Fractional Order Sinusoidal

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2052 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 7, AUGUST 2008

the Barkhausen condition is satisfied [15]. However, circuit de-

signers are still accustomed to applying the Barkhausen condi-

tion 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 derivethe 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 con-

structed with fractance devices. We also show that fractional-

order 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 oscil-

lators 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].

II. OSCILLATORS WITH TWO FRACTANCE DEVICES

In this section, we consider oscillators with two fractance de-vices 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 avalue which satisfies simultaneously the two equations

(5a)

(5b)

where is the determinant of the systemcoefficient matrix.

Proof: Transforming (4) into the -domain, the character-istic equation of the system is obtained as

(6)

1Numerical simulations are carried out using a backward difference inte-gration algorithm based on the Grünwald approximation of (2) with step size

, where is the period of the sinusoid. PSpice simulations are basedon the finite element approximations of [8]–[10].

Let us assume that are two roots of this equation( is a real number and . Using Euler’s identity andsolving separately for the real and imaginary parts yields

(7a)

(7b)

i) Assuming the system is oscillatory thenmust be two roots of the characteristic equation. Substi-tuting and yields the necessary conditionfor oscillation (5a) and (5b).

ii) Assuming (5a) and (5b) are satisfied, then comparing withthe (7) yields and . Hence, it is suf-

ficient to satisfy (5) for the system to admit sinusoidaloscillations.The phase difference between and can be cal-

culated as

(8)

(9)

where sgn . The following special cases are importantones.

1) If then(5b) canbe solved toyieldthe oscillation fre-

quency .Substituting this in (5a) yields the condition for oscilla-tion as and the phase differ-ence is found to be

. For the special case (second-order system), the frequency of oscillation is then

and the o scillation condition is whichare the famous and well-known expressions for any lin-earized second-order oscillator . The phasedifference then reduces to .

2) If then (5b) can be solved2 for yielding

(10)the oscillation condition can then be obtained [by substi-tuting for in (5a)] as

(11)

2Equation (5b) in this case has the formwhich can be solved using the identity

.

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RADWAN et al.: F RACTIONA L-O RDER SINUS OIDAL OSC IL LATORS: DESIGN PROCE DURE AND PRACT IC AL E XAMPL ES 2 05 3

Fig. 1. Oscillations obtained from numerical simulations for examples (1) and (2).

and the phase difference is given by. In

the special case that , andare, respectively, given by

and

. Thiscase corresponds to a sinusoidal oscillator of order 1.5

.

3) If then , the oscillation conditionis and . This case corresponds to aquadrature oscillator which is clearly not possible to realizeunless the oscillator is second-order .

4) If then, the oscillation condition is

and .

The oscillation condition, frequency and phase differenceare all independent of and in this case. Similarly,if the same expressions are obtained replacing

with and interchanging with and vice versa.

5) If or it is impossible to obtain anoscillator.

B. Examples

Example (1): Consider an oscillator withand which belongs to the above

case 3. The characteristic equation of this oscillator is. Fig. 1(a) and (b) shows numerical simula-

tion results for and respectively. Notethat for both figures while and ,respectively.

Example (2): Consider an oscillator withand 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 differentvalues of and . Note from Table I that while this systemcannot oscillate for (second-order oscillator); it canoscillate if either of them is fractional. Fig. 1(c) shows numerical





TABLE INECESSARY VALUE OF TO SATISFY THE OSCILLATION CONDITION FOR EXAMPLE (2) AND THE

CORRESPONDING OSCILLATION FREQUENCY FOR DIFFERENT VALUES OF AND

TABLE IISUMMARY OF STABILITY CONDITIONS FOR

simulation results for where the character-

istic equation in this case becomes .Using the technique developed in [23], it can be shown thatthis characteristic equation has the 6 equivalent complex con- jugate roots

.A root which is visible in the -plane is one with phase

rad. For these 6 pairs, is respectivelyand

hence the only visible pair is the last one with. 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 .

C. Simplified Design Procedure

It is clear that the relationships between and the other vari-ables and are n onlinear and h ence i t is d ifficultto design for a required and . All seven variables can be di-vided into two groups; group (1) contains the set andgroup (2) contains the set . The following pro-cedure can be used if two variables of the first group are givenand only one variable from the second group, most likely isknown. The procedure helps create a design look-up table whichsimplifies the design process. The sign of and the values of

and can be arbitrary chosen.

From (5a) and (5b), isolating it can be shown that

(12a)

(12b)

Hence

(13)

Equation (13) contains five variables and . Let(then must be

met) and define , (13) then becomes

(14)

and using (12b)

(15)

(16)

(17)

The following two steps can now be followed.Step(1):- Related to group (1) variables:

1) If and are known, then (14) can be solved for. 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). Usinglook-up Table IV is then obtained.

3) If and are known, can be found from (17) and thenis 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

(18)





TABLE IIILOOK-UP TABLE FOR WHEN

TABLE IVLOOK-UP TABLE FOR WHEN

Note that the special case yields which is clearin Table III.

D. Circuit Design Examples

In what follows, and for the purpose of performing PSpicesimulations, 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), proposedin [10], shall be used to simulate a fractional capacitor of ar-bitrary order . To realize for ex-

ample, we need branches with and[10]. A number of classical second-order

oscillators, modified to contain two fractance devices of ordersand , where , are presented below.

1) RC Oscillators: Consider the Wien oscillator shown inFig. 2(c) modified to include two fractional capacitors and hencedescribed by

(19)

where, is the saturation voltage of the operational amplifierand the gain .





Fig. 2. Circuit realizations of a (a) fractional capacitor of order [8],(b) fractional capacitor of any order [10], (c) fractional-order Wien oscil-lator, (d) fractional-order negative resistor RC oscillator and (e) fractional-order

LC oscillator.

TABLE VDESIGN EQUATIONS FOR THE WIEN OSCILLATOR

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:

1) substituting for andin (13) the value of is found.

2) substituting with in (12a), can be obtained and henceis found.

If it is required to design for a given and oscillation fre-quency , the simplified procedure explained above can thenbe followed noting from (19) that and henceTables III and IV can be used. Choosing any value for and

the remaining steps are as follows.1) From Tables III and IV find the values of and .

2) Knowing and the desired is calculated from (15)and hence is found.

3) With known, (12b) can be solved forhence is found.

Fig. 3. PSpice simulation of Wien oscillator with k and (a) F F , and(b) F F .

4) With and known, (12a) can be solved forhence 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 theone reported in [20]. Fig. 3 shows two different cases simulatedin PSpice using a generic TL082 opamp. Note from the secondrow in Table V that the oscillation frequencydepends not only on and but also on and can be mademuch higher than the classical since .

Next consider the famous RC oscillator shown in Fig. 2(d).In the opamp linear region, this oscillator is described by

(20)





Fig. 4. PSpice simulation for the oscillator in Fig. 2(d) withk (a) F F and

(b) F F .

Following a procedure similar to the one described for theWien oscillator, we may also design this fractional-order oscil-

lator. Note here that and that if otherwise . Table V can also be

used to design this oscillator applying the minor transformationsand noting that is positive

. Fig. 4shows two different cases of circuit simulation using PSpice.

2) LC Oscillator: An active LC oscillator with fractional-

order inductor and capacitor is shown in Fig. 2(e). The opamp

and associated resistors form a negative resistor and the oscil-

lator is described by

(21)

where are as given by (19) with and

is the inductor internal parasitic resistance.

Fig. 5. PSpice simulation for the LC tank oscillator withk F k and .

In the linear region of operation, it is seen that

and hence the condition for

stability is . It can be shown in the spe-

cial case that the condition for oscillation is

,

which reduces to the famous condition

when . The oscillation frequency is

, which also reduces to the

well-known formula at. 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:

1) let calculate and hence .

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

Thestate space presentation of a linearsystem with three frac-tance devices of orders and is given by:

(22)





A. Theorem 2

The above system can sustain sinusoidal oscillations if there

exists a value for to satisfy the following equation:

(23)

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 character-

istic 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

(25)

from which the phase difference can be easily found. Sim-

ilar relations for and can be derived.

The following special cases are important ones.

1) If (23) and its counterpart reduce to

(26a)

(26b)

where and

. Solving (26b) for yields

(27)

For to be real, the c onditionmust hold. In addition,

for a positive , the conditions if and

if are necessary. For ,

the two conditions and are neces-

sary. Substituting for in (26a) yields the condition for

oscillation as

(28)

2) If (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,

the oscillation condition is and the frequency

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, respec-

tively, are

(29a)

(29b)

Fig. 6(c) and (d) show numerical simulations for the two dif-

ferent cases and .

C. Circuit Design Examples

Few oscillators with three energy storage elements are prac-

tically known. These include the famous RC phase-shift and

Twin-T oscillators and the famous LC Colpitts and Hartley os-

cillators. Two of these oscillators modified to include fractancedevices 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 is as given by (30), shown at

the bottom of the next page, and are as given by (19) after

replacing with .

It is easy to show that

.





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 positiveand

. 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)





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 in (23) yields the

oscillation condition

(33)

2) 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 con-

3using the identity .

Fig. 8. PSpice simulations for the phase-shift oscillator withk and (a)

F F F , and (b) F F 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)





2) Colpitts Oscillator: Fig. 7(b) shows a fractional-order

Colpitts oscillator. Under small-signal conditions, the equiva-

lent circuit in Fig. 7(c) can be drawn and is described by the

following equations:

(35)

where is the bipolar transistor small-signal transconductance

and . 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

(37)

The classical third-order Colpitts oscillator corresponds to set-

ting which results in the famous oscilla-

tion condition and oscillation frequency

where as confirmed by

(36) and (37).

IV. OSCILLATORS WITH FRACTANCE DEVICES

The general state space representation of a linear system withfractance devices is

......

......

......

...(38)

A. Theorem 3

The above system can sustain sinusoidal oscillations if thereexists a value for to satisfy (39), shown at the bottom of thepage, and its counterpart equation obtained by replacing every

term with a term and removing the lastterm. In the special case that , (39)simplifies to

(40)

(36)

(39)





The only application where an oscillator with fractance de-vices may be needed is that of an -phase oscillator.

V. CONCLUSION

In this work, we have generalized the classical marginal sta-

bility 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-orderfilters to the fractional-order domain [25].

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[10] M. Sugi, Y. Hirano, Y. F. Miura, and K. Saito, “Simulation of fractal

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[14] K. Biswas, S. Sen, and P. Dutta, “Realization of a constant phase ele-

ment and its performance study in a differentiator circuits,” IEEE Cir-cuits 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

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89, pp. 197–206, 2002.

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[22] A. G. Radwan, A. M. Soliman, and A. S. Elwakil, “Design equations

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[25] A. G. Radwan, A. M. Soliman, and A. S. Elwakil, “First-order filters

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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 MOSrealizations of some chaotic equations using mathe-

matical transformations”), and the Ph.D. degree (forhis research on “Fractional calculus; stability, frac-tional-oscillators and fractional-order filters”), fromCairo University, Cairo, Egypt, in 1997, 2002, and2006, respectively.

After graduation, he joined the Department of En-gineering Mathematics and Physics at Cairo Univer-

sity where he is now an Assistant Professor. He is the author and coauthor of 13research papers in different scientific journals and his main interest is in chaoticsystems, chaos generation, fractional calculus and applications to circuit designas well as implementation of the circuit blocks for hearing-aid systems.

He received theBestThesis Award from Cairo Universityfor hisM.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 fromthe Department of Electronics and Communications,Cairo University, Cairo, Egypt, and the Ph.D. degreein electrical and electronic engineering from theNational University of Ireland (University CollegeDublin), Dublin, Ireland.

He has actedas an Instructor forseveral courses onVLSI organized by the United Nations University fordeveloping nations. His main research interests are

in the area of analog integrated circuits with particular emphasis on nonlinearcircuits 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 Circuitsand Systems (TCNCAS), a member of IET, and an associate member of theInternational Centre for Theoretical Physics (Trieste, Italy). He has served asan Organizing Committee Member and Track Chair for numerous journals andconferences. 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: Applica-

tions and Algorithms. Dr. Elwakil received the Government of Egypt first-classmedal for achievements in engineering sciences in 2003.





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 fromthe University of Pittsburgh, Pittsburgh, PA., in 1967and 1970, respectively, all in electrical engineering.

He is currently Professor Electronics and Commu-nications Engineering Department, Cairo University,Egypt. From September 1997–September 2003,Dr. Soliman served as Professor and ChairmanElectronics and Communications Engineering De-

partment, Cairo University, Egypt. From 1985–1987, Dr. Soliman served asProfessor and Chairman of the Electrical Engineering Department, United ArabEmirates University, and from 1987–1991 he was the Associate Dean of Engi-

neering at the same University. He has held visiting academic appointments atSan Francisco State University, Florida Atlantic University and the AmericanUniversity in Cairo. He was a Visiting Scholar at Bochum University, Germany(Summer 1985) and with the Technical University of Wien, Austria (Summer1987).

Dr. Soliman served as Associate Editor of the IEEE TRANSACTIONS ON

CIRCUITS AND SYSTEMS—REGULAR PAPERS from December 2001 to De-cember 2003 and is Associate Editor of the Journal of Circuits, Systems and

Signal Processing since January 2004. He is a Member of the Editorial Boardof the IEE Proceedings Circuits, Devices and Systems and a Member of theEditorial 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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