FIRST-ORDER FILTERS GENERALIZED TO THE FRACTIONAL DOMAIN

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1 Journal of ircuits, Systems, an omputers Vol. 7, No. (8) c Worl Scientific Publishing ompany FIST-ODE FILTES GENEALIZED TO THE FATIONAL DOMAIN A. G. ADWAN,,A.M.SOLIMAN, ana.s.elwakil, Department of Engineering Mathematics, Faculty of Engineering, airo University, Egypt Department of Electronics an ommunications, Faculty of Engineering, airo University, Egypt Department of Electrical an omputer Engineering, University of Sharjah, P. O. Box 77, Emirates ahmegom@yahoo.com asoliman@ieee.org elwakil@ieee.org evise 8 July 7 Traitional continuous-time filters are of integer orer. However, using fractional calculus, filters may also be represente by the more general fractional-orer ifferential equations in which case integer-orer filters are only a tight subset of fractional-orer filters. In this work, we show that low-pass, high-pass, ban-pass, an all-pass filters can be realize with circuits incorporating a single fractance evice. We erive expressions for the pole frequencies, the quality factor, the right-phase frequencies, an the halfpower frequencies. Examples of fractional passive filters supporte by numerical an PSpice simulations are given. Keywors: Analog filters; fractional calculus; fractional-orer circuits; filter esign.. Introuction Filter esign is one of the very few areas of electrical engineering for which a complete esign theory exists. Whether passive or active, filters necessarily incorporate inuctors an capacitors, the total number of which ictates the filter orer. However, an inuctor or capacitor is not but a special case of the more general so-calle fractance evice; which is an electrical element whose impeance in the complex frequency omain is given by Z(s) =as α Z(j)=a α e j(πα/). 5 For the special case of α = this element represents an inuctor while for α = itrepresents a capacitor. In the range <α<, this element may generally be consiere to represent a fractional-orer inuctor while for the range <α<, it may be consiere to represent a fractional-orer capacitor. At α =, it represents the well-known frequency-epenent negative resistor (FDN). Although a physical 55

2 56 A. G. awan, A. M. Soliman & A. S. Elwakil fractance evice oes not yet exist in the form of a single commercial evice, it may be emulate via higher-orer passive or L trees, as escribe in efs. 3 for simulation purposes. a However, very recently the authors of ef. 6 have escribe an emonstrate a single apparatus which preforms as a physical fractional-orer capacitor. It is not easy to reconstruct the apparatus escribe in ef. 6, but it is an inication that a fractance evice might soon be commercially available. Hence, it is important to generalize the filter esign theory to the fractional-orer omain. This work is a contribution in this irection. Fractional calculus is the fiel of mathematics which is concerne with the investigation an application of erivatives an integrals of arbitrary (real or complex) orer. 7 The iemann Liouville efinition of a fractional erivative of orer α is given by m t f(τ) D α Γ(m α) t f(t) := m α+ m τ m <α<m, (t τ) () m f(t) α = m. tm A more physical interpretation of a fractional erivative is given by the Grünwal Letnikov approximation b D α f(t) ( t) α m j= Γ(j α) f((m j) t), () Γ( α)γ(j +) where t is the integration step. Applying the Laplace transform is wiely use to escribe electronic circuits in the complex frequency s-omain. Hence, applying the Laplace transform to Eq. (), assuming zero initial conitions, yiels 7 L { α t f(t)} = sα F (s), (3) where α t f(t) = f (t)/t with zero initial conitions. In this work, we explore the characteristics of a filter which inclues a single fractance evice. We erive expressions for the filter center frequency an quality factor an also for the halfpower an right-phase frequencies. PSpice simulations of three passive fractionalorer filters are shown using higher-orer emulation trees. 3 Finally, impeance an frequency scaling are iscusse in the case of fractional-orer filters. It is worth noting that a fractional-orer Wien brige oscillator, which inclues two equalvalue equal-orer fractional capacitors, was stuie in ef.. a The use of a high integer-orer transfer function to emulate a fractional-orer transfer function whose orer is less than was also explaine in hap. 3 of ef. 5. This approximation is base on a Boe-plot approximation but oes not imply equivalence in the state space. b Numerical simulations in this paper are carrie out using a backwar ifference metho base on the Grünwal Letnikov approximation.

3 First-Orer Filters Generalize to the Fractional Domain 57. Single Fractional Element Filters The general transfer function of a filter with one fractional element is T (s) = bsβ + s α + a. (4) From the stability point of view, this system is stable if an only if a>anα< while it will oscillate if an only if a>anα = ; otherwise it is unstable. The location of the poles is important to etermine the filter center frequency o an its quality factor Q. From Eq. (4) it is seen that the poles in the s-plane are locate at s p = a /α e ±j(m+)π/α,m I +. The possible range of the angle in physical s-plane is θ s <πan hence there are no poles in the physical s-plane for α<. For <α<, there are only two poles locate at s, = a /α e ±j(π/α). omparing with a classical secon-orer system whose poles are locate s, = ( o /Q) ± j o (/4Q )= o e ±jδ where δ =cos ( /Q), it can be seen that o an Q are given, respectively, by o = a /α, Q = cos(π/α). (5) From the above equation, Q is negative for α<, an positive for α. At α = (classical first-orer filter), it is seen that Q =.5 as expecte. It is important now to efine the following critical frequencies () m is the frequency at which the magnitue response has a maximum or a minimum an is obtaine by solving the equation (/) T (j) =m =. () h is the half-power frequency at which the power rops to half the passban power, i.e., T (j h )=(/ ) T (j passban ). (3) rp is the right-phase frequency at which the phase T (j rp )=± π/... Fractional-orer low-pass filter onsier the fractional low-pass filter (FLPF) whose transfer function is T FLPF (s) = s α + a. (6) The magnitue an phase of this transfer function are T FLPF (j) = α +a α cos(απ/) + a, (7) T FLPF (j)= tan α sin(απ/) α cos(απ/) + a. The important critical frequencies for this FLPF are foun as m = o ( cos(απ/)) /α, rp = o /( cos(απ/)) /α,an h = o ( +cos (απ/) cos(απ/)) /α, where o is as given by Eq. (5). From these expressions it is seen that both m an rp exist only if α>, in agreement with the quality-factor expression (5). Table summarizes the magnitue an phase values at some important

4 58 A. G. awan, A. M. Soliman & A. S. Elwakil Table. Magnitue an phase values at important frequencies for the FLPF. = T FLPF (j) T FLPF (j) a o απ a cos(απ/4) 4 απ ( α)π m a sin(απ/) h tan sin(απ/) a cos(.5απ)+ p +cos (απ/) rp a cot(απ/) π frequencies. Figure (a) plots the values of m, h,an rp (normalize with respect to o ) for ifferent values of the fractional-orer α. Figure (b) is a plot of transfer function magnitue (normalize with respect to T (j) ) at ifferent critical frequencies versus α. Finally, Fig. (c) shows numerically simulate filter magnitue an phase responses for two ifferent cases α =.4 anα =.6, respectively. Note the peaking in the magnitue response at α =.6 which is classically only possible to observe in a secon-orer (α =)low-passfilter... Fractional-orer high-pass filter onsier the fractional high-pass filter (FHPF) whose transfer function is T FHPF (s) = bsα s α + a. (8) The magnitue an phase of this transfer function are T FHPF (j) = b α α +a α cos(απ/) + a, T FHPF (j)= απ + T FLPF(j). The important critical frequencies are foun as m =( a/cos(απ/)) /α, rp = ( a cos(απ/)) /α,an h = o [cos(απ/) + +cos (απ/)] /α. Figure plots the filter magnitue an phase responses for the two cases α =.4 anα =.6. (9).3. Fractional-orer ban-pass filter onsier the fractional system T FBPF (s) = bsβ s α + a, ()

5 First-Orer Filters Generalize to the Fractional Domain rp o h o m o α (a).5.5 T(j ) h T(j) T(j ) o T(j) T(j ) m T(j) T(j ) rp T(j) α (b).5 T(j).5 α =.4 Phase(T(j)) 5 5 α =.4 h 4 h 4 T(j).5.5 α =.6 Phase(T(j)) 5 5 α =.6 h 4 h 4 (c) Fig.. Numerical simulation for the FLPF representing (a) normalize critical frequencies versus α, (b) normalize magnitue response versus α, an (c) filter magnitue an phase responses at α =.4 anα =.6 assuming a = =4. whose magnitue an phase functions are T FBPF (j) = b β α +a α cos(απ/) + a, T FBPF (j)= π T FLPF(j). ()

6 6 A. G. awan, A. M. Soliman & A. S. Elwakil.5 5 T(j).5 α =.4 Phase(T(j)) 5 α =.4 4 h 4 h T(j).5.5 α =.6 Phase(T(j)) 5 5 α =.6 4 h 4 h Fig.. Magnitue an phase response of the FHPF when a =4anb =forα =.4 an α =.6, respectively. Table. Magnitue an phase values at important frequencies for the FBPF. = T FBPF (j) T FBPF (j) o ba β/α a cos(απ/4) π βπ απ 4 b (β α) (β α)π Table summarizes the magnitue an phase values at important frequencies for this system. It is seen from Table that for β<α lim T (j) =an hence the filter can be a ban-pass filter (BPF). For β = α lim T (j) = b which makes the filter a HPF. The maxima frequency m is equal to o (X) /α where X is given by X = cos(απ/)[(β α)+ α +4β(α β)tan (απ/)] (α β). ()

7 First-Orer Filters Generalize to the Fractional Domain 6 The case α =β yiels m = o. It is easy to see that there is always a maximum point in the magnitue response if α>β. Figure 3(a) shows the value of X versus α for ifferent ratios of β an α. Figure 3(b) shows the magnitue response for the filter when a = b =. Note from Fig. 3(b) that the center frequency o is not β=α X 5 4 β=.7α 3 β=.5α β=.α (a) α T(j) α = β =.3 T(j) α = β =.5.. o 4 o 4 T(j) α = β =.8 T(j) α = β =.. o 4 o 4 (b) Fig. 3. Numerical simulations for the FBPF (a) values of X versus α an (b) magnitue response for ifferent values of α an β (a = b =).

8 6 A. G. awan, A. M. Soliman & A. S. Elwakil necessarily equal to m (where the maxima occurs); which is significantly ifferent from what is known in integer-orer filters..4. Fractional-orer all-pass filter onsier the fractional all-pass filter system T APF (s) = b(sα a) s α + a. (3) The magnitue an phase of this system are, respectively, α a T FAPF (j) = b α cos(απ/) + a α +a α cos(απ/) + a, T FAPF (j)= b (α cos(απ/) a)+j α sin(απ/) ( α cos(απ/) + a)+j α sin(απ/). Table 3 summarizes the magnitue an phase values at important frequencies for this filter. It is seen here that m = rp = o an that at this frequency a minima occurs if α<anamaximaoccursifα> while the magnitue remains flat when α = (classical integer-orer all-pass filter). The half-power frequency is given by h = o [ cos(απ/) + 4cos (απ/) ] /α. Figure 4 shows the magnitue an phase responses for the two ifferent cases α =.4 anα =.6. (4) 3. PSpice Simulations Passive filters are chosen for simulations. In all simulations, we fix the fractance evice to a fractional capacitor of orer α =.4,, or.6. Figure 5(a) is the structure propose in ef. to simulate a fractional capacitor of orer.5 (Y F = F s.5 ; F = /) while the circuit in Fig. 5(b), propose in ef. 3, is use to simulate a fractional capacitor of arbitrary orer α (Y F = F s α ). To realize α =.4, for example, we nee n = 3 branches with n+ / n.5686 an n+ / n To simulate a capacitor of orer α =.6, a floating GI circuit 3 is use. The input impeance of a GI is Z i = Z Z Z 3 /Z 4 Z 5 ;takingz 3 = Z 4 =, Z = Z =/s F,anZ 4 =/s.4 F results in Z i =/s.6 F. Figures 5(c) an 5() show a passive FLPF an PSpice simulation of its magnitue response in three cases α =.4,, an.6, respectively. Figures 5(e) an 5(f) show a passive FHPF an its PSpice magnitue response while Figs. 5(g) an 5(h) Table 3. Magnitue an phase values at important frequencies for the FAPF. = T FAPF (j) T FAPF (j) b π απ π o b tan 4 b

9 First-Orer Filters Generalize to the Fractional Domain 63 T(j).5.5 α =.4 Phase(T(j)) 5 5 α =.4 4 o o 4 3 T(j).5.5 α =.6 Phase(T(j)) 5 5 α = o 4 o Fig. 4. FAPF magnitue an phase responses when α =.4 anα =.6 (a =4,b =). show a passive FAPF an its PSpice simulation results for the same three values of α. Note the peaking in the response for α> which is expecte to increase as the filter approaches α =, i.e., secon-orer filter. We have constructe the fractional low-pass filter of Fig. 5(c) in the laboratory using one of the fractional capacitors onate by the authors of ef. 6. This fractional capacitor has α.5 up to approximately 3 khz. esults are shown in Fig. 6 compare with a normal capacitor. 4. Scaling Impeance an frequency scaling can be use to ajust the filter component values or operating frequency. A fractional-orer filter is similar to an integer-orer one in terms of impeance scaling. However, for frequency scaling an assuming all critical frequencies are to be scale by a factor λ in which the new frequencies equal λ times the ol ones, then the components may be scale accoring to the following equations new = λ α or new = λ α ol. (5)

10 64 A. G. awan, A. M. Soliman & A. S. Elwakil F F p n n (a) (b) + + V i α V o (c) () α + + V i _ V o _ (e) (f) Fig. 5. (a) ealization of a fractional capacitor of orer.5, (b) realization of a fractional capacitor of orer α<, 3 (c) FLPF circuit with transfer function T (s) =/(s α + a), = a = / = 4, () PSpice simulation of the FLPF ( =6.74 kω, = F =37µF), (e) FHPF circuit with transfer function T (s) =s α /(s α + a), =, a =/ = 4, (f) PSpice simulation of the FHPF ( =6.74 kω, = F =37µF), (g) FAPF circuit with transfer function T (s) = (/)(s α a)/(s α + a), a =/ = 4, an (h) PSpice simulation of the FAPF ( =6.74 kω, = F =37µF).

11 First-Orer Filters Generalize to the Fractional Domain 65 + V i _ V o + _ α (g) (h) Fig. 5. (ontinue) NOrmal apacitance Prob apacitance Low-Pass filter Output (mv) 5 5 Frequency (Hz) Fig. 6. Experimental results of a fractional-orer low-pass filter with α.5 comparewitha normal capacitor. 5. onclusion In this work, we have generalize classical first-orer filter networks to be of fractional-orer. We have shown simulation results for filters of orer <α<

12 66 A. G. awan, A. M. Soliman & A. S. Elwakil an <α<. It is clear that more flexibility in shaping the filter response can be obtaine via a fractional-orer filter. It is also clear that the ban-pass filter, classically known to be realizable only through a secon-orer system, can actually be realize by a fractional filter of orer <α<. eferences. M. Nakagawa an K. Sorimachi, Basic characteristics of a fractance evice, IEIE Trans. Funam. Electron. ommun. omput. Sci. E75 (99) K. Saito an M. Sugi, Simulation of power-law relaxations by analog circuits: Fractal istribution of relaxation times an non-integer exponents, IEIE Trans. Funam. Electron. ommun. omput. Sci. E76 (993) M. Sugi, Y. Hirano, Y. F. Miura an K. Saito, Simulation of fractal immittance by analog circuits: An approach to the optimize circuits, IEIE Trans. Funam. Electron. ommun. omput. Sci. E8 (999) A. Abbisso,. aponetto, L. Fortuna an D. Porto, Non-integer-orer integration by using neural networks, Proc. Int. Symp. ircuits an System ISAS, Vol. 3 (), pp P. Arena,. aponetto, L. Fortuna an D. Porto, Nonlinear Noninteger Orer ircuits an Systems (Worl Scientific, ). 6. K. Biswas, S. Sen an P. Dutta, ealization of a constant phase element an its performance stuy in a ifferentiator circuit, IEEE Trans. ircuits Syst.-II 53 (6) K. B. Olham an J. Spanier, Fractional alculus (Acaemic Press, New York, 974). 8. S. G. Samko, A. A. Kilbas an O. I. Marichev, Fractional Integrals an Derivatives: Theory an Application (Goron & Breach, 987). 9. K. S. Miller an B. oss, An Introuction to the Fractional alculus an Fractional Differential Equations (John Wiley & Sons, 993).. T. T. Hartley an. F. Lorenzo, Initialization, conceptualization, an application in the generalize fractional calculus, National Aeronautics an Space Aministration (NASA/TP ) (998).. W. Ahma,. El-Khazali an A. S. Elwakil, Fractional-orer Wien-brige oscillator, Electron. Lett. 37 ().. D. Matignon, Stability results in fractional ifferential equations with applications to control processing, Proc. Multiconf. omputational Engineering in Systems an Application IMIS IEEE-SM, Vol. (996), pp A. Buak, Passive an Active Network Analysis an Synthesis (Wavelan Press Inc., Illinois, 99).