for several courses on VLSI organized by the United Nations University for developing nations. His main research interests are in the area of analog integrated circuits with particular emphasis on nonlinear circuits analysis and design techniques , nonlinear dynamics, and chaos theory. He is author and coauthor of many publications in these areas. Dr. Elwakil is a member of the Technical Committee for Nonlinear Circuits and Systems (TCNCAS), a member of IET, and an associate member of the International Centre for Theoretical Physics (Trieste, Italy). He has served as an Organizing Committee Member and Track Chair for numerous journals and conferences. He is currently on the Editorial Board of the International Journal of Circuit Theory and Applications, and is an Associate Editor of the Journal of Dynamics of Continuous, Discrete, and Impulsive Systems, Series B: Applications and Algorithms. Dr. Elwakil received the Government of Egypt first-class medal for achievements in engineering sciences in 2003.
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Ahmed M. Soliman (SM’77) was born in Cairo, Egypt, on November 22, 1943. He received the B.Sc. degree with honors from Cairo University, Cairo, Egypt, in 1964 ,the M.S. and Ph.D. degrees from the University of Pittsburgh, Pittsburgh, PA., in 1967 and 1970, respectively, all in electrical engineering. He is currently Professor Electronics and Communications Engineering Department, Cairo University, Egypt. From September 1997–September 2003, Dr. Soliman served as Professor and Chairman Electronics and Communications Engineering Department, Cairo University, Egypt. From 1985–1987, Dr. Soliman served as Professor and Chairman of the Electrical Engineering Department, United Arab Emirates University, and from 1987–1991 he was the Associate Dean of Engi-
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neering at the same University. He has held visiting academic appointments at San Francisco State University, Florida Atlantic University and the American University in Cairo. He was a Visiting Scholar at Bochum University, Germany (Summer 1985) and with the Technical University of Wien, Austria (Summer 1987). Dr. Soliman served as Associate Editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—REGULAR PAPERS from December 2001 to December 2003 and is Associate Editor of the Journal of Circuits, Systems and Signal Processing since January 2004. He is a Member of the Editorial Board of the IEE Proceedings Circuits, Devices and Systems and a Member of the Editorial Board of Analog Integrated Circuits and Signal Processing. In 1977, he was decorated with the First Class Science Medal, from the President of Egypt, for his services to the field of Engineering and Engineering Education.
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Fractional-Order Sinusoidal Oscillators: Design Procedure and Practical Examples Ahmed Gomaa Radwan, Ahmed S. Elwakil, Senior Member, IEEE, and Ahmed M. Soliman, Senior Member, IEEE
Abstract—Sinusoidal oscillators are known to be realized using dynamical systems of second-order or higher. Here we derive the Barhkausen condition for a linear noninteger-order (fractional-order) dynamical system to oscillate. We show that the oscillation condition and oscillation frequency of some famous integer-order sinusoidal oscillators can be obtained as special cases from general equations governing their fractional-order counterparts. Examples including fractional-order Wien oscillators, Colpitts oscillator, phase-shift oscillator and LC tank resonator are given supported by numerical and PSpice simulations. Index Terms—Fractional-order circuits, noninteger order systems, oscillators.
I. INTRODUCTION HE classical linear circuit theory upon which electronic circuits are designed today is based on integer-order differential equations which reflect the behavior of the three wellknown elements: the resistor, the capacitor and the inductor in the time domain. Via Laplace transform, integer-order algebraic equations in the complex frequency -domain are also used to describe linear dynamical systems. Accordingly, electronic circuits are traditionally classified as first-order, second-order or th-order circuits where is an integer number. The circuit order is directly proportional to the number of energy storage elements in the circuit. From a fractional calculus mathematical point of view, differential equations are not necessarily of integer-order [1], [2]. The Riemann-Liouville definition of a fractional derivative [3], [4] is given by
T
(1) . A more physical interpretation of a fractional where derivative is the Grünwald approximation given by
(2)
Manuscript received March 18, 2007; revised October 10, 2007. First published February 8, 2008; last published August 13, 2008 (projected). This paper was recommended by Associate Editor T. B. Tarim. A. G. Radwan is with the Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Cairo 12613, Egypt. A. S. Elwakil is with the Department of Electrical and Computer Engineering, University of Sharjah, Emirates (e-mail: [email protected]). A. M. Soliman is with the Department of Electronics and Communications, Faculty of Engineering, Cairo University, Cairo 12613, Egypt (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSI.2008.918196
where
is the integration step and . Applying the Laplace transform to (1), assuming zero initial conditions, yields (3) A fractance device is one whose impedance is proportional to is arbitrary. In such a device, the phase difference between the voltage across its two terminals and the current en. The special cases tering these terminals is correspond respectively to the resistor, inductor and capacitor. Four decades ago, some researchers investigated realizing a fractional-order capacitor [5], [6]. A finite element approximawas reported in [7]. This tion of the special case finite element approximation relies on the possibility of emulating a fractional-order capacitor via semi infinite self-similar RC trees. The technique was later developed by the authors of [8]–[10] for any . Finite element approximations offer a valuable tool by which the effect of a fractance device can be simulated using a standard circuit simulator, or studied experimentally. However, they do not offer a simple practical two-terminal device. Therefore, investigations of fractional-order circuits remained limited and confined mostly to simulations of special case circuits due to the non existence of a real fractance device and hence the lack of practical motivation [11], [12]. Most recently, the authors of [13], [14] reported the fabrication of a real two-terminal fractance device. The fabricated probe described in [13] and [14] is based on a metal-insulatorliquid interface and was used in [14] to realize a fractional-order differentiator circuit. Although this capacitive probe is bulky and relies on a necessary liquid interface, with more research and increasing application motivation, a better device will surely become commercially available in the near future. Should this happen, circuit designers will be faced with the challenge as to how to make use of a fractance device in constructing their application circuits particularly as the available design equations need to be generalized from the narrow integer-order subset to the more general fractional-order domain. In this paper, we focus on sinusoidal oscillators which are key building blocks. We derive the Barkhausen oscillation condition and the oscillation frequency for any linear system with 2, 3 or fractance devices. It is well-known that pure linear systems, whether integer-order or fractional-order, cannot maintain sustained oscillations. An accurate oscillator model requires the modeling differential equations to be necessarily nonlinear. It is also known that the Barkhausen oscillation condition is a necessary but insufficient condition for oscillation. For example, an oscillator might actually latch-up and never oscillate even if
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the Barkhausen condition is satisfied [15]. However, circuit designers are still accustomed to applying the Barkhausen condition to a linearized (with respect to the equilibrium point at the origin) model of their oscillator in order to derive the oscillation condition and oscillation frequency. We therefore opt to derive the general fractional-order Barkhausen oscillation conditions using stability analysis of fractional order systems [16]–[19]. A number of practical examples including fractional-order RC Wien oscillators, LC tank resonator, phase-shift oscillator and Colpitts oscillator are given in this paper. We show that these famous oscillators can still be designed to oscillate if constructed with fractance devices. We also show that fractionalorder oscillators have an advantage which may be exploited. In particular, the oscillation frequency does not only depend on the values of the reactive elements and/or but also on their fractional-order , which adds an extra degree of design freedom. Numerical and PSpice simulation results are shown.1 It is worth noting that some special case fractional-order oscillators were studied in [20] and [21]. It is also worth noting that experimental results of some fractional-order oscillators using the capacitive probe of [14] were very recently reported in [22].
Let us assume that are two roots of this equation . Using Euler’s identity and ( is a real number and solving separately for the real and imaginary parts yields
(7a) (7b) i) Assuming the system is oscillatory then must be two roots of the characteristic equation. Substiand yields the necessary condition tuting for oscillation (5a) and (5b). ii) Assuming (5a) and (5b) are satisfied, then comparing with and . Hence, it is sufthe (7) yields ficient to satisfy (5) for the system to admit sinusoidal oscillations. and can be calThe phase difference between culated as
II. OSCILLATORS WITH TWO FRACTANCE DEVICES
(8)
In this section, we consider oscillators with two fractance devices of fractional orders and . A. Theorem 1 A linear fractional-order system of the form
(4)
can admit sinusoidal oscillations if and only if there exists a value which satisfies simultaneously the two equations
(5a)
(9) where sgn . The following special cases are important ones. then (5b) can be solved to yield the oscillation fre1) If quency . Substituting this in (5a) yields the condition for oscillaand the phase differtion as ence is found to be . For the special case (secondorder system), the frequency of oscillation is then and the oscillation condition is which are the famous and well-known expressions for any lin. The phase earized second-order oscillator difference then reduces to . then (5b) can be solved2 for yielding 2) If
(5b) is the determinant of the system where coefficient matrix. Proof: Transforming (4) into the -domain, the characteristic equation of the system is obtained as
(10) the oscillation condition can then be obtained [by substituting for in (5a)] as
(6)
(11)
1Numerical simulations are carried out using a backward difference integration algorithm based on the Grünwald approximation of (2) with step size , where is the period of the sinusoid. PSpice simulations are based on the finite element approximations of [8]–[10].
2Equation
(5b) in this case has the form which can be solved using the identity .
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Fig. 1. Oscillations obtained from numerical simulations for examples (1) and (2).
and the phase difference is given by . the special case that are, respectively, given by
In , and
and . This case corresponds to a sinusoidal oscillator of order 1.5 . 3) If then , the oscillation condition and . This case corresponds to a is quadrature oscillator which is clearly not possible to realize . unless the oscillator is second-order then 4) If , the oscillation condition is and . The oscillation condition, frequency and phase difference and in this case. Similarly, are all independent of the same expressions are obtained replacing if with and interchanging with and vice versa.
5) If oscillator.
or
it is impossible to obtain an
B. Examples Example (1): Consider an oscillator with and which belongs to the above case 3. The characteristic equation of this oscillator is . Fig. 1(a) and (b) shows numerical simulaand respectively. Note tion results for for both figures while and , that respectively. Example (2): Consider an oscillator with and which belongs to the above case 4. The characteristic equation of this system is . Table I shows the calculated value of necessary to satisfy the oscillation condition and the corresponding for different values of and . Note from Table I that while this system (second-order oscillator); it can cannot oscillate for oscillate if either of them is fractional. Fig. 1(c) shows numerical
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TABLE I NECESSARY VALUE OF TO SATISFY THE OSCILLATION CONDITION FOR EXAMPLE (2) AND THE CORRESPONDING OSCILLATION FREQUENCY FOR DIFFERENT VALUES OF AND
TABLE II SUMMARY OF STABILITY CONDITIONS FOR
(12b) Hence
(13) Equation (13) contains five variables (then , (13) then becomes met) and define simulation results for where the character. istic equation in this case becomes Using the technique developed in [23], it can be shown that this characteristic equation has the 6 equivalent complex conjugate roots . A root which is visible in the -plane is one with phase rad. For these 6 pairs, is respectively and with hence the only visible pair is the last one . This pair is pure imaginary and the system is therefore oscillatory, as confirmed by Fig. 1(c). Table II summarizes the stability conditions and roots of the . characteristic equation
and
. Let must be
(14) and using (12b)
(15) (16) (17)
It is clear that the relationships between and the other variand are nonlinear and hence it is difficult ables to design for a required and . All seven variables can be diand vided into two groups; group (1) contains the set . The following progroup (2) contains the set cedure can be used if two variables of the first group are given and only one variable from the second group, most likely is known. The procedure helps create a design look-up table which and the values of simplifies the design process. The sign of and can be arbitrary chosen. it can be shown that From (5a) and (5b), isolating
The following two steps can now be followed. Step(1):- Related to group (1) variables: and are known, then (14) can be solved for 1) If . Table III is a look-up table for in the case that . A similar table can be constructed when . Knowing is found from (17). 2) If and are known, can be found from (16). Using look-up Table IV is then obtained. 3) If and are known, can be found from (17) and then is found from Table III. Step(2):- related to group (2) variables: Any known parameter from this group enables the rest to be found using the relations
(12a)
(18)
C. Simplified Design Procedure
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TABLE III LOOK-UP TABLE FOR WHEN
TABLE IV LOOK-UP TABLE FOR WHEN
Note that the special case in Table III.
yields
which is clear
1) RC Oscillators: Consider the Wien oscillator shown in Fig. 2(c) modified to include two fractional capacitors and hence described by
D. Circuit Design Examples In what follows, and for the purpose of performing PSpice simulations, we shall use the circuit in Fig. 2(a), proposed in [8], to simulate a fractional capacitor of order 0.5 while the circuit in Fig. 2(b), proposed in [10], shall be used to simulate a fractional capacitor of ar. To realize for exbitrary order branches with and ample, we need [10]. A number of classical second-order oscillators, modified to contain two fractance devices of orders and , where , are presented below.
(19) where, is the saturation voltage of the operational amplifier . and the gain
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Fig. 2. Circuit realizations of a (a) fractional capacitor of order [8], (b) fractional capacitor of any order [10], (c) fractional-order Wien oscillator, (d) fractional-order negative resistor RC oscillator and (e) fractional-order LC oscillator. TABLE V DESIGN EQUATIONS FOR THE WIEN OSCILLATOR
Fig. 3. PSpice simulation of Wien oscillator with and (a) F F F . (b)
4) With
Design is based in the linear region of operation where it is necessary to find the value of needed to start oscillations. If the values of and are all known then can be easily obtained as follows: and 1) substituting for in (13) the value of is found. can be obtained and hence 2) substituting with in (12a), is found. and oscillation freIf it is required to design for a given quency , the simplified procedure explained above can then and hence be followed noting from (19) that and Tables III and IV can be used. Choosing any value for the remaining steps are as follows. 1) From Tables III and IV find the values of and . is calculated from (15) 2) Knowing and the desired is found. and hence known, (12b) can be solved for 3) With hence is found.
F
k , and
and
known, (12a) can be solved for hence the last element is found. Steps 3 and 4 can be considerably simplified by noting that ; i.e., and can be found directly knowing and . Table V summarizes design equations for some important special cases of this oscillator. Note that the case in row 2 is the one reported in [20]. Fig. 3 shows two different cases simulated in PSpice using a generic TL082 opamp. Note from the second row in Table V that the oscillation frequency depends not only on and but also on and can be made since . much higher than the classical Next consider the famous RC oscillator shown in Fig. 2(d). In the opamp linear region, this oscillator is described by
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(20)
RADWAN et al.: FRACTIONAL-ORDER SINUSOIDAL OSCILLATORS: DESIGN PROCEDURE AND PRACTICAL EXAMPLES
Fig. 5. PSpice simulation for the LC tank oscillator with k and F k
Fig. 4. PSpice simulation for the oscillator in Fig. 2(d) with k (a) F (b) F F .
F
and
Following a procedure similar to the one described for the Wien oscillator, we may also design this fractional-order osciland that if lator. Note here that otherwise . Table V can also be used to design this oscillator applying the minor transformations and noting that is positive . Fig. 4 shows two different cases of circuit simulation using PSpice. 2) LC Oscillator: An active LC oscillator with fractionalorder inductor and capacitor is shown in Fig. 2(e). The opamp and associated resistors form a negative resistor and the oscillator is described by
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.
In the linear region of operation, it is seen that and hence the condition for stability is . It can be shown in the spethat the condition for oscillation is cial case , which reduces to the famous condition when . The oscillation frequency is , which also reduces to the at well-known formula . The phase difference between and is , which becomes frequency independent and tends to as . In the general case where , and where the values of and are given, simplified design steps for a desired oscillation frequency can be summarized as follows: calculate and hence . 1) let 2) let ; then find by solving the equation . Finding , the value of directly results. 3) the condition for oscillation is then . PSpice simulation results are shown in Fig. 5, where the fractional inductor, which has impedance was implemented using a standard gyrator [24] and a fractional capacitor.
III. OSCILLATORS WITH THREE FRACTANCE DEVICES The state space presentation of a linear system with three fracand is given by: tance devices of orders
(21) where are as given by (19) with is the inductor internal parasitic resistance.
and
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(22)
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A. Theorem 2 The above system can sustain sinusoidal oscillations if there exists a value for to satisfy the following equation:
the two conditions sary. Substituting for oscillation as
and are necesin (26a) yields the condition for
(28) (23) 2) If and its counterpart equation obtained by replacing every term in (23) with a term and removing the last term. Here, is the determinant of the 3 3 matrix, and . Proof: transforming (23) into the -domain, the characteristic equation of the system is found to be
(24) To achieve oscillatory behavior must satisfy the characteristic equation. Using Euler’s identity and solving separately for the real and imaginary parts yields (23) and its counterpart. The transfer function is given by
(third-order system), (27) reduces to and (28) reduces to .
B. Examples Example (1): Consider a system described by (22) with and . It can be easily shown that and . Hence, applying the equations above, and the frequency the oscillation condition is is . Fig. 6(a) and (b) shows numerical simulation results for two different sets of fractional orders with . Example (2): Consider a system described by (22) with and which results in and . The oscillation condition and frequency, respectively, are (29a)
(25) from which the phase difference can be easily found. Similar relations for and can be derived. The following special cases are important ones. 1) If (23) and its counterpart reduce to
(29b) Fig. 6(c) and (d) show numerical simulations for the two difand . ferent cases C. Circuit Design Examples
(26a)
(26b) where . Solving (26b) for
and yields
(27) to be real, the condition must hold. In addition, for a positive , the conditions if and if are necessary. For ,
For
Few oscillators with three energy storage elements are practically known. These include the famous RC phase-shift and Twin-T oscillators and the famous LC Colpitts and Hartley oscillators. Two of these oscillators modified to include fractance devices are discussed below, while the other two were studied in [22]. 1) Phase-Shift Oscillator: The phase shift oscillator circuit is shown in Fig. 7(a) and can be described by
where , matrix the bottom of the next page, and replacing with It is easy to show that
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is as given by (30), shown at are as given by (19) after .
.
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Fig. 6. Numerical simulation results for examples (1) and (2) of a system with three fractance devices.
Also let
, and . Using (23) and its counterpart equation yields
(31) which has a solution only if all terms are positive and . Given and , (31) can be solved for the oscillation frequency . Substituting with the obtained in (23) yields the oscillation condition. The following are important special cases.
(30)
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Fig. 7. Oscillators with three fractance devices. (a) phase-shift oscillator, (b) Colpitts oscillator, and its (c) small-signal equivalent model.
1)
: equation (31) may be solved3 to give (32), shown at the bottom of the page. For to be real and positive, two conditions must hold: (i) and (ii) . Substituting with oscillation condition
in (23) yields the
(33) and : the oscillation frequency and oscillation condition are, respectively, as shown in (34a) and (34b), at the bottom of the page. 3) and : this case corresponds to the classical third-order phase-shift oscillator whose oscillation frequency and con2)
3using
the identity
.
Fig.
8. PSpice F
simulations F
for the phase-shift oscillator k and (a) F , and (b) F F
with F .
dition are well-known to be and , as confrimed by (34). Fig. 8 shows PSpice simulations of the phase-shift oscillator for two different sets of fractional orders.
(32)
(34a)
(34b)
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2) Colpitts Oscillator: Fig. 7(b) shows a fractional-order Colpitts oscillator. Under small-signal conditions, the equivalent circuit in Fig. 7(c) can be drawn and is described by the following equations:
(35)
where and
is the bipolar transistor small-signal transconductance . Hence, it is seen that and . Applying (23), the condition for oscillation is found to be that of (36), shown at the bottom of the page, and the oscillation frequency is then obtained by solving for the equation
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IV. OSCILLATORS WITH
FRACTANCE DEVICES
The general state space representation of a linear system with fractance devices is
.. .
.. .
.. .
.. .
.. .
.. .
.. .
(38)
A. Theorem 3 The above system can sustain sinusoidal oscillations if there exists a value for to satisfy (39), shown at the bottom of the page, and its counterpart equation obtained by replacing every term with a term and removing the last term. In the special case that , (39) simplifies to
(37) The classical third-order Colpitts oscillator corresponds to setting which results in the famous oscillaand oscillation frequency tion condition where as confirmed by (36) and (37).
(40)
(36)
(39)
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The only application where an oscillator with fractance devices may be needed is that of an -phase oscillator.
V. CONCLUSION In this work, we have generalized the classical marginal stability sinusoidal oscillator design equations to the case where fractional-order elements are used. Several design examples were given. PSpice simulations were based on the finite element fractional capacitor approximation methods proposed in [8] and [10]. We believe the work presented here will prove valuable to circuit designers once a simple to use fractance device becomes commercially available [14]. We emphasize, in particular, the possibility of obtaining very high oscillation frequencies via adjusting the fractional order independent of the values of or .We have also generalized recently the classical first-order filters to the fractional-order domain [25].
REFERENCES [1] K. B. Oldham and J. Spanier, Fractional Calculus. New York: Academic, 1974. [2] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Application. New York: Gordon & Breach, 1987. [3] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley, 1993. [4] T. T. Hartley and C. F. Lorenzo, “Initialization, conceptualization, and application in the generalized fractional calculus,” National Aeronautics and Space Administration (NASA/TP-1998-208415) 1998. [5] G. Carlson and C. Halijak, “Approximation of fractional capacitors by a regular Newton process,” IEEE Trans. Circuits Syst., vol. CAS-11, no. 2, pp. 210–213, Mar. 1964. [6] S. Roy, “On the realization of a constant-argument immitance or fractional operator,” IEEE Trans. Circuits Syst., vol. CAS-14, no. 2, pp. 264–274, Mar. 1967. ,” IEEE Trans. [7] K. Steiglitz, “An RC impedance approximation to Circuits Syst., vol. 11, pp. 160–161, 1964. [8] M. Nakagawa and K. Sorimachi, “Basic characteristics of a fractance device,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci., vol. E75, no. 12, pp. 1814–1819, 1992. [9] K. Saito and M. Sugi, “Simulation of power-law relaxations by analog circuits: Fractal distribution of relaxation times and noninteger exponents,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci., vol. E76, no. 2, pp. 205–209, 1993. [10] M. Sugi, Y. Hirano, Y. F. Miura, and K. Saito, “Simulation of fractal immittance by analog circuits: An approach to the optimized circuits,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci., vol. E82, no. 8, pp. 1627–1634, 1999. [11] A. Abbisso, R. Caponetto, L. Fortuna, and D. Porto, “Noninteger-order integration by using neural networks,” in Proc. Int. Symp. Circuits Syst. ISCAS’01, Sydney, Australia, 2001, vol. 3, pp. 688–691. [12] A. Arena, R. Caponetto, L. Fortuna, and D. Porto, “Nonlinear noninteger order circuits and systems,” World Scientific Ser. Nonlinear Sci., ser. A, vol. 38, 2002. [13] K. Biswas, S. Sen, and P. Dutta, “Modelling of a capacitive probe in a polarizable medium,” Sens. Actuat. Phys., vol. 120, pp. 115–122, 2005. [14] K. Biswas, S. Sen, and P. Dutta, “Realization of a constant phase element and its performance study in a differentiator circuits,” IEEE Circuits Syst. II, Exp. Briefs, vol. 53, no. 8, pp. 802–806, Aug. 2006. [15] A. S. Elwakil and W. Ahmed W., “On the necessary and sufficient conditions for latch-up in sinusoidal oscillators,” Int. J. Electron., vol. 89, pp. 197–206, 2002.
[16] D. Matignon, “Stability results in fractional differential equations with applications to control processing,” in Proc. Multiconf. Comput. Engi. Syst. Appl. IMICS, IEEE-SMC, Nice, France, 1996, vol. 2, pp. 963–968. [17] C. Bonnet and J. Partington, “Coprime factorizations and stability of fractional differential equations,” Syst. Contr. Lett., vol. 41, pp. 167–174, 2000. [18] K. Diethelm and N. Ford, “Analysis of fractional differential equations,” J. Math. Anal. Appl., vol. 265, pp. 229–248, 2002. [19] R. El-Khazali and S. Al-Momani, “Stability analysis of composite fractional systems,” Int. J. Appl. Math., vol. 12, pp. 73–85, 2003. [20] W. Ahmad, R. El Khazali, and A. S. Elwakil, “Fractional-order Wienbridge oscillator,” Electron. Lett., vol. 37, no. 18, pp. 1110–1112, 2001. [21] R. Meilanov, “Features of the phase trajectory of a fractal oscillator,” Tech. Phys. Lett., vol. 28, pp. 30–32, 2002. [22] A. G. Radwan, A. M. Soliman, and A. S. Elwakil, “Design equations for fractional-order sinusoidal oscillators: Four practical design examples,” Int. J. Circuit Theory Appl., vol. 36, pp. 473–492, Jun. 2008. [23] A. G. Radwan, A. M. Soliman, A. S. Elwakil, and A. Sedeek, “On the stability of linear systems with fractional order elements,” Chaos Solitons Fractals, (in press) (available on line doi:10.1016/j.chaos.2007.10. 033). [24] A. Budak, Passive and Active Network Analysis and Synthesis. Chicago, IL: Waveland , 1991. [25] A. G. Radwan, A. M. Soliman, and A. S. Elwakil, “First-order filters generalized to the fractional domain,” J. Circuits Syst. Comput., vol. 17, pp. 55–66, Feb. 2008.
Ahmed Gomaa Radwan received the B.Sc. degree (with honours) in electronics and communications, the M.Sc. degree (for his thesis entitled “New MOS realizations of some chaotic equations using mathematical transformations”), and the Ph.D. degree (for his research on “Fractional calculus; stability, fractional-oscillators and fractional-order filters”), from Cairo University, Cairo, Egypt, in 1997, 2002, and 2006, respectively. After graduation, he joined the Department of Engineering Mathematics and Physics at Cairo University where he is now an Assistant Professor. He is the author and coauthor of 13 research papers in different scientific journals and his main interest is in chaotic systems, chaos generation, fractional calculus and applications to circuit design as well as implementation of the circuit blocks for hearing-aid systems. He received the Best Thesis Award from Cairo University for his M.Sc. thesis.
Ahmed S. Elwakil (SM’03) was born in Cairo, Egypt, in 1972. He received the B.Sc and M.Sc. degrees in electrical and electronic engineering from the Department of Electronics and Communications, Cairo University, Cairo, Egypt, and the Ph.D. degree in electrical and electronic engineering from the National University of Ireland (University College Dublin), Dublin, Ireland. He has acted as an Instructor for several courses on VLSI organized by the United Nations University for developing nations. His main research interests are in the area of analog integrated circuits with particular emphasis on nonlinear circuits analysis and design techniques , nonlinear dynamics, and chaos theory. He is author and coauthor of many publications in these areas. Dr. Elwakil is a member of the Technical Committee for Nonlinear Circuits and Systems (TCNCAS), a member of IET, and an associate member of the International Centre for Theoretical Physics (Trieste, Italy). He has served as an Organizing Committee Member and Track Chair for numerous journals and conferences. He is currently on the Editorial Board of the International Journal of Circuit Theory and Applications, and is an Associate Editor of the Journal of Dynamics of Continuous, Discrete, and Impulsive Systems, Series B: Applications and Algorithms. Dr. Elwakil received the Government of Egypt first-class medal for achievements in engineering sciences in 2003.
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RADWAN et al.: FRACTIONAL-ORDER SINUSOIDAL OSCILLATORS: DESIGN PROCEDURE AND PRACTICAL EXAMPLES
Ahmed M. Soliman (SM’77) was born in Cairo, Egypt, on November 22, 1943. He received the B.Sc. degree with honors from Cairo University, Cairo, Egypt, in 1964 ,the M.S. and Ph.D. degrees from the University of Pittsburgh, Pittsburgh, PA., in 1967 and 1970, respectively, all in electrical engineering. He is currently Professor Electronics and Communications Engineering Department, Cairo University, Egypt. From September 1997–September 2003, Dr. Soliman served as Professor and Chairman Electronics and Communications Engineering Department, Cairo University, Egypt. From 1985–1987, Dr. Soliman served as Professor and Chairman of the Electrical Engineering Department, United Arab Emirates University, and from 1987–1991 he was the Associate Dean of Engi-
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neering at the same University. He has held visiting academic appointments at San Francisco State University, Florida Atlantic University and the American University in Cairo. He was a Visiting Scholar at Bochum University, Germany (Summer 1985) and with the Technical University of Wien, Austria (Summer 1987). Dr. Soliman served as Associate Editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—REGULAR PAPERS from December 2001 to December 2003 and is Associate Editor of the Journal of Circuits, Systems and Signal Processing since January 2004. He is a Member of the Editorial Board of the IEE Proceedings Circuits, Devices and Systems and a Member of the Editorial Board of Analog Integrated Circuits and Signal Processing. In 1977, he was decorated with the First Class Science Medal, from the President of Egypt, for his services to the field of Engineering and Engineering Education.
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