Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 726721 10 pageshttpdxdoiorg1011552013726721

Research ArticleFractional Resonance-Based 119877119871120573119862120572 Filters

Todd J Freeborn1 Brent Maundy1 and Ahmed Elwakil2

1 Department of Electrical and Computer Engineering University of Calgary 2500 University Drive NW CalgaryAB Canada T2N 1N4

2Department of Electrical and Computer Engineering University of Sharjah PO Box 27272 Sharjah UAE

Correspondence should be addressed to Todd J Freeborn toddfreeborngmailcom

Received 17 September 2012 Accepted 20 December 2012

Academic Editor Jozsef Kazmer Tar

Copyright copy 2013 Todd J Freeborn et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We propose the use of a fractional order capacitor and fractional order inductor with orders 0 le 120572 120573 le 1 respectively in afractional 119877119871120573119862120572 series circuit to realize fractional-step lowpass highpass bandpass and bandreject filters MATLAB simulationsof lowpass and highpass responses having orders of (120572+120573) = 11 15 and 19 and bandpass and bandreject responses having ordersof 15 and 19 are given as examples PSPICE simulations of 11 15 and 19 order lowpass and 10 and 14 order bandreject filtersusing approximated fractional order capacitors and fractional order inductors verify the implementations

1 Introduction

Fractional calculus the branch of mathematics concerningdifferentiations and integrations to noninteger order hasbeen steadily migrating from the theoretical realms of math-ematicians into many applied and interdisciplinary branchesof engineering [1] These concepts have been imported intomany broad fields of signal processing having many diverseapplications which include electromagnetics [2] wave prop-agation in human cancellous bone [3] state-of-charge esti-mation in batteries [4] thermal systems [5] and more Fromthe import of these concepts into electronics for analogsignal processing has emerged the field of fractional orderfilters This import into filter design has yielded much recentprogress in theory [6ndash9] noise analysis [10] stability analysis[11] and implementation [12ndash14] These filter circuits haveall been designed using the fractional Laplacian operator 119904120572because the algebraic design of transfer functions are muchsimpler than solving the difficult time domain representa-tions of fractional derivatives A fractional derivative of order120572 is given by the Caputo derivative [15] as

119862119886119863120572119905 119891 (119905) =

1

Γ (120572 minus 119899)int119905

119886

119891(119899) (120591) 119889120591

(119905 minus 120591)120572+1minus119899 (1)

where Γ(sdot) is the gamma function and 119899 minus 1 le 120572 le119899 We use the Caputo definition of a fractional derivativeover other approaches because the initial conditions for thisdefinition take the same form as the more familiar integer-order differential equations Applying the Laplace transformto the fractional derivative of (1) with lower terminal 119886 = 0yields

L 1198620119863120572

119905 119891 (119905) = 119904120572119865 (119904) minus

119899minus1

sum119896=0

119904120572minus119896minus1119891(119896) (0) (2)

where 119904120572 is also referred to as the fractional Laplacianoperator With zero initial conditions (2) can be simplifiedto

L 1198620119863120572

119905 119891 (119905) = 119904120572119865 (119904) (3)

Therefore it becomes possible to define a general fractancedevice with impedance proportional to 119904120572 [16] where thetraditional circuit elements are special cases of the generaldevice when the order is minus1 0 and 1 for a capacitor resistorand inductor respectively The expressions of the voltageacross a traditional capacitor and inductor are defined byinteger order integration and differentiation respectively ofthe current through them We can expand these devices to

2 Mathematical Problems in Engineering

the fractional domain using integrations and differentiationsof non-integer order Then the time domain expressions forthe voltage across the fractional order capacitor and fractionalorder inductor become

119907120572119862 (119905) =1

119862Γ (120572)int119905

0

119894 (120591)

(119905 minus 120591)1minus120572119889120591

119907120573119871 (119905) = 119871

119889120573119894 (119905)

119889119905120573

(4)

where 0 le 120572 120573 le 1 are the fractional orders of the capacitorand inductor respectively 119894(119905) is the current through thedevices 119862 is the capacitance with units Fs1minus120572 119871 is theinductance with units Hs1minus120573 and [s] is a unit of time notto be mistaken with the Laplacian operator Note that we willrefer to the units of these devices as [F] and [H] for simplicity

By applying the Laplace transform to (4) with zero initialconditions the impedances of these fractional order elementsare given as119885120572119862(119904) = 1119904

120572119862 and119885120573119871(119904) = 119904120573119871 for the fractional

order capacitor and fractional order inductor respectivelyUsing these fractional elements in circuits increases the rangeof responses that can be realized expanding them from thenarrow integer subset to the more general fractional domainWhile these devices are not yet commercially available recentresearch regarding their manufacture and production showsvery promising results [17 18] Therefore it is becomingincreasingly important to develop the theory behind usingthese fractional elements so that when they are available theirunique characteristics can be fully taken advantage of

While a thorough stability analysis of the fractional119877119871120573119862120572 circuit has been presented in [11] the full range offilter responses possible with this topology have not In thispaper we examine the responses possible using a fractionalorder capacitor and fractional order inductor with ordersof 0 le 120572 120573 le 1 in a series 119877119871120573119862120572 circuit to realizefractional step filters With this topology fractional lowpasshighpass bandpass and bandreject filters of order (120572 + 120573)are realized MATLAB simulations of lowpass and highpassresponses having orders of (120572 + 120573) = 11 15 and 19and bandpass and bandreject having orders of 15 and 19are presented PSPICE simulations of 11 15 and 19 orderlowpass and 10 and 14 order bandreject filters are presentedusing approximations of both fractional order capacitors andfractional order inductors to verify the 119877119871120573119862120572 circuit and itsimplementation

2 Fractional Responses

The traditional RLC circuit uses standard capacitors andinductors with which only 2nd order filter responses can berealized We can further generalize this filter to the fractionaldomain by introducing fractional orders for both frequency-dependent elements This approach of replacing traditionalcomponents with fractional components has previously beeninvestigated for fractional order capacitors in the Sallen-Key filter Kerwin-Huelsman-Newcombbiquad [7] andTow-Thomas biquad [14] The addition of the two fractionalparameters allows the 119877119871120573119862120572 circuit to realize any order

0 le 120572 + 120573 le 2 With this modification fractional lowpasshighpass bandpass and bandreject filter responses requiringonly rearrangement of the series components are realizableThe topologies to realize these four fractional order filterresponses are shown in Figure 1

21 Fractional Lowpass Filter (FLPF) The circuit shown inFigure 1(a) can be used to realize a lowpass filter responsewith a transfer function given by

119879FLPF (119904) =119881119900 (119904)

119881in (119904)=

1119871119862

119904120572+120573 + 119904120572 (119877119871) + 1119871119862 (5)

This transfer function realizes an FLPF response with DCgain of 1 high frequency gain of zero and fractional atten-uation of minus20(120572 + 120573) dBdecade in the stopband With themagnitude of (5) given as

1003816100381610038161003816119879FLPF (120596)1003816100381610038161003816

= (211987711987111986221205962120572+120573 cos (05120587120573) + 2119877119862120596120572 cos (05120572120587)

+ 2119871119862120596120573+120572 cos (05 (120573 + 120572) 120587)

+119877211986221205962120572 + 119871211986221205962120573+2120572 + 1)minus12

(6)

The MATLAB simulated magnitude responses of (5) withfractional orders of 120572 + 120573 = 11 15 and 19 with 120573 = 1119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated in Figure 2respectively From themagnitude responses of Figure 2we seestopband attenuations and 1205963dB frequencies of minus22 minus30 andminus38 dBdecade and 1623 times 10minus4 03995 and 1209 rads forthe 11 15 and 19 order FLPFs respectively This confirmsthe decreasing fractional step of the stopband attenuation asthe order (120572 + 120573) increases

The half power frequency 1205963dB can be found by numeri-cally solving the following equation

10038161003816100381610038161003816119879FLPF (1205963dB)10038161003816100381610038161003816 =

1

radic2(7)

for1205963dB Solving (7) for fixed values of120573when120572 is varied from01 to 1 in steps of 001 and 119877 = 119871 = 119862 = 1 yields the valuesgiven in Figure 3 These values show a general trend that asboth 120572 and 120573 increase the half power frequency increases Itshould be noted that all subsequentMATLAB simulations areperformed for fixed values of 120573 when 120572 is varied in steps of001 from 01 to 10 with all components fixed with values of119877 = 1Ω 119871 = 1H and 119862 = 1 F

22 Fractional Highpass Filter (FHPF) The circuit shown inFigure 1(b) can be used to realize a highpass filter responsewith a transfer function given by

119879FHPF (119904) =119881119900 (119904)

119881in (119904)=

119904120572+120573

119904120572+120573 + 119904120572 (119877119871) + 1119871119862(8)

Mathematical Problems in Engineering 3

119881in (s)+

119877 119904120573119871 119881119900 (s)

1

119904120572119862

(a)

119877

+

119904120573119871

119881119900 (s)

1

119904120572119862

119881in (s)

(b)

119877

+

119904120573119871

119881119900 (s)

1

119904120572119862

119881in (s)

(c)

119877

+

119881119900 (s)

1

119904120572119862

119881in (s)

(d)

Figure 1 Fractional 119877119871120573119862120572 topologies to realize (120572 + 120573) order (a) FLPF (b) FHPF (c) FBPF and (d) FBRF responses

minus120

minus100

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 01

120572 = 05

120572 = 09

Figure 2 Simulatedmagnitude response of (5) when120573 = 1120572 = 0105 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

10minus4

10minus3

10minus2

10minus1

100

1205963

dB(r

ads

)

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Figure 3 Half power frequencies of (5) for fixed values of 120573 when120572 is varied from 01 to 1 in steps of 001 and 119877 = 1Ω 119871 = 1H and119862 = 1 F

with DC gain of zero high frequency gain of 1 and fractionalattenuation of 20(120572 + 120573) dBdecade in the stopband Themagnitude of (8) is given as

1003816100381610038161003816119879FHPF (120596)1003816100381610038161003816

= 119871119862120596120573+120572 (211987711987111986221205962120572+120573 cos (05120587120573)

+ 2119862119877120596120572 cos (05120572120587)

+ 2119871119862120596120573+120572 cos (05 (120573 + 120572) 120587)

+119877211986221205962120572 + 119871211986221205962120573+2120572 + 1)minus12

(9)

The MATLAB simulated magnitude response of (8) withfractional orders of 120572 + 120573 = 11 15 and 19 when119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated inFigure 4 From the magnitude responses of Figure 4 we seestopband attenuations and 1205963dB frequencies of of 22 30 and38 dBdecade and 1795 1232 and 08570 rads for the 1115 and 19 order FHPFs respectively

The half power frequency 1205963dB can be found by numer-ically solving the equation |119879FHPF(1205963dB )| = 1radic2 for 1205963dB The values calculated with fixed 120573 and varied 120572 are shownin Figure 5 These values show a general trend that as both 120572and 120573 increase the half power frequency decreases

23 Fractional Bandpass Filter (FBPF) The circuit shown inFigure 1(c) can be used to realize a bandpass filter responsewith a transfer function given by

119879FBPF (119904) =119881119900 (119904)

119881in (119904)=

119904120572 (119877119871)

119904120572+120573 + 119904120572 (119877119871) + 1119871119862 (10)

This transfer function realizes an asymmetric FBPF responsewith DC and high frequency gains of zero and fractional

4 Mathematical Problems in Engineering

minus120

minus100

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 01

120572 = 05

120572 = 09

120573 = 1

Figure 4 Simulatedmagnitude response of (8) when120573 = 1120572 = 0105 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

100

101

1205963

dB(r

ads

)

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Figure 5 Half power frequencies of (8) for fixed values of 120573 when120572 is varied from 01 to 1 in steps of 001 and 119877 = 1Ω 119871 = 1H and119862 = 1 F

attenuations of 20120572 and minus20120573 dBdecade for frequencieslower and higher respectively than the maxima frequency(120596119872) With the magnitude of (10) given as

1003816100381610038161003816119879FBPF (120596)1003816100381610038161003816

= 119877119862120596120572 (211987711987111986221205962120572+120573 cos (05120587120573)

+ 2119877119862120596120572 cos (05120572120587)

+ 2119871119862120596120573+120572 cos (05 (120573 + 120572) 120587)

+119877212059621205721198622 + 119871211986221205962120573+2120572 + 1)minus12

(11)

The MATLAB simulated magnitude response of (10) withfractional orders of 120572 + 120573 = 15 and 19 when 120573 = 1119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated inFigure 6 From the simulated magnitude responses we seethat the low frequency stopband has attenuations of 10 and18 dBdecade while the high frequency stopband maintainsan attenuation of minus20 dBdecade The stopband attenuationsclosely match those predicted and confirm that the low andhigh frequency stopband attenuations are independent of

minus60

minus50

minus40

minus30

minus20

minus10

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 05

120572 = 09120573 = 1

Figure 6 Simulated magnitude response of (10) when 120573 = 1 120572 =05 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

Table 1 Maxima frequency and corresponding magnitude halfpower frequencies and quality factors of FBPFs given in Figure 6

120572 120596119872 (rads) |119879(120596119872)| (dB) 1205961 1205962 (rads) 119876

05 1082 minus4747 03173 24171 05151

09 1038 minus1246 05773 17961 08516

Table 2 Minima frequency and corresponding magnitude halfpower frequencies and quality factors of FBRFs given in Figure 10

120572 120596119898 (rads) |119879(120596119898)| (dB) 1205961 1205962 (rads) 119876

05 08974 minus7241 02145 1850 05486

09 09989 minus1735 05309 1701 08539

each other (which is unique for fractional-order bandpassfilters) with low frequency stopband attenuations dependentonly on the order of the fractional capacitor 120572 and thehigh frequency stopband only on the order of the fractionalinductor 120573

The maxima frequency can be found by numericallysolving the following equation

1198891003816100381610038161003816119879FBPF (120596)

1003816100381610038161003816119889120596

= 0 (12)

for120596 Solving (12) for varying120572 andfixed values of120573 yields themaxima frequencies given in Figure 7(a) The quality factor119876 can be found by numerically solving

1003816100381610038161003816119879FBPF (120596)1003816100381610038161003816 =

1003816100381610038161003816119879FBPF (120596119872)1003816100381610038161003816

radic2(13)

for its two real roots 1205961 and 1205962 and then evaluating 119876 =120596119872|1205961minus1205962| Solving these equations numerically for varying120572 and fixed values of 120573 yields the quality factors given inFigure 7(b) While there is no clear trend for 120596119872119876 shows anincrease with increasing order for fixed 120573 when 119877 = 1Ω 119871 =1H and 119862 = 1 F The maxima frequency (120596119872) maximummagnitude (|119879(120596119872)|) half power frequencies (12059612) andquality factors (119876) of the FBPF responses shown in Figure 6solved numerically as described previously are given inTable 1

Mathematical Problems in Engineering 5

0

02

04

06

08

1

120596119872

(rad

s)

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

(a)

0

02

04

06

08

1

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Q

(b)

Figure 7 (a) Maxima frequencies and (b) quality factors of (10) for fixed values of 120573 when 120572 is varied from 01 to 1 in steps of 001 and119877 = 1Ω 119871 = 1H 119862 = 1 F

It is possible to increase the quality factor of these circuitsby decreasing the value of 119877 for fixed order and values of 119871and 119862 The quality factors of the FBPF when 120573 = 1 119871 = 1Hand 119862 = 1 F for 120572 and 119877 varied from 01 to 1 in steps of 001are given in Figure 8 From this figure we can see that theincreasing 119876 is more pronounced with higher orders

24 Fractional Bandreject Filter (FBRF) The circuit shown inFigure 1(d) is able to realize a bandreject filter response witha transfer function given by

119879FBRF (119904) =119881119900 (119904)

119881in (119904)=

119904120572+120573 + 1119871119862

119904120572+120573 + 119904120572 (119877119871) + 1119871119862 (14)

This transfer function realizes an asymmetric FBRF responsewith DC and high frequency gains of 1 With the magnitudeof (14) given as

1003816100381610038161003816119879FBRF (120596)1003816100381610038161003816

= (119871211986221205962120573+2120572 + 2119871119862120596120573+120572 cos (05 (120572 + 120573) 120587) + 1)12

(2119877119862120596120572 cos (05120572120587) + 2119871119862120596120573+120572 cos (05 (120572 + 120573) 120587)

+119862211987721205962120572+211987711987111986221205962120572+120573 cos (05120587120573)+11987121198621205962120573+2120572+1)12

(15)

The minima frequency 120596119898 can be found by numericallysolving 119889|119879FBRF(120596)|119889120596 = 0 for 120596 Solving this equationwhen 119877 = 1Ω 119871 = 1H and 119862 = 1 F for fixed values of120573 when 120572 is varied from 01 to 1 yields the values given inFigure 9(a) while 119876 can be found by numerically solving|119879FBRF(120596)| = 1radic2 for its two real roots 1205961 and 1205962 and thenevaluating119876 = 120596119898|1205961minus1205962|The quality factor calculated forFBRFs while 120572 is varied for fixed 120573 is given in Figure 9(b)respectively again with the general trend showing that 119876increases with increasing order

Qua

lity

fact

or (Q

)

02 0406 08 1 02 04 06 08 1

100

101

102

Resistance (Ω) 120572

Figure 8 Quality factors of (10) for 120573 = 1 119871 = 1H 119862 = 1 F when 120572and 119877 are varied from 01 and 001 respectively to 1 in steps of 001

The MATLAB simulated magnitude responses of (14)with fractional orders of 120572 + 120573 = 15 and 19 when 120573 = 1119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated in Figure 10From this figure we can clearly observe the asymmetricnature of this FBRF with the attenuation of the low and highfrequency stop bands dependent on the element orders of 120572and 120573 respectively The minima frequency (120596119898) minimummagnitude (|119879(120596119898)|) half power frequencies (12059612) andquality factors (119876) of these responses are given in Table 2It is possible to increase the quality factor of the FBRF bydecreasing the value of 119877 for fixed order and values of 119871 and119862 The quality factors of the FBRF when 120573 = 1 119871 = 1H and119862 = 1 F for 120572 varied from 01 to 1 and 119877 varied from 05 to 1in steps of 001 are given in Figure 11 From this figure we cansee that like the FBPF the increasing 119876 is more pronouncedwith higher orders It should be noted though that it is notpossible to calculate a quality factor for FBRFs with bothlow order and resistance The minima peak of these filtersincreases with decreasing order and resistance and when theminima increases above 1radic(2) it is not possible to calculatethe quality factor using the previous definition

6 Mathematical Problems in Engineering

0

02

04

06

08

1

12

01 02 03 04 05 06 07 08 09120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

1205960

(rad

s)

(a)

0

01

02

03

04

05

06

01 02 03 04 05 06 07 08 09120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Q

(b)

Figure 9 (a) Minima frequencies and (b) quality factors of FBRFs for fixed values of 120573 when 120572 is varied from 01 to 1 in steps of 001 and119877 = 1Ω 119871 = 1H and 119862 = 1 F

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 05

120572 = 09

120573 = 1minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

Figure 10 Simulated magnitude response of (14) when 120573 = 1 120572 =05 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

Qua

lity

fact

or (Q

)

050

05

1

2

15

0607

08 0902 04 06 08 1

Resistance (Ω)120572

Figure 11 Quality factors of (14) for 120573 = 1 119871 = 1H119862 = 1 F when 120572and 119877 are varied from 01 and 05 respectively to 1 in steps of 001

3 Circuit Simulation Results

Although there is currently much progress regarding therealization of fractional order capacitors [17 18] there areno commercial devices using these processes available to

implement these circuits As well even though superca-pacitors have been shown to exhibit fractional impedances[19 20] their high capacitance and limited order (05 le120572 le 06) limit their usefulness in signal processing circuitsUntil commercial devices with the desired characteristicsbecome available integer order approximationsmust be usedto realize fractional circuits There are many methods tocreate an approximation of 119904120572 that include continued fractionexpansions (CFEs) aswell as rational approximationmethods[21] These methods present a large array of approximationswith the accuracy and approximated frequency band increas-ing as the order of the approximation increases Here aCFE method [22] was selected to model the fractional ordercapacitors for PSPICE simulations Collecting eight terms ofthe CFE yields a 4th order approximation of the fractionalcapacitor that can be physically realized using the RC laddernetwork in Figure 12

Now while an RC ladder can be used to approximate afractional order capacitor this same topology cannot realizea fractional order inductor as it would require negativecomponent values However we can use a fractional ordercapacitor as a component in a general impedance convertercircuit (GIC) shown in Figure 13 which is used to simulate agrounded impedance [23] Figure 13 realizes the impedance

119885 =119904120573119862119877111987731198775

1198772(16)

simulating a fractional inductor of order 120573 with inductance119871 = 1198621198771119877311987751198772

Using both the approximated fractional order capacitorand fractional order inductor we can realize the FBRFshown in Figure 1(d) with orders 120572 + 120573 = 10 and 120572 +120573 = 14 when 120572 = 05 The realized circuit is given inFigure 14 with the RC ladder shown as the inset for thefractional order capacitor and the GIC circuit as the insetfor the fractional order inductor The 119877 119871 and 119862 values

Mathematical Problems in Engineering 7

119862119890

119862119889

119862119888

119862119887

119877119886

119877119887

119877119888

119877119889

119877119890

asymp 1

119904120572119862

Figure 12 RC ladder structure to realize a 4th order integerapproximation of a fractional order capacitor

minus

minus

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

asymp

Figure 13 GIC topology to simulate a grounded fractional orderinductor using a fractional order capacitor as a component

119877119887

119877119888

119877119889

1198771198901

119904120572119862119862119890

119862119889

119862119888

119862119887

119877119886

minus

minus

+

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

Figure 14 Approximated FBRF circuit realized with RC ladderapproximation of fractional order capacitors and GIC approxima-tion of a fractional order inductor

40

48

64

72

56

80

Impe

danc

e mag

nitu

de (d

B)

Impe

danc

e pha

se (d

egre

e)

Phase

Magnitude

Frequency (Hz)10

110

210

310

410

5

minus43

minus36

minus29

minus22

minus15

minus50

Figure 15 Magnitude and phase response of the approximatedfractional order capacitor (dashed) compared to the ideal (solid)with capacitance of 126 120583F and order 05 after scaling to a centerfrequency of 1 kHz

Table 3 Component values to realize 10 and 14 order FBRFs for120572 = 05 and 120573 = 05 and 09 respectively after magnitude scaled bya factor of 1000 and frequency shifted to 1 kHz

Component Values120573 = 05 120573 = 09

119877 (Ω) 1000 1000

119871 (H) 126 0382

119862 (120583F) 126 126

required to realize these circuits after applying a magnitudescaling of 1000 and frequency scaling to 1 kHz are given inTable 3 The component values required for the 4th orderapproximation of the fractional capacitances with values of126 and 0382 120583F and orders of 05 and 09 respectivelyusing the RC ladder network in Figure 12 shifted to a centerfrequency of 1 kHz are given in Table 4 The magnitude andphase of the ideal (solid line) and 4th order approximated(dashed) fractional order capacitor with capacitance 126 120583Fand order 120572 = 05 shifted to a center frequency of 1 kHzare presented in Figure 15 From this figure we observe thatthe approximation is very good over almost 4 decades from200Hz to 70 kHz for the magnitude and almost 2 decadesfrom 200Hz to 6 kHz for the phase In these regions thedeviation of the approximation from ideal does not exceed123 dB and 023∘ for the magnitude and phase respectivelyThe simulated fractional order inductors of 126 and 0382Hcan be realized using fractional capacitances of 126 and0382 120583F respectively when 1198771 = 1198772 = 1198773 = 1198775 = 1000Ω

Using the component values in Tables 3 and 4 theapproximated FBRF shown in Figure 14 was simulated inPSPICE using MC1458 op amps to realize responses of order(120572 + 120573) = 10 and 14 when 120572 = 05 and 120573 = 05 and 09The PSPICE simulated magnitude responses (dashed lines)compared to the ideal responses (solid lines) are shown inFigure 16 We can clearly see from the magnitude response ofboth theMATLAB and PSPICE simulated responses that thisfilter does realize a fractional band reject response and thatthe simulation shows good agreement with the ideal responseobserving both symmetric and asymmetric characteristicsfor the 10 and 14 order filters respectively Note though

8 Mathematical Problems in Engineering

Table 4 Component values to realize 4th order approximations of fractional order capacitors with a center frequency of 1 kHz

ComponentValues

119862 = 1 120583F 119862 = 1 120583F 119862 = 126 120583F 119862 = 0382 120583F 119862 = 1 120583F120572 = 01 120572 = 05 120572 = 05 120572 = 09 120572 = 09

119877119886 (Ω) 2747 k 1402 k 1111 68 26119877119887 (Ω) 819 k 317 k 2517 433 165119877119888 (Ω) 561 k 478 k 3787 1307 499119877119889 (Ω) 663 k 112 k 8889 6704 2559119877119890 (Ω) 1541 k 929 k 73697 1461897 557898119862119887 (nF) 0165 664 689 705 1846119862119888 (120583F) 00015 0023 0296 113 297119862119889 (120583F) 00052 0043 0537 103 269119862119890 (120583F) 0015 0055 0695 0207 0544

0

Mag

nitu

de (d

B)

Frequency (Hz)

minus7

minus6

minus5

minus4

minus3

minus2

minus1

101

102

103

104

105

120573 = 09

120573 = 05

Figure 16 PSPICE simulations using Figure 14 compared to idealsimulations of (14) as dashed and solid lines respectively to realizeapproximated FBRFs of orders 120572 + 120573 = 10 and 14 when 120572 = 05

that the PSPICE simulations deviate from the MATLABsimulated transfer function at low and high frequencieswhich is a result of using the 4th order approximation ofthe fractional order capacitors We highlight that this filterpresents two characteristics not possible using integer orderfilters (a) bandreject responses have not been possible frominteger order filters with order less than 2 and (b) asymmetricresponses with independent control of stopband attenuationhave never been presented

31 Fractional Bruton Transformation The Bruton transfor-mation applies an impedance transformation to each elementof a passive ladder circuit to create an active circuit realizationusing the concept of frequency-dependent negative resistancethat does not require the use of inductors [24] This methodcan be expanded to the fractional-order domain to removethe fractional order inductor from the 119877119871120573119862120572 circuit Wherethe integer order transformation scales each element by 1119904the fractional Bruton transformation scales each element by1119904120573 Applying this scaling to a resistor transforms it to afractional order capacitor a fractional order inductor to aresistor and a fractional order capacitor to a new fractionalelement of order 0 le 120572 + 120573 le 1 This new fractional elementcan be a resistor capacitor or frequency-dependent negative

119881in (s)

1

119904120573119862119887 119877119887

119881119900 (s)+

1

119904120572+120573119863119887

Figure 17 FLPF of Figure 1(a) after applying the fractional Brutontransformation

minus

+

minus

+

1198772

1198775

1

1199041205731198621

1198774

asymp1

119904120572+1205731198631198871

1199041205721198623

Figure 18 GIC topology to simulate a grounded fractional elementof order 0 le 120572 + 120573 le 2 using two fractional order capacitorscomponents

resistor (FDNR) when 120572 + 120573 = 0 1 and 2 respectivelyNote that we will use the units of [Ω] when referring tothis fractional element in order to remain consistent withthe FDNR whose symbol we are also using to represent thiselementThe FLPF of Figure 1(a) after applying the fractionalBruton transformation is shown in Figure 17 The transferfunction of this transformed circuit is given by

119879FLPF (119904) =119881119900 (119904)

119881in (119904)=

1119877119887119863119887119904120572+120573 + 119904120572119877119887119862119887 + 1119877119887119863119887

(17)

where119862119887 = 1119877119877119887 = 119871 and119863119887 = 119862 for the transfer functionto be equivalent to (5)

Mathematical Problems in Engineering 9

119877119887

119877119888

119877119889

119877119890119862119890

119862119889

119862119887

119877119886

minus

minus

+

+

+

119862119888

1198772

1198775

1198774

1

1199041198621

1

119904119862119887 119877119887

11199041+120572

119863119887

119881119900 (s)

119881in (s)1119872Ω

1

1199041205721198623

Figure 19 Approximated FLPF of Figure 17 realized with RC ladder approximations of fractional order capacitors and GIC approximationof a fractional element of order (120572 + 120573)

Table 5 Component values to realize fractional elements with GICtopology of Figure 18

ComponentValues

119863119887 = 6638 nΩ 119863119887 = 201 nΩ 119863119887 = 6074 pΩOrder = 11 Order = 15 Order = 19

1198621 (120583F) 120573 1 1 1 1 1 1

1198623 (120583F) 120572 1 01 1 05 1 091198772 (Ω) 6638 k 201 k 6074

1198774 (Ω) 1000 1000 1000

1198775 (Ω) 1000 1000 1000

This new fractional element can be realized using a GICwith two fractional order capacitors this topology is shownin Figure 18 and has an impedance

119885119863 =1198775

119904120572+1205731198621119862311987721198774(18)

simulating a fractional element with impedance 119863119887 =11986211198623119877211987741198775 A GIC has also previously been employed torealize a fractional order capacitor of 36 120583F and 120572 = 16 in[7] Using the topology of Figure 17 to simulate the FLPFsof Figure 2 magnitude scaled by 1000 and frequency shiftedto 1 kHz requires 119862119887 = 01592 120583F 119877119887 = 1000Ω and 119863119887 =(6638 n 201 n 6074 p)Ω for the (120572 + 120573) = 11 15 and19 order filters respectively when 120573 = 1 The componentsrequired to simulate the impedance of the fractional elementusing theGIC topology are given in Table 5 with the values ofthe fractional order capacitor 1198623 approximated with the RCladder of Figure 12 given in Table 4

The approximated FLPF shown in Figure 19 was simu-lated in PSPICE usingMC1458 op amps to realize a (120572+120573) =19 order filter when120573 = 1The PSPICE simulatedmagnituderesponses (dashed lines) compared to the ideal responses(solid lines) are shown in Figure 20 Note that a 1MΩ resistorhas been added to bypass the capacitor and provide aDCpathto the noninverting input terminal of the upper op amp in thefractional element realization for its bias current [23]

The PSPICE simulatedmagnitude responses of the FLPFsshowvery good agreementwith theMATLAB simulated ideal

minus60

minus50

minus40

minus30

minus20

minus10

0

Mag

nitu

de (d

B)

101 10

2 103

104

Frequency (Hz)

120573 = 1

120572 = 05

120572 = 09

120572 = 01

Figure 20 PSPICE simulation using Figure 19 compared to idealsimulations of (17) as dashed and solid lines respectively to realizeapproximated FLPF of order 120572 + 120573 = 19 when 120573 = 1

response Verifying the fractional Bruton transformed FLPFsas well as the GIC realizations of a fractional element of order(120572 + 120573) using approximated fractional order capacitors Thedeviations above 20 kHz can be attributed to the approxima-tions of the fractional order capacitors and nonidealities ofthe op amps used to realize the fractional order inductors

4 Conclusion

We have proposed modifying the traditional series RLCcircuit to use a fractional order capacitor and fractionalorder inductor to realize a fractional 119877119871120573119862120572 circuit that iscapable of realizing fractional lowpass highpass bandpassand bandreject filter responses of order 0 lt 120572 + 120573 le 2requiring onlymodification of the element arrangementThistopology can realize bandpass or bandreject responses withorder less than 2 which are not possible using an integerorder circuit In addition these proposed bandpass and ban-dreject filters show asymmetric bandpass characteristics withindependent control of the stopband attenuations throughmanipulation of the fractional elements orders which is noteasily accomplished using integer order filters We have alsoshown how to realize a new fractional element of order (120572 +

10 Mathematical Problems in Engineering

120573) le 2 and a fractional order inductor using fractional ordercapacitors in a GIC circuit PSPICE simulations of FBRFs andFLPFs verify the fractional characteristics of these circuitsas well as the proposed realizations using approximatedfractional elements

Acknowledgments

T J Freeborn would like to acknowledge Canadarsquos NationalSciences and Engineering Research Council (NSERC) Al-berta Innovates-Technology Futures and Alberta AdvancedEducation and Technology for their financial support of thisresearch through their graduate student scholarships

References

[1] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[2] J A Tenreiro Machado I S Jesus A Galhano and JBoaventura Cunha ldquoFractional order electromagneticsrdquo SignalProcessing vol 86 no 10 pp 2637ndash2644 2006

[3] N Sebaa Z E A Fellah W Lauriks and C Depollier ldquoAppli-cation of fractional calculus to ultrasonic wave propagation inhuman cancellous bonerdquo Signal Processing vol 86 no 10 pp2668ndash2677 2006

[4] J Sabatier M Aoun A Oustaloup G Gregoire F Ragot and PRoy ldquoFractional system identification for lead acid battery stateof charge estimationrdquo Signal Processing vol 86 no 10 pp 2645ndash2657 2006

[5] J D Gabano and T Poinot ldquoFractional modelling and identifi-cation of thermal systemsrdquo Signal Processing vol 91 no 3 pp531ndash541 2011

[6] A G Radwan A S Elwakil and A M Soliman ldquoFractional-order sinusoidal oscillators design procedure and practicalexamplesrdquo IEEE Transactions on Circuits and Systems vol 55no 7 pp 2051ndash2063 2008

[7] A G Radwan A S Elwakil and A M Soliman ldquoOn thegeneralization of second-order filters to the fractional-orderdomainrdquo Journal of Circuits Systems and Computers vol 18 no2 pp 361ndash386 2009

[8] A Soltan A G Radwan and A M Soliman ldquoButterworthpassive filter in the fractional orderrdquo in Proceedings of theInternational Conference on Microelectronics (ICM rsquo11) pp 1ndash5Hammamet Tunisia 2011

[9] P Ahmadi BMaundy A S Elwakil and L Belostostski ldquoHigh-quality factor asymmetric-slope band-pass filters a fractional-order capacitor approachrdquo IET Circuits Devices amp Systems vol6 no 3 pp 187ndash197 2012

[10] A Lahiri and T K Rawat ldquoNoise analysis of single stagefractional-order low pass filter using stochastic and fractionalcalculusrdquo ECTI Transactions on Electronics and Communica-tions vol 7 no 2 pp 136ndash143 2009

[11] A Radwan ldquoStability analysis of the fractional-order 119877119871120573119862120572circuitrdquo Journal of Fractional Calculus and Applications vol 3no 1 pp 1ndash15 2012

[12] T J Freeborn B Maundy and A S Elwakil ldquoField pro-grammable analogue array implementations of fractional stepfiltersrdquo IET Circuits Devices amp Systems vol 4 no 6 pp 514ndash524 2010

[13] B Maundy A S Elwakil and T J Freeborn ldquoOn the practicalrealization of higher-order filters with fractional steppingrdquoSignal Processing vol 91 no 3 pp 484ndash491 2011

[14] T J Freeborn B Maundy and A S Elwakil ldquoFractional-step Tow-Thomas biquad filtersrdquo Nonlinear Theory and ItsApplications IEICE (NOLTA) vol 3 no 3 pp 357ndash374 2012

[15] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[16] M Nakagawa and K Sorimachi ldquoBasic characteristics of afractance devicerdquo IEICEmdashTransactions on Fundamentals ofElectronics Communications and Computer Sciences E vol 75pp 1814ndash1819 1992

[17] M Sivarama Krishna S Das K Biswas and B GoswamildquoFabrication of a fractional order capacitor with desired spec-ifications a study on process identification and characteriza-tionrdquo IEEE Transactions on Electron Devices vol 58 no 11 pp4067ndash4073 2011

[18] T Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons Fractals vol 24no 2 pp 479ndash490 2005

[19] J J Quintana A Ramos and I Nuez ldquoIdentification of thefractional impedance of ultracapacitorsrdquo in Proceedings of the2nd Workshop on Fractional Differentiation and Its Applications(IFAC rsquo06) pp 289ndash293 2006

[20] A Dzielinski G Sarwas and D Sierociuk ldquoComparison andvalidation of integer and fractional order ultracapacitor mod-elsrdquo Advances in Difference Equations vol 2011 article 11 2011

[21] I Podlubny I Petras B M Vinagre P OrsquoLeary and Lrsquo DorcakldquoAnalogue realizations of fractional-order controllersrdquo Nonlin-ear Dynamics An International Journal of Nonlinear Dynamicsand Chaos in Engineering Systems vol 29 no 1ndash4 pp 281ndash2962002

[22] B Krishna and K Reddy ldquoActive and passive realization offractance device of order 12rdquo Active and Passive ElectronicComponents vol 2008 Article ID 369421 5 pages 2008

[23] R Schaumann and M E Van Valkenburg Design of AnalogFilters Oxford University Press 2001

[24] L T Bruton ldquoNetwork transfer functions using the conceptof frequency-dependent negative resistancerdquo IEEE Transactionson Circuit Theory vol CT-16 no 3 pp 406ndash408 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

the fractional domain using integrations and differentiationsof non-integer order Then the time domain expressions forthe voltage across the fractional order capacitor and fractionalorder inductor become

119907120572119862 (119905) =1

119862Γ (120572)int119905

0

119894 (120591)

(119905 minus 120591)1minus120572119889120591

119907120573119871 (119905) = 119871

119889120573119894 (119905)

119889119905120573

(4)

where 0 le 120572 120573 le 1 are the fractional orders of the capacitorand inductor respectively 119894(119905) is the current through thedevices 119862 is the capacitance with units Fs1minus120572 119871 is theinductance with units Hs1minus120573 and [s] is a unit of time notto be mistaken with the Laplacian operator Note that we willrefer to the units of these devices as [F] and [H] for simplicity

By applying the Laplace transform to (4) with zero initialconditions the impedances of these fractional order elementsare given as119885120572119862(119904) = 1119904

120572119862 and119885120573119871(119904) = 119904120573119871 for the fractional

order capacitor and fractional order inductor respectivelyUsing these fractional elements in circuits increases the rangeof responses that can be realized expanding them from thenarrow integer subset to the more general fractional domainWhile these devices are not yet commercially available recentresearch regarding their manufacture and production showsvery promising results [17 18] Therefore it is becomingincreasingly important to develop the theory behind usingthese fractional elements so that when they are available theirunique characteristics can be fully taken advantage of

While a thorough stability analysis of the fractional119877119871120573119862120572 circuit has been presented in [11] the full range offilter responses possible with this topology have not In thispaper we examine the responses possible using a fractionalorder capacitor and fractional order inductor with ordersof 0 le 120572 120573 le 1 in a series 119877119871120573119862120572 circuit to realizefractional step filters With this topology fractional lowpasshighpass bandpass and bandreject filters of order (120572 + 120573)are realized MATLAB simulations of lowpass and highpassresponses having orders of (120572 + 120573) = 11 15 and 19and bandpass and bandreject having orders of 15 and 19are presented PSPICE simulations of 11 15 and 19 orderlowpass and 10 and 14 order bandreject filters are presentedusing approximations of both fractional order capacitors andfractional order inductors to verify the 119877119871120573119862120572 circuit and itsimplementation

2 Fractional Responses

The traditional RLC circuit uses standard capacitors andinductors with which only 2nd order filter responses can berealized We can further generalize this filter to the fractionaldomain by introducing fractional orders for both frequency-dependent elements This approach of replacing traditionalcomponents with fractional components has previously beeninvestigated for fractional order capacitors in the Sallen-Key filter Kerwin-Huelsman-Newcombbiquad [7] andTow-Thomas biquad [14] The addition of the two fractionalparameters allows the 119877119871120573119862120572 circuit to realize any order

0 le 120572 + 120573 le 2 With this modification fractional lowpasshighpass bandpass and bandreject filter responses requiringonly rearrangement of the series components are realizableThe topologies to realize these four fractional order filterresponses are shown in Figure 1

21 Fractional Lowpass Filter (FLPF) The circuit shown inFigure 1(a) can be used to realize a lowpass filter responsewith a transfer function given by

119879FLPF (119904) =119881119900 (119904)

119881in (119904)=

1119871119862

119904120572+120573 + 119904120572 (119877119871) + 1119871119862 (5)

This transfer function realizes an FLPF response with DCgain of 1 high frequency gain of zero and fractional atten-uation of minus20(120572 + 120573) dBdecade in the stopband With themagnitude of (5) given as

1003816100381610038161003816119879FLPF (120596)1003816100381610038161003816

= (211987711987111986221205962120572+120573 cos (05120587120573) + 2119877119862120596120572 cos (05120572120587)

+ 2119871119862120596120573+120572 cos (05 (120573 + 120572) 120587)

+119877211986221205962120572 + 119871211986221205962120573+2120572 + 1)minus12

(6)

The MATLAB simulated magnitude responses of (5) withfractional orders of 120572 + 120573 = 11 15 and 19 with 120573 = 1119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated in Figure 2respectively From themagnitude responses of Figure 2we seestopband attenuations and 1205963dB frequencies of minus22 minus30 andminus38 dBdecade and 1623 times 10minus4 03995 and 1209 rads forthe 11 15 and 19 order FLPFs respectively This confirmsthe decreasing fractional step of the stopband attenuation asthe order (120572 + 120573) increases

The half power frequency 1205963dB can be found by numeri-cally solving the following equation

10038161003816100381610038161003816119879FLPF (1205963dB)10038161003816100381610038161003816 =

1

radic2(7)

for1205963dB Solving (7) for fixed values of120573when120572 is varied from01 to 1 in steps of 001 and 119877 = 119871 = 119862 = 1 yields the valuesgiven in Figure 3 These values show a general trend that asboth 120572 and 120573 increase the half power frequency increases Itshould be noted that all subsequentMATLAB simulations areperformed for fixed values of 120573 when 120572 is varied in steps of001 from 01 to 10 with all components fixed with values of119877 = 1Ω 119871 = 1H and 119862 = 1 F

22 Fractional Highpass Filter (FHPF) The circuit shown inFigure 1(b) can be used to realize a highpass filter responsewith a transfer function given by

119879FHPF (119904) =119881119900 (119904)

119881in (119904)=

119904120572+120573

119904120572+120573 + 119904120572 (119877119871) + 1119871119862(8)

Mathematical Problems in Engineering 3

119881in (s)+

119877 119904120573119871 119881119900 (s)

1

119904120572119862

(a)

119877

+

119904120573119871

119881119900 (s)

1

119904120572119862

119881in (s)

(b)

119877

+

119904120573119871

119881119900 (s)

1

119904120572119862

119881in (s)

(c)

119877

+

119881119900 (s)

1

119904120572119862

119881in (s)

(d)

Figure 1 Fractional 119877119871120573119862120572 topologies to realize (120572 + 120573) order (a) FLPF (b) FHPF (c) FBPF and (d) FBRF responses

minus120

minus100

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 01

120572 = 05

120572 = 09

Figure 2 Simulatedmagnitude response of (5) when120573 = 1120572 = 0105 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

10minus4

10minus3

10minus2

10minus1

100

1205963

dB(r

ads

)

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Figure 3 Half power frequencies of (5) for fixed values of 120573 when120572 is varied from 01 to 1 in steps of 001 and 119877 = 1Ω 119871 = 1H and119862 = 1 F

with DC gain of zero high frequency gain of 1 and fractionalattenuation of 20(120572 + 120573) dBdecade in the stopband Themagnitude of (8) is given as

1003816100381610038161003816119879FHPF (120596)1003816100381610038161003816

= 119871119862120596120573+120572 (211987711987111986221205962120572+120573 cos (05120587120573)

+ 2119862119877120596120572 cos (05120572120587)

+ 2119871119862120596120573+120572 cos (05 (120573 + 120572) 120587)

+119877211986221205962120572 + 119871211986221205962120573+2120572 + 1)minus12

(9)

The MATLAB simulated magnitude response of (8) withfractional orders of 120572 + 120573 = 11 15 and 19 when119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated inFigure 4 From the magnitude responses of Figure 4 we seestopband attenuations and 1205963dB frequencies of of 22 30 and38 dBdecade and 1795 1232 and 08570 rads for the 1115 and 19 order FHPFs respectively

The half power frequency 1205963dB can be found by numer-ically solving the equation |119879FHPF(1205963dB )| = 1radic2 for 1205963dB The values calculated with fixed 120573 and varied 120572 are shownin Figure 5 These values show a general trend that as both 120572and 120573 increase the half power frequency decreases

23 Fractional Bandpass Filter (FBPF) The circuit shown inFigure 1(c) can be used to realize a bandpass filter responsewith a transfer function given by

119879FBPF (119904) =119881119900 (119904)

119881in (119904)=

119904120572 (119877119871)

119904120572+120573 + 119904120572 (119877119871) + 1119871119862 (10)

This transfer function realizes an asymmetric FBPF responsewith DC and high frequency gains of zero and fractional

4 Mathematical Problems in Engineering

minus120

minus100

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 01

120572 = 05

120572 = 09

120573 = 1

Figure 4 Simulatedmagnitude response of (8) when120573 = 1120572 = 0105 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

100

101

1205963

dB(r

ads

)

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Figure 5 Half power frequencies of (8) for fixed values of 120573 when120572 is varied from 01 to 1 in steps of 001 and 119877 = 1Ω 119871 = 1H and119862 = 1 F

attenuations of 20120572 and minus20120573 dBdecade for frequencieslower and higher respectively than the maxima frequency(120596119872) With the magnitude of (10) given as

1003816100381610038161003816119879FBPF (120596)1003816100381610038161003816

= 119877119862120596120572 (211987711987111986221205962120572+120573 cos (05120587120573)

+ 2119877119862120596120572 cos (05120572120587)

+ 2119871119862120596120573+120572 cos (05 (120573 + 120572) 120587)

+119877212059621205721198622 + 119871211986221205962120573+2120572 + 1)minus12

(11)

The MATLAB simulated magnitude response of (10) withfractional orders of 120572 + 120573 = 15 and 19 when 120573 = 1119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated inFigure 6 From the simulated magnitude responses we seethat the low frequency stopband has attenuations of 10 and18 dBdecade while the high frequency stopband maintainsan attenuation of minus20 dBdecade The stopband attenuationsclosely match those predicted and confirm that the low andhigh frequency stopband attenuations are independent of

minus60

minus50

minus40

minus30

minus20

minus10

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 05

120572 = 09120573 = 1

Figure 6 Simulated magnitude response of (10) when 120573 = 1 120572 =05 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

Table 1 Maxima frequency and corresponding magnitude halfpower frequencies and quality factors of FBPFs given in Figure 6

120572 120596119872 (rads) |119879(120596119872)| (dB) 1205961 1205962 (rads) 119876

05 1082 minus4747 03173 24171 05151

09 1038 minus1246 05773 17961 08516

Table 2 Minima frequency and corresponding magnitude halfpower frequencies and quality factors of FBRFs given in Figure 10

120572 120596119898 (rads) |119879(120596119898)| (dB) 1205961 1205962 (rads) 119876

05 08974 minus7241 02145 1850 05486

09 09989 minus1735 05309 1701 08539

each other (which is unique for fractional-order bandpassfilters) with low frequency stopband attenuations dependentonly on the order of the fractional capacitor 120572 and thehigh frequency stopband only on the order of the fractionalinductor 120573

The maxima frequency can be found by numericallysolving the following equation

1198891003816100381610038161003816119879FBPF (120596)

1003816100381610038161003816119889120596

= 0 (12)

for120596 Solving (12) for varying120572 andfixed values of120573 yields themaxima frequencies given in Figure 7(a) The quality factor119876 can be found by numerically solving

1003816100381610038161003816119879FBPF (120596)1003816100381610038161003816 =

1003816100381610038161003816119879FBPF (120596119872)1003816100381610038161003816

radic2(13)

for its two real roots 1205961 and 1205962 and then evaluating 119876 =120596119872|1205961minus1205962| Solving these equations numerically for varying120572 and fixed values of 120573 yields the quality factors given inFigure 7(b) While there is no clear trend for 120596119872119876 shows anincrease with increasing order for fixed 120573 when 119877 = 1Ω 119871 =1H and 119862 = 1 F The maxima frequency (120596119872) maximummagnitude (|119879(120596119872)|) half power frequencies (12059612) andquality factors (119876) of the FBPF responses shown in Figure 6solved numerically as described previously are given inTable 1

Mathematical Problems in Engineering 5

0

02

04

06

08

1

120596119872

(rad

s)

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

(a)

0

02

04

06

08

1

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Q

(b)

Figure 7 (a) Maxima frequencies and (b) quality factors of (10) for fixed values of 120573 when 120572 is varied from 01 to 1 in steps of 001 and119877 = 1Ω 119871 = 1H 119862 = 1 F

It is possible to increase the quality factor of these circuitsby decreasing the value of 119877 for fixed order and values of 119871and 119862 The quality factors of the FBPF when 120573 = 1 119871 = 1Hand 119862 = 1 F for 120572 and 119877 varied from 01 to 1 in steps of 001are given in Figure 8 From this figure we can see that theincreasing 119876 is more pronounced with higher orders

24 Fractional Bandreject Filter (FBRF) The circuit shown inFigure 1(d) is able to realize a bandreject filter response witha transfer function given by

119879FBRF (119904) =119881119900 (119904)

119881in (119904)=

119904120572+120573 + 1119871119862

119904120572+120573 + 119904120572 (119877119871) + 1119871119862 (14)

This transfer function realizes an asymmetric FBRF responsewith DC and high frequency gains of 1 With the magnitudeof (14) given as

1003816100381610038161003816119879FBRF (120596)1003816100381610038161003816

= (119871211986221205962120573+2120572 + 2119871119862120596120573+120572 cos (05 (120572 + 120573) 120587) + 1)12

(2119877119862120596120572 cos (05120572120587) + 2119871119862120596120573+120572 cos (05 (120572 + 120573) 120587)

+119862211987721205962120572+211987711987111986221205962120572+120573 cos (05120587120573)+11987121198621205962120573+2120572+1)12

(15)

The minima frequency 120596119898 can be found by numericallysolving 119889|119879FBRF(120596)|119889120596 = 0 for 120596 Solving this equationwhen 119877 = 1Ω 119871 = 1H and 119862 = 1 F for fixed values of120573 when 120572 is varied from 01 to 1 yields the values given inFigure 9(a) while 119876 can be found by numerically solving|119879FBRF(120596)| = 1radic2 for its two real roots 1205961 and 1205962 and thenevaluating119876 = 120596119898|1205961minus1205962|The quality factor calculated forFBRFs while 120572 is varied for fixed 120573 is given in Figure 9(b)respectively again with the general trend showing that 119876increases with increasing order

Qua

lity

fact

or (Q

)

02 0406 08 1 02 04 06 08 1

100

101

102

Resistance (Ω) 120572

Figure 8 Quality factors of (10) for 120573 = 1 119871 = 1H 119862 = 1 F when 120572and 119877 are varied from 01 and 001 respectively to 1 in steps of 001

The MATLAB simulated magnitude responses of (14)with fractional orders of 120572 + 120573 = 15 and 19 when 120573 = 1119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated in Figure 10From this figure we can clearly observe the asymmetricnature of this FBRF with the attenuation of the low and highfrequency stop bands dependent on the element orders of 120572and 120573 respectively The minima frequency (120596119898) minimummagnitude (|119879(120596119898)|) half power frequencies (12059612) andquality factors (119876) of these responses are given in Table 2It is possible to increase the quality factor of the FBRF bydecreasing the value of 119877 for fixed order and values of 119871 and119862 The quality factors of the FBRF when 120573 = 1 119871 = 1H and119862 = 1 F for 120572 varied from 01 to 1 and 119877 varied from 05 to 1in steps of 001 are given in Figure 11 From this figure we cansee that like the FBPF the increasing 119876 is more pronouncedwith higher orders It should be noted though that it is notpossible to calculate a quality factor for FBRFs with bothlow order and resistance The minima peak of these filtersincreases with decreasing order and resistance and when theminima increases above 1radic(2) it is not possible to calculatethe quality factor using the previous definition

6 Mathematical Problems in Engineering

0

02

04

06

08

1

12

01 02 03 04 05 06 07 08 09120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

1205960

(rad

s)

(a)

0

01

02

03

04

05

06

01 02 03 04 05 06 07 08 09120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Q

(b)

Figure 9 (a) Minima frequencies and (b) quality factors of FBRFs for fixed values of 120573 when 120572 is varied from 01 to 1 in steps of 001 and119877 = 1Ω 119871 = 1H and 119862 = 1 F

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 05

120572 = 09

120573 = 1minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

Figure 10 Simulated magnitude response of (14) when 120573 = 1 120572 =05 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

Qua

lity

fact

or (Q

)

050

05

1

2

15

0607

08 0902 04 06 08 1

Resistance (Ω)120572

Figure 11 Quality factors of (14) for 120573 = 1 119871 = 1H119862 = 1 F when 120572and 119877 are varied from 01 and 05 respectively to 1 in steps of 001

3 Circuit Simulation Results

Although there is currently much progress regarding therealization of fractional order capacitors [17 18] there areno commercial devices using these processes available to

implement these circuits As well even though superca-pacitors have been shown to exhibit fractional impedances[19 20] their high capacitance and limited order (05 le120572 le 06) limit their usefulness in signal processing circuitsUntil commercial devices with the desired characteristicsbecome available integer order approximationsmust be usedto realize fractional circuits There are many methods tocreate an approximation of 119904120572 that include continued fractionexpansions (CFEs) aswell as rational approximationmethods[21] These methods present a large array of approximationswith the accuracy and approximated frequency band increas-ing as the order of the approximation increases Here aCFE method [22] was selected to model the fractional ordercapacitors for PSPICE simulations Collecting eight terms ofthe CFE yields a 4th order approximation of the fractionalcapacitor that can be physically realized using the RC laddernetwork in Figure 12

Now while an RC ladder can be used to approximate afractional order capacitor this same topology cannot realizea fractional order inductor as it would require negativecomponent values However we can use a fractional ordercapacitor as a component in a general impedance convertercircuit (GIC) shown in Figure 13 which is used to simulate agrounded impedance [23] Figure 13 realizes the impedance

119885 =119904120573119862119877111987731198775

1198772(16)

simulating a fractional inductor of order 120573 with inductance119871 = 1198621198771119877311987751198772

Using both the approximated fractional order capacitorand fractional order inductor we can realize the FBRFshown in Figure 1(d) with orders 120572 + 120573 = 10 and 120572 +120573 = 14 when 120572 = 05 The realized circuit is given inFigure 14 with the RC ladder shown as the inset for thefractional order capacitor and the GIC circuit as the insetfor the fractional order inductor The 119877 119871 and 119862 values

Mathematical Problems in Engineering 7

119862119890

119862119889

119862119888

119862119887

119877119886

119877119887

119877119888

119877119889

119877119890

asymp 1

119904120572119862

Figure 12 RC ladder structure to realize a 4th order integerapproximation of a fractional order capacitor

minus

minus

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

asymp

Figure 13 GIC topology to simulate a grounded fractional orderinductor using a fractional order capacitor as a component

119877119887

119877119888

119877119889

1198771198901

119904120572119862119862119890

119862119889

119862119888

119862119887

119877119886

minus

minus

+

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

Figure 14 Approximated FBRF circuit realized with RC ladderapproximation of fractional order capacitors and GIC approxima-tion of a fractional order inductor

40

48

64

72

56

80

Impe

danc

e mag

nitu

de (d

B)

Impe

danc

e pha

se (d

egre

e)

Phase

Magnitude

Frequency (Hz)10

110

210

310

410

5

minus43

minus36

minus29

minus22

minus15

minus50

Figure 15 Magnitude and phase response of the approximatedfractional order capacitor (dashed) compared to the ideal (solid)with capacitance of 126 120583F and order 05 after scaling to a centerfrequency of 1 kHz

Table 3 Component values to realize 10 and 14 order FBRFs for120572 = 05 and 120573 = 05 and 09 respectively after magnitude scaled bya factor of 1000 and frequency shifted to 1 kHz

Component Values120573 = 05 120573 = 09

119877 (Ω) 1000 1000

119871 (H) 126 0382

119862 (120583F) 126 126

required to realize these circuits after applying a magnitudescaling of 1000 and frequency scaling to 1 kHz are given inTable 3 The component values required for the 4th orderapproximation of the fractional capacitances with values of126 and 0382 120583F and orders of 05 and 09 respectivelyusing the RC ladder network in Figure 12 shifted to a centerfrequency of 1 kHz are given in Table 4 The magnitude andphase of the ideal (solid line) and 4th order approximated(dashed) fractional order capacitor with capacitance 126 120583Fand order 120572 = 05 shifted to a center frequency of 1 kHzare presented in Figure 15 From this figure we observe thatthe approximation is very good over almost 4 decades from200Hz to 70 kHz for the magnitude and almost 2 decadesfrom 200Hz to 6 kHz for the phase In these regions thedeviation of the approximation from ideal does not exceed123 dB and 023∘ for the magnitude and phase respectivelyThe simulated fractional order inductors of 126 and 0382Hcan be realized using fractional capacitances of 126 and0382 120583F respectively when 1198771 = 1198772 = 1198773 = 1198775 = 1000Ω

Using the component values in Tables 3 and 4 theapproximated FBRF shown in Figure 14 was simulated inPSPICE using MC1458 op amps to realize responses of order(120572 + 120573) = 10 and 14 when 120572 = 05 and 120573 = 05 and 09The PSPICE simulated magnitude responses (dashed lines)compared to the ideal responses (solid lines) are shown inFigure 16 We can clearly see from the magnitude response ofboth theMATLAB and PSPICE simulated responses that thisfilter does realize a fractional band reject response and thatthe simulation shows good agreement with the ideal responseobserving both symmetric and asymmetric characteristicsfor the 10 and 14 order filters respectively Note though

8 Mathematical Problems in Engineering

Table 4 Component values to realize 4th order approximations of fractional order capacitors with a center frequency of 1 kHz

ComponentValues

119862 = 1 120583F 119862 = 1 120583F 119862 = 126 120583F 119862 = 0382 120583F 119862 = 1 120583F120572 = 01 120572 = 05 120572 = 05 120572 = 09 120572 = 09

119877119886 (Ω) 2747 k 1402 k 1111 68 26119877119887 (Ω) 819 k 317 k 2517 433 165119877119888 (Ω) 561 k 478 k 3787 1307 499119877119889 (Ω) 663 k 112 k 8889 6704 2559119877119890 (Ω) 1541 k 929 k 73697 1461897 557898119862119887 (nF) 0165 664 689 705 1846119862119888 (120583F) 00015 0023 0296 113 297119862119889 (120583F) 00052 0043 0537 103 269119862119890 (120583F) 0015 0055 0695 0207 0544

0

Mag

nitu

de (d

B)

Frequency (Hz)

minus7

minus6

minus5

minus4

minus3

minus2

minus1

101

102

103

104

105

120573 = 09

120573 = 05

Figure 16 PSPICE simulations using Figure 14 compared to idealsimulations of (14) as dashed and solid lines respectively to realizeapproximated FBRFs of orders 120572 + 120573 = 10 and 14 when 120572 = 05

that the PSPICE simulations deviate from the MATLABsimulated transfer function at low and high frequencieswhich is a result of using the 4th order approximation ofthe fractional order capacitors We highlight that this filterpresents two characteristics not possible using integer orderfilters (a) bandreject responses have not been possible frominteger order filters with order less than 2 and (b) asymmetricresponses with independent control of stopband attenuationhave never been presented

31 Fractional Bruton Transformation The Bruton transfor-mation applies an impedance transformation to each elementof a passive ladder circuit to create an active circuit realizationusing the concept of frequency-dependent negative resistancethat does not require the use of inductors [24] This methodcan be expanded to the fractional-order domain to removethe fractional order inductor from the 119877119871120573119862120572 circuit Wherethe integer order transformation scales each element by 1119904the fractional Bruton transformation scales each element by1119904120573 Applying this scaling to a resistor transforms it to afractional order capacitor a fractional order inductor to aresistor and a fractional order capacitor to a new fractionalelement of order 0 le 120572 + 120573 le 1 This new fractional elementcan be a resistor capacitor or frequency-dependent negative

119881in (s)

1

119904120573119862119887 119877119887

119881119900 (s)+

1

119904120572+120573119863119887

Figure 17 FLPF of Figure 1(a) after applying the fractional Brutontransformation

minus

+

minus

+

1198772

1198775

1

1199041205731198621

1198774

asymp1

119904120572+1205731198631198871

1199041205721198623

Figure 18 GIC topology to simulate a grounded fractional elementof order 0 le 120572 + 120573 le 2 using two fractional order capacitorscomponents

resistor (FDNR) when 120572 + 120573 = 0 1 and 2 respectivelyNote that we will use the units of [Ω] when referring tothis fractional element in order to remain consistent withthe FDNR whose symbol we are also using to represent thiselementThe FLPF of Figure 1(a) after applying the fractionalBruton transformation is shown in Figure 17 The transferfunction of this transformed circuit is given by

119879FLPF (119904) =119881119900 (119904)

119881in (119904)=

1119877119887119863119887119904120572+120573 + 119904120572119877119887119862119887 + 1119877119887119863119887

(17)

where119862119887 = 1119877119877119887 = 119871 and119863119887 = 119862 for the transfer functionto be equivalent to (5)

Mathematical Problems in Engineering 9

119877119887

119877119888

119877119889

119877119890119862119890

119862119889

119862119887

119877119886

minus

minus

+

+

+

119862119888

1198772

1198775

1198774

1

1199041198621

1

119904119862119887 119877119887

11199041+120572

119863119887

119881119900 (s)

119881in (s)1119872Ω

1

1199041205721198623

Figure 19 Approximated FLPF of Figure 17 realized with RC ladder approximations of fractional order capacitors and GIC approximationof a fractional element of order (120572 + 120573)

Table 5 Component values to realize fractional elements with GICtopology of Figure 18

ComponentValues

119863119887 = 6638 nΩ 119863119887 = 201 nΩ 119863119887 = 6074 pΩOrder = 11 Order = 15 Order = 19

1198621 (120583F) 120573 1 1 1 1 1 1

1198623 (120583F) 120572 1 01 1 05 1 091198772 (Ω) 6638 k 201 k 6074

1198774 (Ω) 1000 1000 1000

1198775 (Ω) 1000 1000 1000

This new fractional element can be realized using a GICwith two fractional order capacitors this topology is shownin Figure 18 and has an impedance

119885119863 =1198775

119904120572+1205731198621119862311987721198774(18)

simulating a fractional element with impedance 119863119887 =11986211198623119877211987741198775 A GIC has also previously been employed torealize a fractional order capacitor of 36 120583F and 120572 = 16 in[7] Using the topology of Figure 17 to simulate the FLPFsof Figure 2 magnitude scaled by 1000 and frequency shiftedto 1 kHz requires 119862119887 = 01592 120583F 119877119887 = 1000Ω and 119863119887 =(6638 n 201 n 6074 p)Ω for the (120572 + 120573) = 11 15 and19 order filters respectively when 120573 = 1 The componentsrequired to simulate the impedance of the fractional elementusing theGIC topology are given in Table 5 with the values ofthe fractional order capacitor 1198623 approximated with the RCladder of Figure 12 given in Table 4

The approximated FLPF shown in Figure 19 was simu-lated in PSPICE usingMC1458 op amps to realize a (120572+120573) =19 order filter when120573 = 1The PSPICE simulatedmagnituderesponses (dashed lines) compared to the ideal responses(solid lines) are shown in Figure 20 Note that a 1MΩ resistorhas been added to bypass the capacitor and provide aDCpathto the noninverting input terminal of the upper op amp in thefractional element realization for its bias current [23]

The PSPICE simulatedmagnitude responses of the FLPFsshowvery good agreementwith theMATLAB simulated ideal

minus60

minus50

minus40

minus30

minus20

minus10

0

Mag

nitu

de (d

B)

101 10

2 103

104

Frequency (Hz)

120573 = 1

120572 = 05

120572 = 09

120572 = 01

Figure 20 PSPICE simulation using Figure 19 compared to idealsimulations of (17) as dashed and solid lines respectively to realizeapproximated FLPF of order 120572 + 120573 = 19 when 120573 = 1

response Verifying the fractional Bruton transformed FLPFsas well as the GIC realizations of a fractional element of order(120572 + 120573) using approximated fractional order capacitors Thedeviations above 20 kHz can be attributed to the approxima-tions of the fractional order capacitors and nonidealities ofthe op amps used to realize the fractional order inductors

4 Conclusion

We have proposed modifying the traditional series RLCcircuit to use a fractional order capacitor and fractionalorder inductor to realize a fractional 119877119871120573119862120572 circuit that iscapable of realizing fractional lowpass highpass bandpassand bandreject filter responses of order 0 lt 120572 + 120573 le 2requiring onlymodification of the element arrangementThistopology can realize bandpass or bandreject responses withorder less than 2 which are not possible using an integerorder circuit In addition these proposed bandpass and ban-dreject filters show asymmetric bandpass characteristics withindependent control of the stopband attenuations throughmanipulation of the fractional elements orders which is noteasily accomplished using integer order filters We have alsoshown how to realize a new fractional element of order (120572 +

10 Mathematical Problems in Engineering

120573) le 2 and a fractional order inductor using fractional ordercapacitors in a GIC circuit PSPICE simulations of FBRFs andFLPFs verify the fractional characteristics of these circuitsas well as the proposed realizations using approximatedfractional elements

Acknowledgments

T J Freeborn would like to acknowledge Canadarsquos NationalSciences and Engineering Research Council (NSERC) Al-berta Innovates-Technology Futures and Alberta AdvancedEducation and Technology for their financial support of thisresearch through their graduate student scholarships

References

[1] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[2] J A Tenreiro Machado I S Jesus A Galhano and JBoaventura Cunha ldquoFractional order electromagneticsrdquo SignalProcessing vol 86 no 10 pp 2637ndash2644 2006

[3] N Sebaa Z E A Fellah W Lauriks and C Depollier ldquoAppli-cation of fractional calculus to ultrasonic wave propagation inhuman cancellous bonerdquo Signal Processing vol 86 no 10 pp2668ndash2677 2006

[4] J Sabatier M Aoun A Oustaloup G Gregoire F Ragot and PRoy ldquoFractional system identification for lead acid battery stateof charge estimationrdquo Signal Processing vol 86 no 10 pp 2645ndash2657 2006

[5] J D Gabano and T Poinot ldquoFractional modelling and identifi-cation of thermal systemsrdquo Signal Processing vol 91 no 3 pp531ndash541 2011

[6] A G Radwan A S Elwakil and A M Soliman ldquoFractional-order sinusoidal oscillators design procedure and practicalexamplesrdquo IEEE Transactions on Circuits and Systems vol 55no 7 pp 2051ndash2063 2008

[7] A G Radwan A S Elwakil and A M Soliman ldquoOn thegeneralization of second-order filters to the fractional-orderdomainrdquo Journal of Circuits Systems and Computers vol 18 no2 pp 361ndash386 2009

[8] A Soltan A G Radwan and A M Soliman ldquoButterworthpassive filter in the fractional orderrdquo in Proceedings of theInternational Conference on Microelectronics (ICM rsquo11) pp 1ndash5Hammamet Tunisia 2011

[9] P Ahmadi BMaundy A S Elwakil and L Belostostski ldquoHigh-quality factor asymmetric-slope band-pass filters a fractional-order capacitor approachrdquo IET Circuits Devices amp Systems vol6 no 3 pp 187ndash197 2012

[10] A Lahiri and T K Rawat ldquoNoise analysis of single stagefractional-order low pass filter using stochastic and fractionalcalculusrdquo ECTI Transactions on Electronics and Communica-tions vol 7 no 2 pp 136ndash143 2009

[11] A Radwan ldquoStability analysis of the fractional-order 119877119871120573119862120572circuitrdquo Journal of Fractional Calculus and Applications vol 3no 1 pp 1ndash15 2012

[12] T J Freeborn B Maundy and A S Elwakil ldquoField pro-grammable analogue array implementations of fractional stepfiltersrdquo IET Circuits Devices amp Systems vol 4 no 6 pp 514ndash524 2010

[13] B Maundy A S Elwakil and T J Freeborn ldquoOn the practicalrealization of higher-order filters with fractional steppingrdquoSignal Processing vol 91 no 3 pp 484ndash491 2011

[14] T J Freeborn B Maundy and A S Elwakil ldquoFractional-step Tow-Thomas biquad filtersrdquo Nonlinear Theory and ItsApplications IEICE (NOLTA) vol 3 no 3 pp 357ndash374 2012

[15] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[16] M Nakagawa and K Sorimachi ldquoBasic characteristics of afractance devicerdquo IEICEmdashTransactions on Fundamentals ofElectronics Communications and Computer Sciences E vol 75pp 1814ndash1819 1992

[17] M Sivarama Krishna S Das K Biswas and B GoswamildquoFabrication of a fractional order capacitor with desired spec-ifications a study on process identification and characteriza-tionrdquo IEEE Transactions on Electron Devices vol 58 no 11 pp4067ndash4073 2011

[18] T Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons Fractals vol 24no 2 pp 479ndash490 2005

[19] J J Quintana A Ramos and I Nuez ldquoIdentification of thefractional impedance of ultracapacitorsrdquo in Proceedings of the2nd Workshop on Fractional Differentiation and Its Applications(IFAC rsquo06) pp 289ndash293 2006

[20] A Dzielinski G Sarwas and D Sierociuk ldquoComparison andvalidation of integer and fractional order ultracapacitor mod-elsrdquo Advances in Difference Equations vol 2011 article 11 2011

[21] I Podlubny I Petras B M Vinagre P OrsquoLeary and Lrsquo DorcakldquoAnalogue realizations of fractional-order controllersrdquo Nonlin-ear Dynamics An International Journal of Nonlinear Dynamicsand Chaos in Engineering Systems vol 29 no 1ndash4 pp 281ndash2962002

[22] B Krishna and K Reddy ldquoActive and passive realization offractance device of order 12rdquo Active and Passive ElectronicComponents vol 2008 Article ID 369421 5 pages 2008

[23] R Schaumann and M E Van Valkenburg Design of AnalogFilters Oxford University Press 2001

[24] L T Bruton ldquoNetwork transfer functions using the conceptof frequency-dependent negative resistancerdquo IEEE Transactionson Circuit Theory vol CT-16 no 3 pp 406ndash408 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

119881in (s)+

119877 119904120573119871 119881119900 (s)

1

119904120572119862

(a)

119877

+

119904120573119871

119881119900 (s)

1

119904120572119862

119881in (s)

(b)

119877

+

119904120573119871

119881119900 (s)

1

119904120572119862

119881in (s)

(c)

119877

+

119881119900 (s)

1

119904120572119862

119881in (s)

(d)

Figure 1 Fractional 119877119871120573119862120572 topologies to realize (120572 + 120573) order (a) FLPF (b) FHPF (c) FBPF and (d) FBRF responses

minus120

minus100

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 01

120572 = 05

120572 = 09

Figure 2 Simulatedmagnitude response of (5) when120573 = 1120572 = 0105 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

10minus4

10minus3

10minus2

10minus1

100

1205963

dB(r

ads

)

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Figure 3 Half power frequencies of (5) for fixed values of 120573 when120572 is varied from 01 to 1 in steps of 001 and 119877 = 1Ω 119871 = 1H and119862 = 1 F

with DC gain of zero high frequency gain of 1 and fractionalattenuation of 20(120572 + 120573) dBdecade in the stopband Themagnitude of (8) is given as

1003816100381610038161003816119879FHPF (120596)1003816100381610038161003816

= 119871119862120596120573+120572 (211987711987111986221205962120572+120573 cos (05120587120573)

+ 2119862119877120596120572 cos (05120572120587)

+ 2119871119862120596120573+120572 cos (05 (120573 + 120572) 120587)

+119877211986221205962120572 + 119871211986221205962120573+2120572 + 1)minus12

(9)

The MATLAB simulated magnitude response of (8) withfractional orders of 120572 + 120573 = 11 15 and 19 when119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated inFigure 4 From the magnitude responses of Figure 4 we seestopband attenuations and 1205963dB frequencies of of 22 30 and38 dBdecade and 1795 1232 and 08570 rads for the 1115 and 19 order FHPFs respectively

The half power frequency 1205963dB can be found by numer-ically solving the equation |119879FHPF(1205963dB )| = 1radic2 for 1205963dB The values calculated with fixed 120573 and varied 120572 are shownin Figure 5 These values show a general trend that as both 120572and 120573 increase the half power frequency decreases

23 Fractional Bandpass Filter (FBPF) The circuit shown inFigure 1(c) can be used to realize a bandpass filter responsewith a transfer function given by

119879FBPF (119904) =119881119900 (119904)

119881in (119904)=

119904120572 (119877119871)

119904120572+120573 + 119904120572 (119877119871) + 1119871119862 (10)

This transfer function realizes an asymmetric FBPF responsewith DC and high frequency gains of zero and fractional

4 Mathematical Problems in Engineering

minus120

minus100

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 01

120572 = 05

120572 = 09

120573 = 1

Figure 4 Simulatedmagnitude response of (8) when120573 = 1120572 = 0105 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

100

101

1205963

dB(r

ads

)

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Figure 5 Half power frequencies of (8) for fixed values of 120573 when120572 is varied from 01 to 1 in steps of 001 and 119877 = 1Ω 119871 = 1H and119862 = 1 F

attenuations of 20120572 and minus20120573 dBdecade for frequencieslower and higher respectively than the maxima frequency(120596119872) With the magnitude of (10) given as

1003816100381610038161003816119879FBPF (120596)1003816100381610038161003816

= 119877119862120596120572 (211987711987111986221205962120572+120573 cos (05120587120573)

+ 2119877119862120596120572 cos (05120572120587)

+ 2119871119862120596120573+120572 cos (05 (120573 + 120572) 120587)

+119877212059621205721198622 + 119871211986221205962120573+2120572 + 1)minus12

(11)

The MATLAB simulated magnitude response of (10) withfractional orders of 120572 + 120573 = 15 and 19 when 120573 = 1119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated inFigure 6 From the simulated magnitude responses we seethat the low frequency stopband has attenuations of 10 and18 dBdecade while the high frequency stopband maintainsan attenuation of minus20 dBdecade The stopband attenuationsclosely match those predicted and confirm that the low andhigh frequency stopband attenuations are independent of

minus60

minus50

minus40

minus30

minus20

minus10

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 05

120572 = 09120573 = 1

Figure 6 Simulated magnitude response of (10) when 120573 = 1 120572 =05 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

Table 1 Maxima frequency and corresponding magnitude halfpower frequencies and quality factors of FBPFs given in Figure 6

120572 120596119872 (rads) |119879(120596119872)| (dB) 1205961 1205962 (rads) 119876

05 1082 minus4747 03173 24171 05151

09 1038 minus1246 05773 17961 08516

Table 2 Minima frequency and corresponding magnitude halfpower frequencies and quality factors of FBRFs given in Figure 10

120572 120596119898 (rads) |119879(120596119898)| (dB) 1205961 1205962 (rads) 119876

05 08974 minus7241 02145 1850 05486

09 09989 minus1735 05309 1701 08539

each other (which is unique for fractional-order bandpassfilters) with low frequency stopband attenuations dependentonly on the order of the fractional capacitor 120572 and thehigh frequency stopband only on the order of the fractionalinductor 120573

The maxima frequency can be found by numericallysolving the following equation

1198891003816100381610038161003816119879FBPF (120596)

1003816100381610038161003816119889120596

= 0 (12)

for120596 Solving (12) for varying120572 andfixed values of120573 yields themaxima frequencies given in Figure 7(a) The quality factor119876 can be found by numerically solving

1003816100381610038161003816119879FBPF (120596)1003816100381610038161003816 =

1003816100381610038161003816119879FBPF (120596119872)1003816100381610038161003816

radic2(13)

for its two real roots 1205961 and 1205962 and then evaluating 119876 =120596119872|1205961minus1205962| Solving these equations numerically for varying120572 and fixed values of 120573 yields the quality factors given inFigure 7(b) While there is no clear trend for 120596119872119876 shows anincrease with increasing order for fixed 120573 when 119877 = 1Ω 119871 =1H and 119862 = 1 F The maxima frequency (120596119872) maximummagnitude (|119879(120596119872)|) half power frequencies (12059612) andquality factors (119876) of the FBPF responses shown in Figure 6solved numerically as described previously are given inTable 1

Mathematical Problems in Engineering 5

0

02

04

06

08

1

120596119872

(rad

s)

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

(a)

0

02

04

06

08

1

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Q

(b)

Figure 7 (a) Maxima frequencies and (b) quality factors of (10) for fixed values of 120573 when 120572 is varied from 01 to 1 in steps of 001 and119877 = 1Ω 119871 = 1H 119862 = 1 F

It is possible to increase the quality factor of these circuitsby decreasing the value of 119877 for fixed order and values of 119871and 119862 The quality factors of the FBPF when 120573 = 1 119871 = 1Hand 119862 = 1 F for 120572 and 119877 varied from 01 to 1 in steps of 001are given in Figure 8 From this figure we can see that theincreasing 119876 is more pronounced with higher orders

24 Fractional Bandreject Filter (FBRF) The circuit shown inFigure 1(d) is able to realize a bandreject filter response witha transfer function given by

119879FBRF (119904) =119881119900 (119904)

119881in (119904)=

119904120572+120573 + 1119871119862

119904120572+120573 + 119904120572 (119877119871) + 1119871119862 (14)

This transfer function realizes an asymmetric FBRF responsewith DC and high frequency gains of 1 With the magnitudeof (14) given as

1003816100381610038161003816119879FBRF (120596)1003816100381610038161003816

= (119871211986221205962120573+2120572 + 2119871119862120596120573+120572 cos (05 (120572 + 120573) 120587) + 1)12

(2119877119862120596120572 cos (05120572120587) + 2119871119862120596120573+120572 cos (05 (120572 + 120573) 120587)

+119862211987721205962120572+211987711987111986221205962120572+120573 cos (05120587120573)+11987121198621205962120573+2120572+1)12

(15)

The minima frequency 120596119898 can be found by numericallysolving 119889|119879FBRF(120596)|119889120596 = 0 for 120596 Solving this equationwhen 119877 = 1Ω 119871 = 1H and 119862 = 1 F for fixed values of120573 when 120572 is varied from 01 to 1 yields the values given inFigure 9(a) while 119876 can be found by numerically solving|119879FBRF(120596)| = 1radic2 for its two real roots 1205961 and 1205962 and thenevaluating119876 = 120596119898|1205961minus1205962|The quality factor calculated forFBRFs while 120572 is varied for fixed 120573 is given in Figure 9(b)respectively again with the general trend showing that 119876increases with increasing order

Qua

lity

fact

or (Q

)

02 0406 08 1 02 04 06 08 1

100

101

102

Resistance (Ω) 120572

Figure 8 Quality factors of (10) for 120573 = 1 119871 = 1H 119862 = 1 F when 120572and 119877 are varied from 01 and 001 respectively to 1 in steps of 001

The MATLAB simulated magnitude responses of (14)with fractional orders of 120572 + 120573 = 15 and 19 when 120573 = 1119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated in Figure 10From this figure we can clearly observe the asymmetricnature of this FBRF with the attenuation of the low and highfrequency stop bands dependent on the element orders of 120572and 120573 respectively The minima frequency (120596119898) minimummagnitude (|119879(120596119898)|) half power frequencies (12059612) andquality factors (119876) of these responses are given in Table 2It is possible to increase the quality factor of the FBRF bydecreasing the value of 119877 for fixed order and values of 119871 and119862 The quality factors of the FBRF when 120573 = 1 119871 = 1H and119862 = 1 F for 120572 varied from 01 to 1 and 119877 varied from 05 to 1in steps of 001 are given in Figure 11 From this figure we cansee that like the FBPF the increasing 119876 is more pronouncedwith higher orders It should be noted though that it is notpossible to calculate a quality factor for FBRFs with bothlow order and resistance The minima peak of these filtersincreases with decreasing order and resistance and when theminima increases above 1radic(2) it is not possible to calculatethe quality factor using the previous definition

6 Mathematical Problems in Engineering

0

02

04

06

08

1

12

01 02 03 04 05 06 07 08 09120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

1205960

(rad

s)

(a)

0

01

02

03

04

05

06

01 02 03 04 05 06 07 08 09120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Q

(b)

Figure 9 (a) Minima frequencies and (b) quality factors of FBRFs for fixed values of 120573 when 120572 is varied from 01 to 1 in steps of 001 and119877 = 1Ω 119871 = 1H and 119862 = 1 F

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 05

120572 = 09

120573 = 1minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

Figure 10 Simulated magnitude response of (14) when 120573 = 1 120572 =05 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

Qua

lity

fact

or (Q

)

050

05

1

2

15

0607

08 0902 04 06 08 1

Resistance (Ω)120572

Figure 11 Quality factors of (14) for 120573 = 1 119871 = 1H119862 = 1 F when 120572and 119877 are varied from 01 and 05 respectively to 1 in steps of 001

3 Circuit Simulation Results

Although there is currently much progress regarding therealization of fractional order capacitors [17 18] there areno commercial devices using these processes available to

implement these circuits As well even though superca-pacitors have been shown to exhibit fractional impedances[19 20] their high capacitance and limited order (05 le120572 le 06) limit their usefulness in signal processing circuitsUntil commercial devices with the desired characteristicsbecome available integer order approximationsmust be usedto realize fractional circuits There are many methods tocreate an approximation of 119904120572 that include continued fractionexpansions (CFEs) aswell as rational approximationmethods[21] These methods present a large array of approximationswith the accuracy and approximated frequency band increas-ing as the order of the approximation increases Here aCFE method [22] was selected to model the fractional ordercapacitors for PSPICE simulations Collecting eight terms ofthe CFE yields a 4th order approximation of the fractionalcapacitor that can be physically realized using the RC laddernetwork in Figure 12

Now while an RC ladder can be used to approximate afractional order capacitor this same topology cannot realizea fractional order inductor as it would require negativecomponent values However we can use a fractional ordercapacitor as a component in a general impedance convertercircuit (GIC) shown in Figure 13 which is used to simulate agrounded impedance [23] Figure 13 realizes the impedance

119885 =119904120573119862119877111987731198775

1198772(16)

simulating a fractional inductor of order 120573 with inductance119871 = 1198621198771119877311987751198772

Using both the approximated fractional order capacitorand fractional order inductor we can realize the FBRFshown in Figure 1(d) with orders 120572 + 120573 = 10 and 120572 +120573 = 14 when 120572 = 05 The realized circuit is given inFigure 14 with the RC ladder shown as the inset for thefractional order capacitor and the GIC circuit as the insetfor the fractional order inductor The 119877 119871 and 119862 values

Mathematical Problems in Engineering 7

119862119890

119862119889

119862119888

119862119887

119877119886

119877119887

119877119888

119877119889

119877119890

asymp 1

119904120572119862

Figure 12 RC ladder structure to realize a 4th order integerapproximation of a fractional order capacitor

minus

minus

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

asymp

Figure 13 GIC topology to simulate a grounded fractional orderinductor using a fractional order capacitor as a component

119877119887

119877119888

119877119889

1198771198901

119904120572119862119862119890

119862119889

119862119888

119862119887

119877119886

minus

minus

+

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

Figure 14 Approximated FBRF circuit realized with RC ladderapproximation of fractional order capacitors and GIC approxima-tion of a fractional order inductor

40

48

64

72

56

80

Impe

danc

e mag

nitu

de (d

B)

Impe

danc

e pha

se (d

egre

e)

Phase

Magnitude

Frequency (Hz)10

110

210

310

410

5

minus43

minus36

minus29

minus22

minus15

minus50

Figure 15 Magnitude and phase response of the approximatedfractional order capacitor (dashed) compared to the ideal (solid)with capacitance of 126 120583F and order 05 after scaling to a centerfrequency of 1 kHz

Table 3 Component values to realize 10 and 14 order FBRFs for120572 = 05 and 120573 = 05 and 09 respectively after magnitude scaled bya factor of 1000 and frequency shifted to 1 kHz

Component Values120573 = 05 120573 = 09

119877 (Ω) 1000 1000

119871 (H) 126 0382

119862 (120583F) 126 126

required to realize these circuits after applying a magnitudescaling of 1000 and frequency scaling to 1 kHz are given inTable 3 The component values required for the 4th orderapproximation of the fractional capacitances with values of126 and 0382 120583F and orders of 05 and 09 respectivelyusing the RC ladder network in Figure 12 shifted to a centerfrequency of 1 kHz are given in Table 4 The magnitude andphase of the ideal (solid line) and 4th order approximated(dashed) fractional order capacitor with capacitance 126 120583Fand order 120572 = 05 shifted to a center frequency of 1 kHzare presented in Figure 15 From this figure we observe thatthe approximation is very good over almost 4 decades from200Hz to 70 kHz for the magnitude and almost 2 decadesfrom 200Hz to 6 kHz for the phase In these regions thedeviation of the approximation from ideal does not exceed123 dB and 023∘ for the magnitude and phase respectivelyThe simulated fractional order inductors of 126 and 0382Hcan be realized using fractional capacitances of 126 and0382 120583F respectively when 1198771 = 1198772 = 1198773 = 1198775 = 1000Ω

Using the component values in Tables 3 and 4 theapproximated FBRF shown in Figure 14 was simulated inPSPICE using MC1458 op amps to realize responses of order(120572 + 120573) = 10 and 14 when 120572 = 05 and 120573 = 05 and 09The PSPICE simulated magnitude responses (dashed lines)compared to the ideal responses (solid lines) are shown inFigure 16 We can clearly see from the magnitude response ofboth theMATLAB and PSPICE simulated responses that thisfilter does realize a fractional band reject response and thatthe simulation shows good agreement with the ideal responseobserving both symmetric and asymmetric characteristicsfor the 10 and 14 order filters respectively Note though

8 Mathematical Problems in Engineering

Table 4 Component values to realize 4th order approximations of fractional order capacitors with a center frequency of 1 kHz

ComponentValues

119862 = 1 120583F 119862 = 1 120583F 119862 = 126 120583F 119862 = 0382 120583F 119862 = 1 120583F120572 = 01 120572 = 05 120572 = 05 120572 = 09 120572 = 09

119877119886 (Ω) 2747 k 1402 k 1111 68 26119877119887 (Ω) 819 k 317 k 2517 433 165119877119888 (Ω) 561 k 478 k 3787 1307 499119877119889 (Ω) 663 k 112 k 8889 6704 2559119877119890 (Ω) 1541 k 929 k 73697 1461897 557898119862119887 (nF) 0165 664 689 705 1846119862119888 (120583F) 00015 0023 0296 113 297119862119889 (120583F) 00052 0043 0537 103 269119862119890 (120583F) 0015 0055 0695 0207 0544

0

Mag

nitu

de (d

B)

Frequency (Hz)

minus7

minus6

minus5

minus4

minus3

minus2

minus1

101

102

103

104

105

120573 = 09

120573 = 05

Figure 16 PSPICE simulations using Figure 14 compared to idealsimulations of (14) as dashed and solid lines respectively to realizeapproximated FBRFs of orders 120572 + 120573 = 10 and 14 when 120572 = 05

that the PSPICE simulations deviate from the MATLABsimulated transfer function at low and high frequencieswhich is a result of using the 4th order approximation ofthe fractional order capacitors We highlight that this filterpresents two characteristics not possible using integer orderfilters (a) bandreject responses have not been possible frominteger order filters with order less than 2 and (b) asymmetricresponses with independent control of stopband attenuationhave never been presented

31 Fractional Bruton Transformation The Bruton transfor-mation applies an impedance transformation to each elementof a passive ladder circuit to create an active circuit realizationusing the concept of frequency-dependent negative resistancethat does not require the use of inductors [24] This methodcan be expanded to the fractional-order domain to removethe fractional order inductor from the 119877119871120573119862120572 circuit Wherethe integer order transformation scales each element by 1119904the fractional Bruton transformation scales each element by1119904120573 Applying this scaling to a resistor transforms it to afractional order capacitor a fractional order inductor to aresistor and a fractional order capacitor to a new fractionalelement of order 0 le 120572 + 120573 le 1 This new fractional elementcan be a resistor capacitor or frequency-dependent negative

119881in (s)

1

119904120573119862119887 119877119887

119881119900 (s)+

1

119904120572+120573119863119887

Figure 17 FLPF of Figure 1(a) after applying the fractional Brutontransformation

minus

+

minus

+

1198772

1198775

1

1199041205731198621

1198774

asymp1

119904120572+1205731198631198871

1199041205721198623

Figure 18 GIC topology to simulate a grounded fractional elementof order 0 le 120572 + 120573 le 2 using two fractional order capacitorscomponents

resistor (FDNR) when 120572 + 120573 = 0 1 and 2 respectivelyNote that we will use the units of [Ω] when referring tothis fractional element in order to remain consistent withthe FDNR whose symbol we are also using to represent thiselementThe FLPF of Figure 1(a) after applying the fractionalBruton transformation is shown in Figure 17 The transferfunction of this transformed circuit is given by

119879FLPF (119904) =119881119900 (119904)

119881in (119904)=

1119877119887119863119887119904120572+120573 + 119904120572119877119887119862119887 + 1119877119887119863119887

(17)

where119862119887 = 1119877119877119887 = 119871 and119863119887 = 119862 for the transfer functionto be equivalent to (5)

Mathematical Problems in Engineering 9

119877119887

119877119888

119877119889

119877119890119862119890

119862119889

119862119887

119877119886

minus

minus

+

+

+

119862119888

1198772

1198775

1198774

1

1199041198621

1

119904119862119887 119877119887

11199041+120572

119863119887

119881119900 (s)

119881in (s)1119872Ω

1

1199041205721198623

Figure 19 Approximated FLPF of Figure 17 realized with RC ladder approximations of fractional order capacitors and GIC approximationof a fractional element of order (120572 + 120573)

Table 5 Component values to realize fractional elements with GICtopology of Figure 18

ComponentValues

119863119887 = 6638 nΩ 119863119887 = 201 nΩ 119863119887 = 6074 pΩOrder = 11 Order = 15 Order = 19

1198621 (120583F) 120573 1 1 1 1 1 1

1198623 (120583F) 120572 1 01 1 05 1 091198772 (Ω) 6638 k 201 k 6074

1198774 (Ω) 1000 1000 1000

1198775 (Ω) 1000 1000 1000

This new fractional element can be realized using a GICwith two fractional order capacitors this topology is shownin Figure 18 and has an impedance

119885119863 =1198775

119904120572+1205731198621119862311987721198774(18)

simulating a fractional element with impedance 119863119887 =11986211198623119877211987741198775 A GIC has also previously been employed torealize a fractional order capacitor of 36 120583F and 120572 = 16 in[7] Using the topology of Figure 17 to simulate the FLPFsof Figure 2 magnitude scaled by 1000 and frequency shiftedto 1 kHz requires 119862119887 = 01592 120583F 119877119887 = 1000Ω and 119863119887 =(6638 n 201 n 6074 p)Ω for the (120572 + 120573) = 11 15 and19 order filters respectively when 120573 = 1 The componentsrequired to simulate the impedance of the fractional elementusing theGIC topology are given in Table 5 with the values ofthe fractional order capacitor 1198623 approximated with the RCladder of Figure 12 given in Table 4

The approximated FLPF shown in Figure 19 was simu-lated in PSPICE usingMC1458 op amps to realize a (120572+120573) =19 order filter when120573 = 1The PSPICE simulatedmagnituderesponses (dashed lines) compared to the ideal responses(solid lines) are shown in Figure 20 Note that a 1MΩ resistorhas been added to bypass the capacitor and provide aDCpathto the noninverting input terminal of the upper op amp in thefractional element realization for its bias current [23]

The PSPICE simulatedmagnitude responses of the FLPFsshowvery good agreementwith theMATLAB simulated ideal

minus60

minus50

minus40

minus30

minus20

minus10

0

Mag

nitu

de (d

B)

101 10

2 103

104

Frequency (Hz)

120573 = 1

120572 = 05

120572 = 09

120572 = 01

Figure 20 PSPICE simulation using Figure 19 compared to idealsimulations of (17) as dashed and solid lines respectively to realizeapproximated FLPF of order 120572 + 120573 = 19 when 120573 = 1

response Verifying the fractional Bruton transformed FLPFsas well as the GIC realizations of a fractional element of order(120572 + 120573) using approximated fractional order capacitors Thedeviations above 20 kHz can be attributed to the approxima-tions of the fractional order capacitors and nonidealities ofthe op amps used to realize the fractional order inductors

4 Conclusion

We have proposed modifying the traditional series RLCcircuit to use a fractional order capacitor and fractionalorder inductor to realize a fractional 119877119871120573119862120572 circuit that iscapable of realizing fractional lowpass highpass bandpassand bandreject filter responses of order 0 lt 120572 + 120573 le 2requiring onlymodification of the element arrangementThistopology can realize bandpass or bandreject responses withorder less than 2 which are not possible using an integerorder circuit In addition these proposed bandpass and ban-dreject filters show asymmetric bandpass characteristics withindependent control of the stopband attenuations throughmanipulation of the fractional elements orders which is noteasily accomplished using integer order filters We have alsoshown how to realize a new fractional element of order (120572 +

10 Mathematical Problems in Engineering

120573) le 2 and a fractional order inductor using fractional ordercapacitors in a GIC circuit PSPICE simulations of FBRFs andFLPFs verify the fractional characteristics of these circuitsas well as the proposed realizations using approximatedfractional elements

Acknowledgments

T J Freeborn would like to acknowledge Canadarsquos NationalSciences and Engineering Research Council (NSERC) Al-berta Innovates-Technology Futures and Alberta AdvancedEducation and Technology for their financial support of thisresearch through their graduate student scholarships

References

[1] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[2] J A Tenreiro Machado I S Jesus A Galhano and JBoaventura Cunha ldquoFractional order electromagneticsrdquo SignalProcessing vol 86 no 10 pp 2637ndash2644 2006

[3] N Sebaa Z E A Fellah W Lauriks and C Depollier ldquoAppli-cation of fractional calculus to ultrasonic wave propagation inhuman cancellous bonerdquo Signal Processing vol 86 no 10 pp2668ndash2677 2006

[4] J Sabatier M Aoun A Oustaloup G Gregoire F Ragot and PRoy ldquoFractional system identification for lead acid battery stateof charge estimationrdquo Signal Processing vol 86 no 10 pp 2645ndash2657 2006

[5] J D Gabano and T Poinot ldquoFractional modelling and identifi-cation of thermal systemsrdquo Signal Processing vol 91 no 3 pp531ndash541 2011

[6] A G Radwan A S Elwakil and A M Soliman ldquoFractional-order sinusoidal oscillators design procedure and practicalexamplesrdquo IEEE Transactions on Circuits and Systems vol 55no 7 pp 2051ndash2063 2008

[7] A G Radwan A S Elwakil and A M Soliman ldquoOn thegeneralization of second-order filters to the fractional-orderdomainrdquo Journal of Circuits Systems and Computers vol 18 no2 pp 361ndash386 2009

[8] A Soltan A G Radwan and A M Soliman ldquoButterworthpassive filter in the fractional orderrdquo in Proceedings of theInternational Conference on Microelectronics (ICM rsquo11) pp 1ndash5Hammamet Tunisia 2011

[9] P Ahmadi BMaundy A S Elwakil and L Belostostski ldquoHigh-quality factor asymmetric-slope band-pass filters a fractional-order capacitor approachrdquo IET Circuits Devices amp Systems vol6 no 3 pp 187ndash197 2012

[10] A Lahiri and T K Rawat ldquoNoise analysis of single stagefractional-order low pass filter using stochastic and fractionalcalculusrdquo ECTI Transactions on Electronics and Communica-tions vol 7 no 2 pp 136ndash143 2009

[11] A Radwan ldquoStability analysis of the fractional-order 119877119871120573119862120572circuitrdquo Journal of Fractional Calculus and Applications vol 3no 1 pp 1ndash15 2012

[12] T J Freeborn B Maundy and A S Elwakil ldquoField pro-grammable analogue array implementations of fractional stepfiltersrdquo IET Circuits Devices amp Systems vol 4 no 6 pp 514ndash524 2010

[13] B Maundy A S Elwakil and T J Freeborn ldquoOn the practicalrealization of higher-order filters with fractional steppingrdquoSignal Processing vol 91 no 3 pp 484ndash491 2011

[14] T J Freeborn B Maundy and A S Elwakil ldquoFractional-step Tow-Thomas biquad filtersrdquo Nonlinear Theory and ItsApplications IEICE (NOLTA) vol 3 no 3 pp 357ndash374 2012

[15] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[16] M Nakagawa and K Sorimachi ldquoBasic characteristics of afractance devicerdquo IEICEmdashTransactions on Fundamentals ofElectronics Communications and Computer Sciences E vol 75pp 1814ndash1819 1992

[17] M Sivarama Krishna S Das K Biswas and B GoswamildquoFabrication of a fractional order capacitor with desired spec-ifications a study on process identification and characteriza-tionrdquo IEEE Transactions on Electron Devices vol 58 no 11 pp4067ndash4073 2011

[18] T Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons Fractals vol 24no 2 pp 479ndash490 2005

[19] J J Quintana A Ramos and I Nuez ldquoIdentification of thefractional impedance of ultracapacitorsrdquo in Proceedings of the2nd Workshop on Fractional Differentiation and Its Applications(IFAC rsquo06) pp 289ndash293 2006

[20] A Dzielinski G Sarwas and D Sierociuk ldquoComparison andvalidation of integer and fractional order ultracapacitor mod-elsrdquo Advances in Difference Equations vol 2011 article 11 2011

[21] I Podlubny I Petras B M Vinagre P OrsquoLeary and Lrsquo DorcakldquoAnalogue realizations of fractional-order controllersrdquo Nonlin-ear Dynamics An International Journal of Nonlinear Dynamicsand Chaos in Engineering Systems vol 29 no 1ndash4 pp 281ndash2962002

[22] B Krishna and K Reddy ldquoActive and passive realization offractance device of order 12rdquo Active and Passive ElectronicComponents vol 2008 Article ID 369421 5 pages 2008

[23] R Schaumann and M E Van Valkenburg Design of AnalogFilters Oxford University Press 2001

[24] L T Bruton ldquoNetwork transfer functions using the conceptof frequency-dependent negative resistancerdquo IEEE Transactionson Circuit Theory vol CT-16 no 3 pp 406ndash408 1969

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

minus120

minus100

minus80

minus60

minus40

minus20

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 01

120572 = 05

120572 = 09

120573 = 1

Figure 4 Simulatedmagnitude response of (8) when120573 = 1120572 = 0105 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

100

101

1205963

dB(r

ads

)

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Figure 5 Half power frequencies of (8) for fixed values of 120573 when120572 is varied from 01 to 1 in steps of 001 and 119877 = 1Ω 119871 = 1H and119862 = 1 F

attenuations of 20120572 and minus20120573 dBdecade for frequencieslower and higher respectively than the maxima frequency(120596119872) With the magnitude of (10) given as

1003816100381610038161003816119879FBPF (120596)1003816100381610038161003816

= 119877119862120596120572 (211987711987111986221205962120572+120573 cos (05120587120573)

+ 2119877119862120596120572 cos (05120572120587)

+ 2119871119862120596120573+120572 cos (05 (120573 + 120572) 120587)

+119877212059621205721198622 + 119871211986221205962120573+2120572 + 1)minus12

(11)

The MATLAB simulated magnitude response of (10) withfractional orders of 120572 + 120573 = 15 and 19 when 120573 = 1119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated inFigure 6 From the simulated magnitude responses we seethat the low frequency stopband has attenuations of 10 and18 dBdecade while the high frequency stopband maintainsan attenuation of minus20 dBdecade The stopband attenuationsclosely match those predicted and confirm that the low andhigh frequency stopband attenuations are independent of

minus60

minus50

minus40

minus30

minus20

minus10

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 05

120572 = 09120573 = 1

Figure 6 Simulated magnitude response of (10) when 120573 = 1 120572 =05 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

Table 1 Maxima frequency and corresponding magnitude halfpower frequencies and quality factors of FBPFs given in Figure 6

120572 120596119872 (rads) |119879(120596119872)| (dB) 1205961 1205962 (rads) 119876

05 1082 minus4747 03173 24171 05151

09 1038 minus1246 05773 17961 08516

Table 2 Minima frequency and corresponding magnitude halfpower frequencies and quality factors of FBRFs given in Figure 10

120572 120596119898 (rads) |119879(120596119898)| (dB) 1205961 1205962 (rads) 119876

05 08974 minus7241 02145 1850 05486

09 09989 minus1735 05309 1701 08539

each other (which is unique for fractional-order bandpassfilters) with low frequency stopband attenuations dependentonly on the order of the fractional capacitor 120572 and thehigh frequency stopband only on the order of the fractionalinductor 120573

The maxima frequency can be found by numericallysolving the following equation

1198891003816100381610038161003816119879FBPF (120596)

1003816100381610038161003816119889120596

= 0 (12)

for120596 Solving (12) for varying120572 andfixed values of120573 yields themaxima frequencies given in Figure 7(a) The quality factor119876 can be found by numerically solving

1003816100381610038161003816119879FBPF (120596)1003816100381610038161003816 =

1003816100381610038161003816119879FBPF (120596119872)1003816100381610038161003816

radic2(13)

for its two real roots 1205961 and 1205962 and then evaluating 119876 =120596119872|1205961minus1205962| Solving these equations numerically for varying120572 and fixed values of 120573 yields the quality factors given inFigure 7(b) While there is no clear trend for 120596119872119876 shows anincrease with increasing order for fixed 120573 when 119877 = 1Ω 119871 =1H and 119862 = 1 F The maxima frequency (120596119872) maximummagnitude (|119879(120596119872)|) half power frequencies (12059612) andquality factors (119876) of the FBPF responses shown in Figure 6solved numerically as described previously are given inTable 1

Mathematical Problems in Engineering 5

0

02

04

06

08

1

120596119872

(rad

s)

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

(a)

0

02

04

06

08

1

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Q

(b)

Figure 7 (a) Maxima frequencies and (b) quality factors of (10) for fixed values of 120573 when 120572 is varied from 01 to 1 in steps of 001 and119877 = 1Ω 119871 = 1H 119862 = 1 F

It is possible to increase the quality factor of these circuitsby decreasing the value of 119877 for fixed order and values of 119871and 119862 The quality factors of the FBPF when 120573 = 1 119871 = 1Hand 119862 = 1 F for 120572 and 119877 varied from 01 to 1 in steps of 001are given in Figure 8 From this figure we can see that theincreasing 119876 is more pronounced with higher orders

24 Fractional Bandreject Filter (FBRF) The circuit shown inFigure 1(d) is able to realize a bandreject filter response witha transfer function given by

119879FBRF (119904) =119881119900 (119904)

119881in (119904)=

119904120572+120573 + 1119871119862

119904120572+120573 + 119904120572 (119877119871) + 1119871119862 (14)

This transfer function realizes an asymmetric FBRF responsewith DC and high frequency gains of 1 With the magnitudeof (14) given as

1003816100381610038161003816119879FBRF (120596)1003816100381610038161003816

= (119871211986221205962120573+2120572 + 2119871119862120596120573+120572 cos (05 (120572 + 120573) 120587) + 1)12

(2119877119862120596120572 cos (05120572120587) + 2119871119862120596120573+120572 cos (05 (120572 + 120573) 120587)

+119862211987721205962120572+211987711987111986221205962120572+120573 cos (05120587120573)+11987121198621205962120573+2120572+1)12

(15)

The minima frequency 120596119898 can be found by numericallysolving 119889|119879FBRF(120596)|119889120596 = 0 for 120596 Solving this equationwhen 119877 = 1Ω 119871 = 1H and 119862 = 1 F for fixed values of120573 when 120572 is varied from 01 to 1 yields the values given inFigure 9(a) while 119876 can be found by numerically solving|119879FBRF(120596)| = 1radic2 for its two real roots 1205961 and 1205962 and thenevaluating119876 = 120596119898|1205961minus1205962|The quality factor calculated forFBRFs while 120572 is varied for fixed 120573 is given in Figure 9(b)respectively again with the general trend showing that 119876increases with increasing order

Qua

lity

fact

or (Q

)

02 0406 08 1 02 04 06 08 1

100

101

102

Resistance (Ω) 120572

Figure 8 Quality factors of (10) for 120573 = 1 119871 = 1H 119862 = 1 F when 120572and 119877 are varied from 01 and 001 respectively to 1 in steps of 001

The MATLAB simulated magnitude responses of (14)with fractional orders of 120572 + 120573 = 15 and 19 when 120573 = 1119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated in Figure 10From this figure we can clearly observe the asymmetricnature of this FBRF with the attenuation of the low and highfrequency stop bands dependent on the element orders of 120572and 120573 respectively The minima frequency (120596119898) minimummagnitude (|119879(120596119898)|) half power frequencies (12059612) andquality factors (119876) of these responses are given in Table 2It is possible to increase the quality factor of the FBRF bydecreasing the value of 119877 for fixed order and values of 119871 and119862 The quality factors of the FBRF when 120573 = 1 119871 = 1H and119862 = 1 F for 120572 varied from 01 to 1 and 119877 varied from 05 to 1in steps of 001 are given in Figure 11 From this figure we cansee that like the FBPF the increasing 119876 is more pronouncedwith higher orders It should be noted though that it is notpossible to calculate a quality factor for FBRFs with bothlow order and resistance The minima peak of these filtersincreases with decreasing order and resistance and when theminima increases above 1radic(2) it is not possible to calculatethe quality factor using the previous definition

6 Mathematical Problems in Engineering

0

02

04

06

08

1

12

01 02 03 04 05 06 07 08 09120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

1205960

(rad

s)

(a)

0

01

02

03

04

05

06

01 02 03 04 05 06 07 08 09120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Q

(b)

Figure 9 (a) Minima frequencies and (b) quality factors of FBRFs for fixed values of 120573 when 120572 is varied from 01 to 1 in steps of 001 and119877 = 1Ω 119871 = 1H and 119862 = 1 F

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 05

120572 = 09

120573 = 1minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

Figure 10 Simulated magnitude response of (14) when 120573 = 1 120572 =05 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

Qua

lity

fact

or (Q

)

050

05

1

2

15

0607

08 0902 04 06 08 1

Resistance (Ω)120572

Figure 11 Quality factors of (14) for 120573 = 1 119871 = 1H119862 = 1 F when 120572and 119877 are varied from 01 and 05 respectively to 1 in steps of 001

3 Circuit Simulation Results

Although there is currently much progress regarding therealization of fractional order capacitors [17 18] there areno commercial devices using these processes available to

implement these circuits As well even though superca-pacitors have been shown to exhibit fractional impedances[19 20] their high capacitance and limited order (05 le120572 le 06) limit their usefulness in signal processing circuitsUntil commercial devices with the desired characteristicsbecome available integer order approximationsmust be usedto realize fractional circuits There are many methods tocreate an approximation of 119904120572 that include continued fractionexpansions (CFEs) aswell as rational approximationmethods[21] These methods present a large array of approximationswith the accuracy and approximated frequency band increas-ing as the order of the approximation increases Here aCFE method [22] was selected to model the fractional ordercapacitors for PSPICE simulations Collecting eight terms ofthe CFE yields a 4th order approximation of the fractionalcapacitor that can be physically realized using the RC laddernetwork in Figure 12

Now while an RC ladder can be used to approximate afractional order capacitor this same topology cannot realizea fractional order inductor as it would require negativecomponent values However we can use a fractional ordercapacitor as a component in a general impedance convertercircuit (GIC) shown in Figure 13 which is used to simulate agrounded impedance [23] Figure 13 realizes the impedance

119885 =119904120573119862119877111987731198775

1198772(16)

simulating a fractional inductor of order 120573 with inductance119871 = 1198621198771119877311987751198772

Using both the approximated fractional order capacitorand fractional order inductor we can realize the FBRFshown in Figure 1(d) with orders 120572 + 120573 = 10 and 120572 +120573 = 14 when 120572 = 05 The realized circuit is given inFigure 14 with the RC ladder shown as the inset for thefractional order capacitor and the GIC circuit as the insetfor the fractional order inductor The 119877 119871 and 119862 values

Mathematical Problems in Engineering 7

119862119890

119862119889

119862119888

119862119887

119877119886

119877119887

119877119888

119877119889

119877119890

asymp 1

119904120572119862

Figure 12 RC ladder structure to realize a 4th order integerapproximation of a fractional order capacitor

minus

minus

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

asymp

Figure 13 GIC topology to simulate a grounded fractional orderinductor using a fractional order capacitor as a component

119877119887

119877119888

119877119889

1198771198901

119904120572119862119862119890

119862119889

119862119888

119862119887

119877119886

minus

minus

+

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

Figure 14 Approximated FBRF circuit realized with RC ladderapproximation of fractional order capacitors and GIC approxima-tion of a fractional order inductor

40

48

64

72

56

80

Impe

danc

e mag

nitu

de (d

B)

Impe

danc

e pha

se (d

egre

e)

Phase

Magnitude

Frequency (Hz)10

110

210

310

410

5

minus43

minus36

minus29

minus22

minus15

minus50

Figure 15 Magnitude and phase response of the approximatedfractional order capacitor (dashed) compared to the ideal (solid)with capacitance of 126 120583F and order 05 after scaling to a centerfrequency of 1 kHz

Table 3 Component values to realize 10 and 14 order FBRFs for120572 = 05 and 120573 = 05 and 09 respectively after magnitude scaled bya factor of 1000 and frequency shifted to 1 kHz

Component Values120573 = 05 120573 = 09

119877 (Ω) 1000 1000

119871 (H) 126 0382

119862 (120583F) 126 126

required to realize these circuits after applying a magnitudescaling of 1000 and frequency scaling to 1 kHz are given inTable 3 The component values required for the 4th orderapproximation of the fractional capacitances with values of126 and 0382 120583F and orders of 05 and 09 respectivelyusing the RC ladder network in Figure 12 shifted to a centerfrequency of 1 kHz are given in Table 4 The magnitude andphase of the ideal (solid line) and 4th order approximated(dashed) fractional order capacitor with capacitance 126 120583Fand order 120572 = 05 shifted to a center frequency of 1 kHzare presented in Figure 15 From this figure we observe thatthe approximation is very good over almost 4 decades from200Hz to 70 kHz for the magnitude and almost 2 decadesfrom 200Hz to 6 kHz for the phase In these regions thedeviation of the approximation from ideal does not exceed123 dB and 023∘ for the magnitude and phase respectivelyThe simulated fractional order inductors of 126 and 0382Hcan be realized using fractional capacitances of 126 and0382 120583F respectively when 1198771 = 1198772 = 1198773 = 1198775 = 1000Ω

Using the component values in Tables 3 and 4 theapproximated FBRF shown in Figure 14 was simulated inPSPICE using MC1458 op amps to realize responses of order(120572 + 120573) = 10 and 14 when 120572 = 05 and 120573 = 05 and 09The PSPICE simulated magnitude responses (dashed lines)compared to the ideal responses (solid lines) are shown inFigure 16 We can clearly see from the magnitude response ofboth theMATLAB and PSPICE simulated responses that thisfilter does realize a fractional band reject response and thatthe simulation shows good agreement with the ideal responseobserving both symmetric and asymmetric characteristicsfor the 10 and 14 order filters respectively Note though

8 Mathematical Problems in Engineering

Table 4 Component values to realize 4th order approximations of fractional order capacitors with a center frequency of 1 kHz

ComponentValues

119862 = 1 120583F 119862 = 1 120583F 119862 = 126 120583F 119862 = 0382 120583F 119862 = 1 120583F120572 = 01 120572 = 05 120572 = 05 120572 = 09 120572 = 09

119877119886 (Ω) 2747 k 1402 k 1111 68 26119877119887 (Ω) 819 k 317 k 2517 433 165119877119888 (Ω) 561 k 478 k 3787 1307 499119877119889 (Ω) 663 k 112 k 8889 6704 2559119877119890 (Ω) 1541 k 929 k 73697 1461897 557898119862119887 (nF) 0165 664 689 705 1846119862119888 (120583F) 00015 0023 0296 113 297119862119889 (120583F) 00052 0043 0537 103 269119862119890 (120583F) 0015 0055 0695 0207 0544

0

Mag

nitu

de (d

B)

Frequency (Hz)

minus7

minus6

minus5

minus4

minus3

minus2

minus1

101

102

103

104

105

120573 = 09

120573 = 05

Figure 16 PSPICE simulations using Figure 14 compared to idealsimulations of (14) as dashed and solid lines respectively to realizeapproximated FBRFs of orders 120572 + 120573 = 10 and 14 when 120572 = 05

that the PSPICE simulations deviate from the MATLABsimulated transfer function at low and high frequencieswhich is a result of using the 4th order approximation ofthe fractional order capacitors We highlight that this filterpresents two characteristics not possible using integer orderfilters (a) bandreject responses have not been possible frominteger order filters with order less than 2 and (b) asymmetricresponses with independent control of stopband attenuationhave never been presented

31 Fractional Bruton Transformation The Bruton transfor-mation applies an impedance transformation to each elementof a passive ladder circuit to create an active circuit realizationusing the concept of frequency-dependent negative resistancethat does not require the use of inductors [24] This methodcan be expanded to the fractional-order domain to removethe fractional order inductor from the 119877119871120573119862120572 circuit Wherethe integer order transformation scales each element by 1119904the fractional Bruton transformation scales each element by1119904120573 Applying this scaling to a resistor transforms it to afractional order capacitor a fractional order inductor to aresistor and a fractional order capacitor to a new fractionalelement of order 0 le 120572 + 120573 le 1 This new fractional elementcan be a resistor capacitor or frequency-dependent negative

119881in (s)

1

119904120573119862119887 119877119887

119881119900 (s)+

1

119904120572+120573119863119887

Figure 17 FLPF of Figure 1(a) after applying the fractional Brutontransformation

minus

+

minus

+

1198772

1198775

1

1199041205731198621

1198774

asymp1

119904120572+1205731198631198871

1199041205721198623

Figure 18 GIC topology to simulate a grounded fractional elementof order 0 le 120572 + 120573 le 2 using two fractional order capacitorscomponents

resistor (FDNR) when 120572 + 120573 = 0 1 and 2 respectivelyNote that we will use the units of [Ω] when referring tothis fractional element in order to remain consistent withthe FDNR whose symbol we are also using to represent thiselementThe FLPF of Figure 1(a) after applying the fractionalBruton transformation is shown in Figure 17 The transferfunction of this transformed circuit is given by

119879FLPF (119904) =119881119900 (119904)

119881in (119904)=

1119877119887119863119887119904120572+120573 + 119904120572119877119887119862119887 + 1119877119887119863119887

(17)

where119862119887 = 1119877119877119887 = 119871 and119863119887 = 119862 for the transfer functionto be equivalent to (5)

Mathematical Problems in Engineering 9

119877119887

119877119888

119877119889

119877119890119862119890

119862119889

119862119887

119877119886

minus

minus

+

+

+

119862119888

1198772

1198775

1198774

1

1199041198621

1

119904119862119887 119877119887

11199041+120572

119863119887

119881119900 (s)

119881in (s)1119872Ω

1

1199041205721198623

Figure 19 Approximated FLPF of Figure 17 realized with RC ladder approximations of fractional order capacitors and GIC approximationof a fractional element of order (120572 + 120573)

Table 5 Component values to realize fractional elements with GICtopology of Figure 18

ComponentValues

119863119887 = 6638 nΩ 119863119887 = 201 nΩ 119863119887 = 6074 pΩOrder = 11 Order = 15 Order = 19

1198621 (120583F) 120573 1 1 1 1 1 1

1198623 (120583F) 120572 1 01 1 05 1 091198772 (Ω) 6638 k 201 k 6074

1198774 (Ω) 1000 1000 1000

1198775 (Ω) 1000 1000 1000

This new fractional element can be realized using a GICwith two fractional order capacitors this topology is shownin Figure 18 and has an impedance

119885119863 =1198775

119904120572+1205731198621119862311987721198774(18)

simulating a fractional element with impedance 119863119887 =11986211198623119877211987741198775 A GIC has also previously been employed torealize a fractional order capacitor of 36 120583F and 120572 = 16 in[7] Using the topology of Figure 17 to simulate the FLPFsof Figure 2 magnitude scaled by 1000 and frequency shiftedto 1 kHz requires 119862119887 = 01592 120583F 119877119887 = 1000Ω and 119863119887 =(6638 n 201 n 6074 p)Ω for the (120572 + 120573) = 11 15 and19 order filters respectively when 120573 = 1 The componentsrequired to simulate the impedance of the fractional elementusing theGIC topology are given in Table 5 with the values ofthe fractional order capacitor 1198623 approximated with the RCladder of Figure 12 given in Table 4

The approximated FLPF shown in Figure 19 was simu-lated in PSPICE usingMC1458 op amps to realize a (120572+120573) =19 order filter when120573 = 1The PSPICE simulatedmagnituderesponses (dashed lines) compared to the ideal responses(solid lines) are shown in Figure 20 Note that a 1MΩ resistorhas been added to bypass the capacitor and provide aDCpathto the noninverting input terminal of the upper op amp in thefractional element realization for its bias current [23]

The PSPICE simulatedmagnitude responses of the FLPFsshowvery good agreementwith theMATLAB simulated ideal

minus60

minus50

minus40

minus30

minus20

minus10

0

Mag

nitu

de (d

B)

101 10

2 103

104

Frequency (Hz)

120573 = 1

120572 = 05

120572 = 09

120572 = 01

Figure 20 PSPICE simulation using Figure 19 compared to idealsimulations of (17) as dashed and solid lines respectively to realizeapproximated FLPF of order 120572 + 120573 = 19 when 120573 = 1

response Verifying the fractional Bruton transformed FLPFsas well as the GIC realizations of a fractional element of order(120572 + 120573) using approximated fractional order capacitors Thedeviations above 20 kHz can be attributed to the approxima-tions of the fractional order capacitors and nonidealities ofthe op amps used to realize the fractional order inductors

4 Conclusion

We have proposed modifying the traditional series RLCcircuit to use a fractional order capacitor and fractionalorder inductor to realize a fractional 119877119871120573119862120572 circuit that iscapable of realizing fractional lowpass highpass bandpassand bandreject filter responses of order 0 lt 120572 + 120573 le 2requiring onlymodification of the element arrangementThistopology can realize bandpass or bandreject responses withorder less than 2 which are not possible using an integerorder circuit In addition these proposed bandpass and ban-dreject filters show asymmetric bandpass characteristics withindependent control of the stopband attenuations throughmanipulation of the fractional elements orders which is noteasily accomplished using integer order filters We have alsoshown how to realize a new fractional element of order (120572 +

10 Mathematical Problems in Engineering

120573) le 2 and a fractional order inductor using fractional ordercapacitors in a GIC circuit PSPICE simulations of FBRFs andFLPFs verify the fractional characteristics of these circuitsas well as the proposed realizations using approximatedfractional elements

Acknowledgments

T J Freeborn would like to acknowledge Canadarsquos NationalSciences and Engineering Research Council (NSERC) Al-berta Innovates-Technology Futures and Alberta AdvancedEducation and Technology for their financial support of thisresearch through their graduate student scholarships

References

[1] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[2] J A Tenreiro Machado I S Jesus A Galhano and JBoaventura Cunha ldquoFractional order electromagneticsrdquo SignalProcessing vol 86 no 10 pp 2637ndash2644 2006

[3] N Sebaa Z E A Fellah W Lauriks and C Depollier ldquoAppli-cation of fractional calculus to ultrasonic wave propagation inhuman cancellous bonerdquo Signal Processing vol 86 no 10 pp2668ndash2677 2006

[4] J Sabatier M Aoun A Oustaloup G Gregoire F Ragot and PRoy ldquoFractional system identification for lead acid battery stateof charge estimationrdquo Signal Processing vol 86 no 10 pp 2645ndash2657 2006

[5] J D Gabano and T Poinot ldquoFractional modelling and identifi-cation of thermal systemsrdquo Signal Processing vol 91 no 3 pp531ndash541 2011

[6] A G Radwan A S Elwakil and A M Soliman ldquoFractional-order sinusoidal oscillators design procedure and practicalexamplesrdquo IEEE Transactions on Circuits and Systems vol 55no 7 pp 2051ndash2063 2008

[7] A G Radwan A S Elwakil and A M Soliman ldquoOn thegeneralization of second-order filters to the fractional-orderdomainrdquo Journal of Circuits Systems and Computers vol 18 no2 pp 361ndash386 2009

[8] A Soltan A G Radwan and A M Soliman ldquoButterworthpassive filter in the fractional orderrdquo in Proceedings of theInternational Conference on Microelectronics (ICM rsquo11) pp 1ndash5Hammamet Tunisia 2011

[9] P Ahmadi BMaundy A S Elwakil and L Belostostski ldquoHigh-quality factor asymmetric-slope band-pass filters a fractional-order capacitor approachrdquo IET Circuits Devices amp Systems vol6 no 3 pp 187ndash197 2012

[10] A Lahiri and T K Rawat ldquoNoise analysis of single stagefractional-order low pass filter using stochastic and fractionalcalculusrdquo ECTI Transactions on Electronics and Communica-tions vol 7 no 2 pp 136ndash143 2009

[11] A Radwan ldquoStability analysis of the fractional-order 119877119871120573119862120572circuitrdquo Journal of Fractional Calculus and Applications vol 3no 1 pp 1ndash15 2012

[12] T J Freeborn B Maundy and A S Elwakil ldquoField pro-grammable analogue array implementations of fractional stepfiltersrdquo IET Circuits Devices amp Systems vol 4 no 6 pp 514ndash524 2010

[13] B Maundy A S Elwakil and T J Freeborn ldquoOn the practicalrealization of higher-order filters with fractional steppingrdquoSignal Processing vol 91 no 3 pp 484ndash491 2011

[14] T J Freeborn B Maundy and A S Elwakil ldquoFractional-step Tow-Thomas biquad filtersrdquo Nonlinear Theory and ItsApplications IEICE (NOLTA) vol 3 no 3 pp 357ndash374 2012

[15] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[16] M Nakagawa and K Sorimachi ldquoBasic characteristics of afractance devicerdquo IEICEmdashTransactions on Fundamentals ofElectronics Communications and Computer Sciences E vol 75pp 1814ndash1819 1992

[17] M Sivarama Krishna S Das K Biswas and B GoswamildquoFabrication of a fractional order capacitor with desired spec-ifications a study on process identification and characteriza-tionrdquo IEEE Transactions on Electron Devices vol 58 no 11 pp4067ndash4073 2011

[18] T Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons Fractals vol 24no 2 pp 479ndash490 2005

[19] J J Quintana A Ramos and I Nuez ldquoIdentification of thefractional impedance of ultracapacitorsrdquo in Proceedings of the2nd Workshop on Fractional Differentiation and Its Applications(IFAC rsquo06) pp 289ndash293 2006

[20] A Dzielinski G Sarwas and D Sierociuk ldquoComparison andvalidation of integer and fractional order ultracapacitor mod-elsrdquo Advances in Difference Equations vol 2011 article 11 2011

[21] I Podlubny I Petras B M Vinagre P OrsquoLeary and Lrsquo DorcakldquoAnalogue realizations of fractional-order controllersrdquo Nonlin-ear Dynamics An International Journal of Nonlinear Dynamicsand Chaos in Engineering Systems vol 29 no 1ndash4 pp 281ndash2962002

[22] B Krishna and K Reddy ldquoActive and passive realization offractance device of order 12rdquo Active and Passive ElectronicComponents vol 2008 Article ID 369421 5 pages 2008

[23] R Schaumann and M E Van Valkenburg Design of AnalogFilters Oxford University Press 2001

[24] L T Bruton ldquoNetwork transfer functions using the conceptof frequency-dependent negative resistancerdquo IEEE Transactionson Circuit Theory vol CT-16 no 3 pp 406ndash408 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0

02

04

06

08

1

120596119872

(rad

s)

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

(a)

0

02

04

06

08

1

01 02 03 04 05 06 07 08 09 1120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Q

(b)

Figure 7 (a) Maxima frequencies and (b) quality factors of (10) for fixed values of 120573 when 120572 is varied from 01 to 1 in steps of 001 and119877 = 1Ω 119871 = 1H 119862 = 1 F

It is possible to increase the quality factor of these circuitsby decreasing the value of 119877 for fixed order and values of 119871and 119862 The quality factors of the FBPF when 120573 = 1 119871 = 1Hand 119862 = 1 F for 120572 and 119877 varied from 01 to 1 in steps of 001are given in Figure 8 From this figure we can see that theincreasing 119876 is more pronounced with higher orders

24 Fractional Bandreject Filter (FBRF) The circuit shown inFigure 1(d) is able to realize a bandreject filter response witha transfer function given by

119879FBRF (119904) =119881119900 (119904)

119881in (119904)=

119904120572+120573 + 1119871119862

119904120572+120573 + 119904120572 (119877119871) + 1119871119862 (14)

This transfer function realizes an asymmetric FBRF responsewith DC and high frequency gains of 1 With the magnitudeof (14) given as

1003816100381610038161003816119879FBRF (120596)1003816100381610038161003816

= (119871211986221205962120573+2120572 + 2119871119862120596120573+120572 cos (05 (120572 + 120573) 120587) + 1)12

(2119877119862120596120572 cos (05120572120587) + 2119871119862120596120573+120572 cos (05 (120572 + 120573) 120587)

+119862211987721205962120572+211987711987111986221205962120572+120573 cos (05120587120573)+11987121198621205962120573+2120572+1)12

(15)

The minima frequency 120596119898 can be found by numericallysolving 119889|119879FBRF(120596)|119889120596 = 0 for 120596 Solving this equationwhen 119877 = 1Ω 119871 = 1H and 119862 = 1 F for fixed values of120573 when 120572 is varied from 01 to 1 yields the values given inFigure 9(a) while 119876 can be found by numerically solving|119879FBRF(120596)| = 1radic2 for its two real roots 1205961 and 1205962 and thenevaluating119876 = 120596119898|1205961minus1205962|The quality factor calculated forFBRFs while 120572 is varied for fixed 120573 is given in Figure 9(b)respectively again with the general trend showing that 119876increases with increasing order

Qua

lity

fact

or (Q

)

02 0406 08 1 02 04 06 08 1

100

101

102

Resistance (Ω) 120572

Figure 8 Quality factors of (10) for 120573 = 1 119871 = 1H 119862 = 1 F when 120572and 119877 are varied from 01 and 001 respectively to 1 in steps of 001

The MATLAB simulated magnitude responses of (14)with fractional orders of 120572 + 120573 = 15 and 19 when 120573 = 1119877 = 1Ω 119871 = 1H and 119862 = 1 F are illustrated in Figure 10From this figure we can clearly observe the asymmetricnature of this FBRF with the attenuation of the low and highfrequency stop bands dependent on the element orders of 120572and 120573 respectively The minima frequency (120596119898) minimummagnitude (|119879(120596119898)|) half power frequencies (12059612) andquality factors (119876) of these responses are given in Table 2It is possible to increase the quality factor of the FBRF bydecreasing the value of 119877 for fixed order and values of 119871 and119862 The quality factors of the FBRF when 120573 = 1 119871 = 1H and119862 = 1 F for 120572 varied from 01 to 1 and 119877 varied from 05 to 1in steps of 001 are given in Figure 11 From this figure we cansee that like the FBPF the increasing 119876 is more pronouncedwith higher orders It should be noted though that it is notpossible to calculate a quality factor for FBRFs with bothlow order and resistance The minima peak of these filtersincreases with decreasing order and resistance and when theminima increases above 1radic(2) it is not possible to calculatethe quality factor using the previous definition

6 Mathematical Problems in Engineering

0

02

04

06

08

1

12

01 02 03 04 05 06 07 08 09120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

1205960

(rad

s)

(a)

0

01

02

03

04

05

06

01 02 03 04 05 06 07 08 09120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Q

(b)

Figure 9 (a) Minima frequencies and (b) quality factors of FBRFs for fixed values of 120573 when 120572 is varied from 01 to 1 in steps of 001 and119877 = 1Ω 119871 = 1H and 119862 = 1 F

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 05

120572 = 09

120573 = 1minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

Figure 10 Simulated magnitude response of (14) when 120573 = 1 120572 =05 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

Qua

lity

fact

or (Q

)

050

05

1

2

15

0607

08 0902 04 06 08 1

Resistance (Ω)120572

Figure 11 Quality factors of (14) for 120573 = 1 119871 = 1H119862 = 1 F when 120572and 119877 are varied from 01 and 05 respectively to 1 in steps of 001

3 Circuit Simulation Results

Although there is currently much progress regarding therealization of fractional order capacitors [17 18] there areno commercial devices using these processes available to

implement these circuits As well even though superca-pacitors have been shown to exhibit fractional impedances[19 20] their high capacitance and limited order (05 le120572 le 06) limit their usefulness in signal processing circuitsUntil commercial devices with the desired characteristicsbecome available integer order approximationsmust be usedto realize fractional circuits There are many methods tocreate an approximation of 119904120572 that include continued fractionexpansions (CFEs) aswell as rational approximationmethods[21] These methods present a large array of approximationswith the accuracy and approximated frequency band increas-ing as the order of the approximation increases Here aCFE method [22] was selected to model the fractional ordercapacitors for PSPICE simulations Collecting eight terms ofthe CFE yields a 4th order approximation of the fractionalcapacitor that can be physically realized using the RC laddernetwork in Figure 12

Now while an RC ladder can be used to approximate afractional order capacitor this same topology cannot realizea fractional order inductor as it would require negativecomponent values However we can use a fractional ordercapacitor as a component in a general impedance convertercircuit (GIC) shown in Figure 13 which is used to simulate agrounded impedance [23] Figure 13 realizes the impedance

119885 =119904120573119862119877111987731198775

1198772(16)

simulating a fractional inductor of order 120573 with inductance119871 = 1198621198771119877311987751198772

Using both the approximated fractional order capacitorand fractional order inductor we can realize the FBRFshown in Figure 1(d) with orders 120572 + 120573 = 10 and 120572 +120573 = 14 when 120572 = 05 The realized circuit is given inFigure 14 with the RC ladder shown as the inset for thefractional order capacitor and the GIC circuit as the insetfor the fractional order inductor The 119877 119871 and 119862 values

Mathematical Problems in Engineering 7

119862119890

119862119889

119862119888

119862119887

119877119886

119877119887

119877119888

119877119889

119877119890

asymp 1

119904120572119862

Figure 12 RC ladder structure to realize a 4th order integerapproximation of a fractional order capacitor

minus

minus

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

asymp

Figure 13 GIC topology to simulate a grounded fractional orderinductor using a fractional order capacitor as a component

119877119887

119877119888

119877119889

1198771198901

119904120572119862119862119890

119862119889

119862119888

119862119887

119877119886

minus

minus

+

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

Figure 14 Approximated FBRF circuit realized with RC ladderapproximation of fractional order capacitors and GIC approxima-tion of a fractional order inductor

40

48

64

72

56

80

Impe

danc

e mag

nitu

de (d

B)

Impe

danc

e pha

se (d

egre

e)

Phase

Magnitude

Frequency (Hz)10

110

210

310

410

5

minus43

minus36

minus29

minus22

minus15

minus50

Figure 15 Magnitude and phase response of the approximatedfractional order capacitor (dashed) compared to the ideal (solid)with capacitance of 126 120583F and order 05 after scaling to a centerfrequency of 1 kHz

Table 3 Component values to realize 10 and 14 order FBRFs for120572 = 05 and 120573 = 05 and 09 respectively after magnitude scaled bya factor of 1000 and frequency shifted to 1 kHz

Component Values120573 = 05 120573 = 09

119877 (Ω) 1000 1000

119871 (H) 126 0382

119862 (120583F) 126 126

required to realize these circuits after applying a magnitudescaling of 1000 and frequency scaling to 1 kHz are given inTable 3 The component values required for the 4th orderapproximation of the fractional capacitances with values of126 and 0382 120583F and orders of 05 and 09 respectivelyusing the RC ladder network in Figure 12 shifted to a centerfrequency of 1 kHz are given in Table 4 The magnitude andphase of the ideal (solid line) and 4th order approximated(dashed) fractional order capacitor with capacitance 126 120583Fand order 120572 = 05 shifted to a center frequency of 1 kHzare presented in Figure 15 From this figure we observe thatthe approximation is very good over almost 4 decades from200Hz to 70 kHz for the magnitude and almost 2 decadesfrom 200Hz to 6 kHz for the phase In these regions thedeviation of the approximation from ideal does not exceed123 dB and 023∘ for the magnitude and phase respectivelyThe simulated fractional order inductors of 126 and 0382Hcan be realized using fractional capacitances of 126 and0382 120583F respectively when 1198771 = 1198772 = 1198773 = 1198775 = 1000Ω

Using the component values in Tables 3 and 4 theapproximated FBRF shown in Figure 14 was simulated inPSPICE using MC1458 op amps to realize responses of order(120572 + 120573) = 10 and 14 when 120572 = 05 and 120573 = 05 and 09The PSPICE simulated magnitude responses (dashed lines)compared to the ideal responses (solid lines) are shown inFigure 16 We can clearly see from the magnitude response ofboth theMATLAB and PSPICE simulated responses that thisfilter does realize a fractional band reject response and thatthe simulation shows good agreement with the ideal responseobserving both symmetric and asymmetric characteristicsfor the 10 and 14 order filters respectively Note though

8 Mathematical Problems in Engineering

Table 4 Component values to realize 4th order approximations of fractional order capacitors with a center frequency of 1 kHz

ComponentValues

119862 = 1 120583F 119862 = 1 120583F 119862 = 126 120583F 119862 = 0382 120583F 119862 = 1 120583F120572 = 01 120572 = 05 120572 = 05 120572 = 09 120572 = 09

119877119886 (Ω) 2747 k 1402 k 1111 68 26119877119887 (Ω) 819 k 317 k 2517 433 165119877119888 (Ω) 561 k 478 k 3787 1307 499119877119889 (Ω) 663 k 112 k 8889 6704 2559119877119890 (Ω) 1541 k 929 k 73697 1461897 557898119862119887 (nF) 0165 664 689 705 1846119862119888 (120583F) 00015 0023 0296 113 297119862119889 (120583F) 00052 0043 0537 103 269119862119890 (120583F) 0015 0055 0695 0207 0544

0

Mag

nitu

de (d

B)

Frequency (Hz)

minus7

minus6

minus5

minus4

minus3

minus2

minus1

101

102

103

104

105

120573 = 09

120573 = 05

Figure 16 PSPICE simulations using Figure 14 compared to idealsimulations of (14) as dashed and solid lines respectively to realizeapproximated FBRFs of orders 120572 + 120573 = 10 and 14 when 120572 = 05

that the PSPICE simulations deviate from the MATLABsimulated transfer function at low and high frequencieswhich is a result of using the 4th order approximation ofthe fractional order capacitors We highlight that this filterpresents two characteristics not possible using integer orderfilters (a) bandreject responses have not been possible frominteger order filters with order less than 2 and (b) asymmetricresponses with independent control of stopband attenuationhave never been presented

31 Fractional Bruton Transformation The Bruton transfor-mation applies an impedance transformation to each elementof a passive ladder circuit to create an active circuit realizationusing the concept of frequency-dependent negative resistancethat does not require the use of inductors [24] This methodcan be expanded to the fractional-order domain to removethe fractional order inductor from the 119877119871120573119862120572 circuit Wherethe integer order transformation scales each element by 1119904the fractional Bruton transformation scales each element by1119904120573 Applying this scaling to a resistor transforms it to afractional order capacitor a fractional order inductor to aresistor and a fractional order capacitor to a new fractionalelement of order 0 le 120572 + 120573 le 1 This new fractional elementcan be a resistor capacitor or frequency-dependent negative

119881in (s)

1

119904120573119862119887 119877119887

119881119900 (s)+

1

119904120572+120573119863119887

Figure 17 FLPF of Figure 1(a) after applying the fractional Brutontransformation

minus

+

minus

+

1198772

1198775

1

1199041205731198621

1198774

asymp1

119904120572+1205731198631198871

1199041205721198623

Figure 18 GIC topology to simulate a grounded fractional elementof order 0 le 120572 + 120573 le 2 using two fractional order capacitorscomponents

resistor (FDNR) when 120572 + 120573 = 0 1 and 2 respectivelyNote that we will use the units of [Ω] when referring tothis fractional element in order to remain consistent withthe FDNR whose symbol we are also using to represent thiselementThe FLPF of Figure 1(a) after applying the fractionalBruton transformation is shown in Figure 17 The transferfunction of this transformed circuit is given by

119879FLPF (119904) =119881119900 (119904)

119881in (119904)=

1119877119887119863119887119904120572+120573 + 119904120572119877119887119862119887 + 1119877119887119863119887

(17)

where119862119887 = 1119877119877119887 = 119871 and119863119887 = 119862 for the transfer functionto be equivalent to (5)

Mathematical Problems in Engineering 9

119877119887

119877119888

119877119889

119877119890119862119890

119862119889

119862119887

119877119886

minus

minus

+

+

+

119862119888

1198772

1198775

1198774

1

1199041198621

1

119904119862119887 119877119887

11199041+120572

119863119887

119881119900 (s)

119881in (s)1119872Ω

1

1199041205721198623

Figure 19 Approximated FLPF of Figure 17 realized with RC ladder approximations of fractional order capacitors and GIC approximationof a fractional element of order (120572 + 120573)

Table 5 Component values to realize fractional elements with GICtopology of Figure 18

ComponentValues

119863119887 = 6638 nΩ 119863119887 = 201 nΩ 119863119887 = 6074 pΩOrder = 11 Order = 15 Order = 19

1198621 (120583F) 120573 1 1 1 1 1 1

1198623 (120583F) 120572 1 01 1 05 1 091198772 (Ω) 6638 k 201 k 6074

1198774 (Ω) 1000 1000 1000

1198775 (Ω) 1000 1000 1000

This new fractional element can be realized using a GICwith two fractional order capacitors this topology is shownin Figure 18 and has an impedance

119885119863 =1198775

119904120572+1205731198621119862311987721198774(18)

simulating a fractional element with impedance 119863119887 =11986211198623119877211987741198775 A GIC has also previously been employed torealize a fractional order capacitor of 36 120583F and 120572 = 16 in[7] Using the topology of Figure 17 to simulate the FLPFsof Figure 2 magnitude scaled by 1000 and frequency shiftedto 1 kHz requires 119862119887 = 01592 120583F 119877119887 = 1000Ω and 119863119887 =(6638 n 201 n 6074 p)Ω for the (120572 + 120573) = 11 15 and19 order filters respectively when 120573 = 1 The componentsrequired to simulate the impedance of the fractional elementusing theGIC topology are given in Table 5 with the values ofthe fractional order capacitor 1198623 approximated with the RCladder of Figure 12 given in Table 4

The approximated FLPF shown in Figure 19 was simu-lated in PSPICE usingMC1458 op amps to realize a (120572+120573) =19 order filter when120573 = 1The PSPICE simulatedmagnituderesponses (dashed lines) compared to the ideal responses(solid lines) are shown in Figure 20 Note that a 1MΩ resistorhas been added to bypass the capacitor and provide aDCpathto the noninverting input terminal of the upper op amp in thefractional element realization for its bias current [23]

The PSPICE simulatedmagnitude responses of the FLPFsshowvery good agreementwith theMATLAB simulated ideal

minus60

minus50

minus40

minus30

minus20

minus10

0

Mag

nitu

de (d

B)

101 10

2 103

104

Frequency (Hz)

120573 = 1

120572 = 05

120572 = 09

120572 = 01

Figure 20 PSPICE simulation using Figure 19 compared to idealsimulations of (17) as dashed and solid lines respectively to realizeapproximated FLPF of order 120572 + 120573 = 19 when 120573 = 1

response Verifying the fractional Bruton transformed FLPFsas well as the GIC realizations of a fractional element of order(120572 + 120573) using approximated fractional order capacitors Thedeviations above 20 kHz can be attributed to the approxima-tions of the fractional order capacitors and nonidealities ofthe op amps used to realize the fractional order inductors

4 Conclusion

We have proposed modifying the traditional series RLCcircuit to use a fractional order capacitor and fractionalorder inductor to realize a fractional 119877119871120573119862120572 circuit that iscapable of realizing fractional lowpass highpass bandpassand bandreject filter responses of order 0 lt 120572 + 120573 le 2requiring onlymodification of the element arrangementThistopology can realize bandpass or bandreject responses withorder less than 2 which are not possible using an integerorder circuit In addition these proposed bandpass and ban-dreject filters show asymmetric bandpass characteristics withindependent control of the stopband attenuations throughmanipulation of the fractional elements orders which is noteasily accomplished using integer order filters We have alsoshown how to realize a new fractional element of order (120572 +

10 Mathematical Problems in Engineering

120573) le 2 and a fractional order inductor using fractional ordercapacitors in a GIC circuit PSPICE simulations of FBRFs andFLPFs verify the fractional characteristics of these circuitsas well as the proposed realizations using approximatedfractional elements

Acknowledgments

T J Freeborn would like to acknowledge Canadarsquos NationalSciences and Engineering Research Council (NSERC) Al-berta Innovates-Technology Futures and Alberta AdvancedEducation and Technology for their financial support of thisresearch through their graduate student scholarships

References

[1] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[2] J A Tenreiro Machado I S Jesus A Galhano and JBoaventura Cunha ldquoFractional order electromagneticsrdquo SignalProcessing vol 86 no 10 pp 2637ndash2644 2006

[3] N Sebaa Z E A Fellah W Lauriks and C Depollier ldquoAppli-cation of fractional calculus to ultrasonic wave propagation inhuman cancellous bonerdquo Signal Processing vol 86 no 10 pp2668ndash2677 2006

[4] J Sabatier M Aoun A Oustaloup G Gregoire F Ragot and PRoy ldquoFractional system identification for lead acid battery stateof charge estimationrdquo Signal Processing vol 86 no 10 pp 2645ndash2657 2006

[5] J D Gabano and T Poinot ldquoFractional modelling and identifi-cation of thermal systemsrdquo Signal Processing vol 91 no 3 pp531ndash541 2011

[6] A G Radwan A S Elwakil and A M Soliman ldquoFractional-order sinusoidal oscillators design procedure and practicalexamplesrdquo IEEE Transactions on Circuits and Systems vol 55no 7 pp 2051ndash2063 2008

[7] A G Radwan A S Elwakil and A M Soliman ldquoOn thegeneralization of second-order filters to the fractional-orderdomainrdquo Journal of Circuits Systems and Computers vol 18 no2 pp 361ndash386 2009

[8] A Soltan A G Radwan and A M Soliman ldquoButterworthpassive filter in the fractional orderrdquo in Proceedings of theInternational Conference on Microelectronics (ICM rsquo11) pp 1ndash5Hammamet Tunisia 2011

[9] P Ahmadi BMaundy A S Elwakil and L Belostostski ldquoHigh-quality factor asymmetric-slope band-pass filters a fractional-order capacitor approachrdquo IET Circuits Devices amp Systems vol6 no 3 pp 187ndash197 2012

[10] A Lahiri and T K Rawat ldquoNoise analysis of single stagefractional-order low pass filter using stochastic and fractionalcalculusrdquo ECTI Transactions on Electronics and Communica-tions vol 7 no 2 pp 136ndash143 2009

[11] A Radwan ldquoStability analysis of the fractional-order 119877119871120573119862120572circuitrdquo Journal of Fractional Calculus and Applications vol 3no 1 pp 1ndash15 2012

[12] T J Freeborn B Maundy and A S Elwakil ldquoField pro-grammable analogue array implementations of fractional stepfiltersrdquo IET Circuits Devices amp Systems vol 4 no 6 pp 514ndash524 2010

[13] B Maundy A S Elwakil and T J Freeborn ldquoOn the practicalrealization of higher-order filters with fractional steppingrdquoSignal Processing vol 91 no 3 pp 484ndash491 2011

[14] T J Freeborn B Maundy and A S Elwakil ldquoFractional-step Tow-Thomas biquad filtersrdquo Nonlinear Theory and ItsApplications IEICE (NOLTA) vol 3 no 3 pp 357ndash374 2012

[15] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[16] M Nakagawa and K Sorimachi ldquoBasic characteristics of afractance devicerdquo IEICEmdashTransactions on Fundamentals ofElectronics Communications and Computer Sciences E vol 75pp 1814ndash1819 1992

[17] M Sivarama Krishna S Das K Biswas and B GoswamildquoFabrication of a fractional order capacitor with desired spec-ifications a study on process identification and characteriza-tionrdquo IEEE Transactions on Electron Devices vol 58 no 11 pp4067ndash4073 2011

[18] T Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons Fractals vol 24no 2 pp 479ndash490 2005

[19] J J Quintana A Ramos and I Nuez ldquoIdentification of thefractional impedance of ultracapacitorsrdquo in Proceedings of the2nd Workshop on Fractional Differentiation and Its Applications(IFAC rsquo06) pp 289ndash293 2006

[20] A Dzielinski G Sarwas and D Sierociuk ldquoComparison andvalidation of integer and fractional order ultracapacitor mod-elsrdquo Advances in Difference Equations vol 2011 article 11 2011

[21] I Podlubny I Petras B M Vinagre P OrsquoLeary and Lrsquo DorcakldquoAnalogue realizations of fractional-order controllersrdquo Nonlin-ear Dynamics An International Journal of Nonlinear Dynamicsand Chaos in Engineering Systems vol 29 no 1ndash4 pp 281ndash2962002

[22] B Krishna and K Reddy ldquoActive and passive realization offractance device of order 12rdquo Active and Passive ElectronicComponents vol 2008 Article ID 369421 5 pages 2008

[23] R Schaumann and M E Van Valkenburg Design of AnalogFilters Oxford University Press 2001

[24] L T Bruton ldquoNetwork transfer functions using the conceptof frequency-dependent negative resistancerdquo IEEE Transactionson Circuit Theory vol CT-16 no 3 pp 406ndash408 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0

02

04

06

08

1

12

01 02 03 04 05 06 07 08 09120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

1205960

(rad

s)

(a)

0

01

02

03

04

05

06

01 02 03 04 05 06 07 08 09120572

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 1

Q

(b)

Figure 9 (a) Minima frequencies and (b) quality factors of FBRFs for fixed values of 120573 when 120572 is varied from 01 to 1 in steps of 001 and119877 = 1Ω 119871 = 1H and 119862 = 1 F

0

Mag

nitu

de (d

B)

10minus3

10minus2

10minus1

100

101

102

103

Frequency (rads)

120572 = 05

120572 = 09

120573 = 1minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

Figure 10 Simulated magnitude response of (14) when 120573 = 1 120572 =05 and 09 for 119877 = 1Ω 119871 = 1H and 119862 = 1 F

Qua

lity

fact

or (Q

)

050

05

1

2

15

0607

08 0902 04 06 08 1

Resistance (Ω)120572

Figure 11 Quality factors of (14) for 120573 = 1 119871 = 1H119862 = 1 F when 120572and 119877 are varied from 01 and 05 respectively to 1 in steps of 001

3 Circuit Simulation Results

Although there is currently much progress regarding therealization of fractional order capacitors [17 18] there areno commercial devices using these processes available to

implement these circuits As well even though superca-pacitors have been shown to exhibit fractional impedances[19 20] their high capacitance and limited order (05 le120572 le 06) limit their usefulness in signal processing circuitsUntil commercial devices with the desired characteristicsbecome available integer order approximationsmust be usedto realize fractional circuits There are many methods tocreate an approximation of 119904120572 that include continued fractionexpansions (CFEs) aswell as rational approximationmethods[21] These methods present a large array of approximationswith the accuracy and approximated frequency band increas-ing as the order of the approximation increases Here aCFE method [22] was selected to model the fractional ordercapacitors for PSPICE simulations Collecting eight terms ofthe CFE yields a 4th order approximation of the fractionalcapacitor that can be physically realized using the RC laddernetwork in Figure 12

Now while an RC ladder can be used to approximate afractional order capacitor this same topology cannot realizea fractional order inductor as it would require negativecomponent values However we can use a fractional ordercapacitor as a component in a general impedance convertercircuit (GIC) shown in Figure 13 which is used to simulate agrounded impedance [23] Figure 13 realizes the impedance

119885 =119904120573119862119877111987731198775

1198772(16)

simulating a fractional inductor of order 120573 with inductance119871 = 1198621198771119877311987751198772

Using both the approximated fractional order capacitorand fractional order inductor we can realize the FBRFshown in Figure 1(d) with orders 120572 + 120573 = 10 and 120572 +120573 = 14 when 120572 = 05 The realized circuit is given inFigure 14 with the RC ladder shown as the inset for thefractional order capacitor and the GIC circuit as the insetfor the fractional order inductor The 119877 119871 and 119862 values

Mathematical Problems in Engineering 7

119862119890

119862119889

119862119888

119862119887

119877119886

119877119887

119877119888

119877119889

119877119890

asymp 1

119904120572119862

Figure 12 RC ladder structure to realize a 4th order integerapproximation of a fractional order capacitor

minus

minus

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

asymp

Figure 13 GIC topology to simulate a grounded fractional orderinductor using a fractional order capacitor as a component

119877119887

119877119888

119877119889

1198771198901

119904120572119862119862119890

119862119889

119862119888

119862119887

119877119886

minus

minus

+

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

Figure 14 Approximated FBRF circuit realized with RC ladderapproximation of fractional order capacitors and GIC approxima-tion of a fractional order inductor

40

48

64

72

56

80

Impe

danc

e mag

nitu

de (d

B)

Impe

danc

e pha

se (d

egre

e)

Phase

Magnitude

Frequency (Hz)10

110

210

310

410

5

minus43

minus36

minus29

minus22

minus15

minus50

Figure 15 Magnitude and phase response of the approximatedfractional order capacitor (dashed) compared to the ideal (solid)with capacitance of 126 120583F and order 05 after scaling to a centerfrequency of 1 kHz

Table 3 Component values to realize 10 and 14 order FBRFs for120572 = 05 and 120573 = 05 and 09 respectively after magnitude scaled bya factor of 1000 and frequency shifted to 1 kHz

Component Values120573 = 05 120573 = 09

119877 (Ω) 1000 1000

119871 (H) 126 0382

119862 (120583F) 126 126

required to realize these circuits after applying a magnitudescaling of 1000 and frequency scaling to 1 kHz are given inTable 3 The component values required for the 4th orderapproximation of the fractional capacitances with values of126 and 0382 120583F and orders of 05 and 09 respectivelyusing the RC ladder network in Figure 12 shifted to a centerfrequency of 1 kHz are given in Table 4 The magnitude andphase of the ideal (solid line) and 4th order approximated(dashed) fractional order capacitor with capacitance 126 120583Fand order 120572 = 05 shifted to a center frequency of 1 kHzare presented in Figure 15 From this figure we observe thatthe approximation is very good over almost 4 decades from200Hz to 70 kHz for the magnitude and almost 2 decadesfrom 200Hz to 6 kHz for the phase In these regions thedeviation of the approximation from ideal does not exceed123 dB and 023∘ for the magnitude and phase respectivelyThe simulated fractional order inductors of 126 and 0382Hcan be realized using fractional capacitances of 126 and0382 120583F respectively when 1198771 = 1198772 = 1198773 = 1198775 = 1000Ω

Using the component values in Tables 3 and 4 theapproximated FBRF shown in Figure 14 was simulated inPSPICE using MC1458 op amps to realize responses of order(120572 + 120573) = 10 and 14 when 120572 = 05 and 120573 = 05 and 09The PSPICE simulated magnitude responses (dashed lines)compared to the ideal responses (solid lines) are shown inFigure 16 We can clearly see from the magnitude response ofboth theMATLAB and PSPICE simulated responses that thisfilter does realize a fractional band reject response and thatthe simulation shows good agreement with the ideal responseobserving both symmetric and asymmetric characteristicsfor the 10 and 14 order filters respectively Note though

8 Mathematical Problems in Engineering

Table 4 Component values to realize 4th order approximations of fractional order capacitors with a center frequency of 1 kHz

ComponentValues

119862 = 1 120583F 119862 = 1 120583F 119862 = 126 120583F 119862 = 0382 120583F 119862 = 1 120583F120572 = 01 120572 = 05 120572 = 05 120572 = 09 120572 = 09

119877119886 (Ω) 2747 k 1402 k 1111 68 26119877119887 (Ω) 819 k 317 k 2517 433 165119877119888 (Ω) 561 k 478 k 3787 1307 499119877119889 (Ω) 663 k 112 k 8889 6704 2559119877119890 (Ω) 1541 k 929 k 73697 1461897 557898119862119887 (nF) 0165 664 689 705 1846119862119888 (120583F) 00015 0023 0296 113 297119862119889 (120583F) 00052 0043 0537 103 269119862119890 (120583F) 0015 0055 0695 0207 0544

0

Mag

nitu

de (d

B)

Frequency (Hz)

minus7

minus6

minus5

minus4

minus3

minus2

minus1

101

102

103

104

105

120573 = 09

120573 = 05

Figure 16 PSPICE simulations using Figure 14 compared to idealsimulations of (14) as dashed and solid lines respectively to realizeapproximated FBRFs of orders 120572 + 120573 = 10 and 14 when 120572 = 05

that the PSPICE simulations deviate from the MATLABsimulated transfer function at low and high frequencieswhich is a result of using the 4th order approximation ofthe fractional order capacitors We highlight that this filterpresents two characteristics not possible using integer orderfilters (a) bandreject responses have not been possible frominteger order filters with order less than 2 and (b) asymmetricresponses with independent control of stopband attenuationhave never been presented

31 Fractional Bruton Transformation The Bruton transfor-mation applies an impedance transformation to each elementof a passive ladder circuit to create an active circuit realizationusing the concept of frequency-dependent negative resistancethat does not require the use of inductors [24] This methodcan be expanded to the fractional-order domain to removethe fractional order inductor from the 119877119871120573119862120572 circuit Wherethe integer order transformation scales each element by 1119904the fractional Bruton transformation scales each element by1119904120573 Applying this scaling to a resistor transforms it to afractional order capacitor a fractional order inductor to aresistor and a fractional order capacitor to a new fractionalelement of order 0 le 120572 + 120573 le 1 This new fractional elementcan be a resistor capacitor or frequency-dependent negative

119881in (s)

1

119904120573119862119887 119877119887

119881119900 (s)+

1

119904120572+120573119863119887

Figure 17 FLPF of Figure 1(a) after applying the fractional Brutontransformation

minus

+

minus

+

1198772

1198775

1

1199041205731198621

1198774

asymp1

119904120572+1205731198631198871

1199041205721198623

Figure 18 GIC topology to simulate a grounded fractional elementof order 0 le 120572 + 120573 le 2 using two fractional order capacitorscomponents

resistor (FDNR) when 120572 + 120573 = 0 1 and 2 respectivelyNote that we will use the units of [Ω] when referring tothis fractional element in order to remain consistent withthe FDNR whose symbol we are also using to represent thiselementThe FLPF of Figure 1(a) after applying the fractionalBruton transformation is shown in Figure 17 The transferfunction of this transformed circuit is given by

119879FLPF (119904) =119881119900 (119904)

119881in (119904)=

1119877119887119863119887119904120572+120573 + 119904120572119877119887119862119887 + 1119877119887119863119887

(17)

where119862119887 = 1119877119877119887 = 119871 and119863119887 = 119862 for the transfer functionto be equivalent to (5)

Mathematical Problems in Engineering 9

119877119887

119877119888

119877119889

119877119890119862119890

119862119889

119862119887

119877119886

minus

minus

+

+

+

119862119888

1198772

1198775

1198774

1

1199041198621

1

119904119862119887 119877119887

11199041+120572

119863119887

119881119900 (s)

119881in (s)1119872Ω

1

1199041205721198623

Figure 19 Approximated FLPF of Figure 17 realized with RC ladder approximations of fractional order capacitors and GIC approximationof a fractional element of order (120572 + 120573)

Table 5 Component values to realize fractional elements with GICtopology of Figure 18

ComponentValues

119863119887 = 6638 nΩ 119863119887 = 201 nΩ 119863119887 = 6074 pΩOrder = 11 Order = 15 Order = 19

1198621 (120583F) 120573 1 1 1 1 1 1

1198623 (120583F) 120572 1 01 1 05 1 091198772 (Ω) 6638 k 201 k 6074

1198774 (Ω) 1000 1000 1000

1198775 (Ω) 1000 1000 1000

This new fractional element can be realized using a GICwith two fractional order capacitors this topology is shownin Figure 18 and has an impedance

119885119863 =1198775

119904120572+1205731198621119862311987721198774(18)

simulating a fractional element with impedance 119863119887 =11986211198623119877211987741198775 A GIC has also previously been employed torealize a fractional order capacitor of 36 120583F and 120572 = 16 in[7] Using the topology of Figure 17 to simulate the FLPFsof Figure 2 magnitude scaled by 1000 and frequency shiftedto 1 kHz requires 119862119887 = 01592 120583F 119877119887 = 1000Ω and 119863119887 =(6638 n 201 n 6074 p)Ω for the (120572 + 120573) = 11 15 and19 order filters respectively when 120573 = 1 The componentsrequired to simulate the impedance of the fractional elementusing theGIC topology are given in Table 5 with the values ofthe fractional order capacitor 1198623 approximated with the RCladder of Figure 12 given in Table 4

The approximated FLPF shown in Figure 19 was simu-lated in PSPICE usingMC1458 op amps to realize a (120572+120573) =19 order filter when120573 = 1The PSPICE simulatedmagnituderesponses (dashed lines) compared to the ideal responses(solid lines) are shown in Figure 20 Note that a 1MΩ resistorhas been added to bypass the capacitor and provide aDCpathto the noninverting input terminal of the upper op amp in thefractional element realization for its bias current [23]

The PSPICE simulatedmagnitude responses of the FLPFsshowvery good agreementwith theMATLAB simulated ideal

minus60

minus50

minus40

minus30

minus20

minus10

0

Mag

nitu

de (d

B)

101 10

2 103

104

Frequency (Hz)

120573 = 1

120572 = 05

120572 = 09

120572 = 01

Figure 20 PSPICE simulation using Figure 19 compared to idealsimulations of (17) as dashed and solid lines respectively to realizeapproximated FLPF of order 120572 + 120573 = 19 when 120573 = 1

response Verifying the fractional Bruton transformed FLPFsas well as the GIC realizations of a fractional element of order(120572 + 120573) using approximated fractional order capacitors Thedeviations above 20 kHz can be attributed to the approxima-tions of the fractional order capacitors and nonidealities ofthe op amps used to realize the fractional order inductors

4 Conclusion

We have proposed modifying the traditional series RLCcircuit to use a fractional order capacitor and fractionalorder inductor to realize a fractional 119877119871120573119862120572 circuit that iscapable of realizing fractional lowpass highpass bandpassand bandreject filter responses of order 0 lt 120572 + 120573 le 2requiring onlymodification of the element arrangementThistopology can realize bandpass or bandreject responses withorder less than 2 which are not possible using an integerorder circuit In addition these proposed bandpass and ban-dreject filters show asymmetric bandpass characteristics withindependent control of the stopband attenuations throughmanipulation of the fractional elements orders which is noteasily accomplished using integer order filters We have alsoshown how to realize a new fractional element of order (120572 +

10 Mathematical Problems in Engineering

120573) le 2 and a fractional order inductor using fractional ordercapacitors in a GIC circuit PSPICE simulations of FBRFs andFLPFs verify the fractional characteristics of these circuitsas well as the proposed realizations using approximatedfractional elements

Acknowledgments

T J Freeborn would like to acknowledge Canadarsquos NationalSciences and Engineering Research Council (NSERC) Al-berta Innovates-Technology Futures and Alberta AdvancedEducation and Technology for their financial support of thisresearch through their graduate student scholarships

References

[1] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[2] J A Tenreiro Machado I S Jesus A Galhano and JBoaventura Cunha ldquoFractional order electromagneticsrdquo SignalProcessing vol 86 no 10 pp 2637ndash2644 2006

[3] N Sebaa Z E A Fellah W Lauriks and C Depollier ldquoAppli-cation of fractional calculus to ultrasonic wave propagation inhuman cancellous bonerdquo Signal Processing vol 86 no 10 pp2668ndash2677 2006

[4] J Sabatier M Aoun A Oustaloup G Gregoire F Ragot and PRoy ldquoFractional system identification for lead acid battery stateof charge estimationrdquo Signal Processing vol 86 no 10 pp 2645ndash2657 2006

[5] J D Gabano and T Poinot ldquoFractional modelling and identifi-cation of thermal systemsrdquo Signal Processing vol 91 no 3 pp531ndash541 2011

[6] A G Radwan A S Elwakil and A M Soliman ldquoFractional-order sinusoidal oscillators design procedure and practicalexamplesrdquo IEEE Transactions on Circuits and Systems vol 55no 7 pp 2051ndash2063 2008

[7] A G Radwan A S Elwakil and A M Soliman ldquoOn thegeneralization of second-order filters to the fractional-orderdomainrdquo Journal of Circuits Systems and Computers vol 18 no2 pp 361ndash386 2009

[8] A Soltan A G Radwan and A M Soliman ldquoButterworthpassive filter in the fractional orderrdquo in Proceedings of theInternational Conference on Microelectronics (ICM rsquo11) pp 1ndash5Hammamet Tunisia 2011

[9] P Ahmadi BMaundy A S Elwakil and L Belostostski ldquoHigh-quality factor asymmetric-slope band-pass filters a fractional-order capacitor approachrdquo IET Circuits Devices amp Systems vol6 no 3 pp 187ndash197 2012

[10] A Lahiri and T K Rawat ldquoNoise analysis of single stagefractional-order low pass filter using stochastic and fractionalcalculusrdquo ECTI Transactions on Electronics and Communica-tions vol 7 no 2 pp 136ndash143 2009

[11] A Radwan ldquoStability analysis of the fractional-order 119877119871120573119862120572circuitrdquo Journal of Fractional Calculus and Applications vol 3no 1 pp 1ndash15 2012

[12] T J Freeborn B Maundy and A S Elwakil ldquoField pro-grammable analogue array implementations of fractional stepfiltersrdquo IET Circuits Devices amp Systems vol 4 no 6 pp 514ndash524 2010

[13] B Maundy A S Elwakil and T J Freeborn ldquoOn the practicalrealization of higher-order filters with fractional steppingrdquoSignal Processing vol 91 no 3 pp 484ndash491 2011

[14] T J Freeborn B Maundy and A S Elwakil ldquoFractional-step Tow-Thomas biquad filtersrdquo Nonlinear Theory and ItsApplications IEICE (NOLTA) vol 3 no 3 pp 357ndash374 2012

[15] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[16] M Nakagawa and K Sorimachi ldquoBasic characteristics of afractance devicerdquo IEICEmdashTransactions on Fundamentals ofElectronics Communications and Computer Sciences E vol 75pp 1814ndash1819 1992

[17] M Sivarama Krishna S Das K Biswas and B GoswamildquoFabrication of a fractional order capacitor with desired spec-ifications a study on process identification and characteriza-tionrdquo IEEE Transactions on Electron Devices vol 58 no 11 pp4067ndash4073 2011

[18] T Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons Fractals vol 24no 2 pp 479ndash490 2005

[19] J J Quintana A Ramos and I Nuez ldquoIdentification of thefractional impedance of ultracapacitorsrdquo in Proceedings of the2nd Workshop on Fractional Differentiation and Its Applications(IFAC rsquo06) pp 289ndash293 2006

[20] A Dzielinski G Sarwas and D Sierociuk ldquoComparison andvalidation of integer and fractional order ultracapacitor mod-elsrdquo Advances in Difference Equations vol 2011 article 11 2011

[21] I Podlubny I Petras B M Vinagre P OrsquoLeary and Lrsquo DorcakldquoAnalogue realizations of fractional-order controllersrdquo Nonlin-ear Dynamics An International Journal of Nonlinear Dynamicsand Chaos in Engineering Systems vol 29 no 1ndash4 pp 281ndash2962002

[22] B Krishna and K Reddy ldquoActive and passive realization offractance device of order 12rdquo Active and Passive ElectronicComponents vol 2008 Article ID 369421 5 pages 2008

[23] R Schaumann and M E Van Valkenburg Design of AnalogFilters Oxford University Press 2001

[24] L T Bruton ldquoNetwork transfer functions using the conceptof frequency-dependent negative resistancerdquo IEEE Transactionson Circuit Theory vol CT-16 no 3 pp 406ndash408 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

119862119890

119862119889

119862119888

119862119887

119877119886

119877119887

119877119888

119877119889

119877119890

asymp 1

119904120572119862

Figure 12 RC ladder structure to realize a 4th order integerapproximation of a fractional order capacitor

minus

minus

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

asymp

Figure 13 GIC topology to simulate a grounded fractional orderinductor using a fractional order capacitor as a component

119877119887

119877119888

119877119889

1198771198901

119904120572119862119862119890

119862119889

119862119888

119862119887

119877119886

minus

minus

+

+

+

1198771

1198772

1198773

1198775

119904120573119871

1

119904120573119862

Figure 14 Approximated FBRF circuit realized with RC ladderapproximation of fractional order capacitors and GIC approxima-tion of a fractional order inductor

40

48

64

72

56

80

Impe

danc

e mag

nitu

de (d

B)

Impe

danc

e pha

se (d

egre

e)

Phase

Magnitude

Frequency (Hz)10

110

210

310

410

5

minus43

minus36

minus29

minus22

minus15

minus50

Figure 15 Magnitude and phase response of the approximatedfractional order capacitor (dashed) compared to the ideal (solid)with capacitance of 126 120583F and order 05 after scaling to a centerfrequency of 1 kHz

Table 3 Component values to realize 10 and 14 order FBRFs for120572 = 05 and 120573 = 05 and 09 respectively after magnitude scaled bya factor of 1000 and frequency shifted to 1 kHz

Component Values120573 = 05 120573 = 09

119877 (Ω) 1000 1000

119871 (H) 126 0382

119862 (120583F) 126 126

required to realize these circuits after applying a magnitudescaling of 1000 and frequency scaling to 1 kHz are given inTable 3 The component values required for the 4th orderapproximation of the fractional capacitances with values of126 and 0382 120583F and orders of 05 and 09 respectivelyusing the RC ladder network in Figure 12 shifted to a centerfrequency of 1 kHz are given in Table 4 The magnitude andphase of the ideal (solid line) and 4th order approximated(dashed) fractional order capacitor with capacitance 126 120583Fand order 120572 = 05 shifted to a center frequency of 1 kHzare presented in Figure 15 From this figure we observe thatthe approximation is very good over almost 4 decades from200Hz to 70 kHz for the magnitude and almost 2 decadesfrom 200Hz to 6 kHz for the phase In these regions thedeviation of the approximation from ideal does not exceed123 dB and 023∘ for the magnitude and phase respectivelyThe simulated fractional order inductors of 126 and 0382Hcan be realized using fractional capacitances of 126 and0382 120583F respectively when 1198771 = 1198772 = 1198773 = 1198775 = 1000Ω

Using the component values in Tables 3 and 4 theapproximated FBRF shown in Figure 14 was simulated inPSPICE using MC1458 op amps to realize responses of order(120572 + 120573) = 10 and 14 when 120572 = 05 and 120573 = 05 and 09The PSPICE simulated magnitude responses (dashed lines)compared to the ideal responses (solid lines) are shown inFigure 16 We can clearly see from the magnitude response ofboth theMATLAB and PSPICE simulated responses that thisfilter does realize a fractional band reject response and thatthe simulation shows good agreement with the ideal responseobserving both symmetric and asymmetric characteristicsfor the 10 and 14 order filters respectively Note though

8 Mathematical Problems in Engineering

Table 4 Component values to realize 4th order approximations of fractional order capacitors with a center frequency of 1 kHz

ComponentValues

119862 = 1 120583F 119862 = 1 120583F 119862 = 126 120583F 119862 = 0382 120583F 119862 = 1 120583F120572 = 01 120572 = 05 120572 = 05 120572 = 09 120572 = 09

119877119886 (Ω) 2747 k 1402 k 1111 68 26119877119887 (Ω) 819 k 317 k 2517 433 165119877119888 (Ω) 561 k 478 k 3787 1307 499119877119889 (Ω) 663 k 112 k 8889 6704 2559119877119890 (Ω) 1541 k 929 k 73697 1461897 557898119862119887 (nF) 0165 664 689 705 1846119862119888 (120583F) 00015 0023 0296 113 297119862119889 (120583F) 00052 0043 0537 103 269119862119890 (120583F) 0015 0055 0695 0207 0544

0

Mag

nitu

de (d

B)

Frequency (Hz)

minus7

minus6

minus5

minus4

minus3

minus2

minus1

101

102

103

104

105

120573 = 09

120573 = 05

Figure 16 PSPICE simulations using Figure 14 compared to idealsimulations of (14) as dashed and solid lines respectively to realizeapproximated FBRFs of orders 120572 + 120573 = 10 and 14 when 120572 = 05

that the PSPICE simulations deviate from the MATLABsimulated transfer function at low and high frequencieswhich is a result of using the 4th order approximation ofthe fractional order capacitors We highlight that this filterpresents two characteristics not possible using integer orderfilters (a) bandreject responses have not been possible frominteger order filters with order less than 2 and (b) asymmetricresponses with independent control of stopband attenuationhave never been presented

31 Fractional Bruton Transformation The Bruton transfor-mation applies an impedance transformation to each elementof a passive ladder circuit to create an active circuit realizationusing the concept of frequency-dependent negative resistancethat does not require the use of inductors [24] This methodcan be expanded to the fractional-order domain to removethe fractional order inductor from the 119877119871120573119862120572 circuit Wherethe integer order transformation scales each element by 1119904the fractional Bruton transformation scales each element by1119904120573 Applying this scaling to a resistor transforms it to afractional order capacitor a fractional order inductor to aresistor and a fractional order capacitor to a new fractionalelement of order 0 le 120572 + 120573 le 1 This new fractional elementcan be a resistor capacitor or frequency-dependent negative

119881in (s)

1

119904120573119862119887 119877119887

119881119900 (s)+

1

119904120572+120573119863119887

Figure 17 FLPF of Figure 1(a) after applying the fractional Brutontransformation

minus

+

minus

+

1198772

1198775

1

1199041205731198621

1198774

asymp1

119904120572+1205731198631198871

1199041205721198623

Figure 18 GIC topology to simulate a grounded fractional elementof order 0 le 120572 + 120573 le 2 using two fractional order capacitorscomponents

resistor (FDNR) when 120572 + 120573 = 0 1 and 2 respectivelyNote that we will use the units of [Ω] when referring tothis fractional element in order to remain consistent withthe FDNR whose symbol we are also using to represent thiselementThe FLPF of Figure 1(a) after applying the fractionalBruton transformation is shown in Figure 17 The transferfunction of this transformed circuit is given by

119879FLPF (119904) =119881119900 (119904)

119881in (119904)=

1119877119887119863119887119904120572+120573 + 119904120572119877119887119862119887 + 1119877119887119863119887

(17)

where119862119887 = 1119877119877119887 = 119871 and119863119887 = 119862 for the transfer functionto be equivalent to (5)

Mathematical Problems in Engineering 9

119877119887

119877119888

119877119889

119877119890119862119890

119862119889

119862119887

119877119886

minus

minus

+

+

+

119862119888

1198772

1198775

1198774

1

1199041198621

1

119904119862119887 119877119887

11199041+120572

119863119887

119881119900 (s)

119881in (s)1119872Ω

1

1199041205721198623

Figure 19 Approximated FLPF of Figure 17 realized with RC ladder approximations of fractional order capacitors and GIC approximationof a fractional element of order (120572 + 120573)

Table 5 Component values to realize fractional elements with GICtopology of Figure 18

ComponentValues

119863119887 = 6638 nΩ 119863119887 = 201 nΩ 119863119887 = 6074 pΩOrder = 11 Order = 15 Order = 19

1198621 (120583F) 120573 1 1 1 1 1 1

1198623 (120583F) 120572 1 01 1 05 1 091198772 (Ω) 6638 k 201 k 6074

1198774 (Ω) 1000 1000 1000

1198775 (Ω) 1000 1000 1000

This new fractional element can be realized using a GICwith two fractional order capacitors this topology is shownin Figure 18 and has an impedance

119885119863 =1198775

119904120572+1205731198621119862311987721198774(18)

simulating a fractional element with impedance 119863119887 =11986211198623119877211987741198775 A GIC has also previously been employed torealize a fractional order capacitor of 36 120583F and 120572 = 16 in[7] Using the topology of Figure 17 to simulate the FLPFsof Figure 2 magnitude scaled by 1000 and frequency shiftedto 1 kHz requires 119862119887 = 01592 120583F 119877119887 = 1000Ω and 119863119887 =(6638 n 201 n 6074 p)Ω for the (120572 + 120573) = 11 15 and19 order filters respectively when 120573 = 1 The componentsrequired to simulate the impedance of the fractional elementusing theGIC topology are given in Table 5 with the values ofthe fractional order capacitor 1198623 approximated with the RCladder of Figure 12 given in Table 4

The approximated FLPF shown in Figure 19 was simu-lated in PSPICE usingMC1458 op amps to realize a (120572+120573) =19 order filter when120573 = 1The PSPICE simulatedmagnituderesponses (dashed lines) compared to the ideal responses(solid lines) are shown in Figure 20 Note that a 1MΩ resistorhas been added to bypass the capacitor and provide aDCpathto the noninverting input terminal of the upper op amp in thefractional element realization for its bias current [23]

The PSPICE simulatedmagnitude responses of the FLPFsshowvery good agreementwith theMATLAB simulated ideal

minus60

minus50

minus40

minus30

minus20

minus10

0

Mag

nitu

de (d

B)

101 10

2 103

104

Frequency (Hz)

120573 = 1

120572 = 05

120572 = 09

120572 = 01

Figure 20 PSPICE simulation using Figure 19 compared to idealsimulations of (17) as dashed and solid lines respectively to realizeapproximated FLPF of order 120572 + 120573 = 19 when 120573 = 1

response Verifying the fractional Bruton transformed FLPFsas well as the GIC realizations of a fractional element of order(120572 + 120573) using approximated fractional order capacitors Thedeviations above 20 kHz can be attributed to the approxima-tions of the fractional order capacitors and nonidealities ofthe op amps used to realize the fractional order inductors

4 Conclusion

We have proposed modifying the traditional series RLCcircuit to use a fractional order capacitor and fractionalorder inductor to realize a fractional 119877119871120573119862120572 circuit that iscapable of realizing fractional lowpass highpass bandpassand bandreject filter responses of order 0 lt 120572 + 120573 le 2requiring onlymodification of the element arrangementThistopology can realize bandpass or bandreject responses withorder less than 2 which are not possible using an integerorder circuit In addition these proposed bandpass and ban-dreject filters show asymmetric bandpass characteristics withindependent control of the stopband attenuations throughmanipulation of the fractional elements orders which is noteasily accomplished using integer order filters We have alsoshown how to realize a new fractional element of order (120572 +

10 Mathematical Problems in Engineering

120573) le 2 and a fractional order inductor using fractional ordercapacitors in a GIC circuit PSPICE simulations of FBRFs andFLPFs verify the fractional characteristics of these circuitsas well as the proposed realizations using approximatedfractional elements

Acknowledgments

T J Freeborn would like to acknowledge Canadarsquos NationalSciences and Engineering Research Council (NSERC) Al-berta Innovates-Technology Futures and Alberta AdvancedEducation and Technology for their financial support of thisresearch through their graduate student scholarships

References

[1] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[2] J A Tenreiro Machado I S Jesus A Galhano and JBoaventura Cunha ldquoFractional order electromagneticsrdquo SignalProcessing vol 86 no 10 pp 2637ndash2644 2006

[3] N Sebaa Z E A Fellah W Lauriks and C Depollier ldquoAppli-cation of fractional calculus to ultrasonic wave propagation inhuman cancellous bonerdquo Signal Processing vol 86 no 10 pp2668ndash2677 2006

[4] J Sabatier M Aoun A Oustaloup G Gregoire F Ragot and PRoy ldquoFractional system identification for lead acid battery stateof charge estimationrdquo Signal Processing vol 86 no 10 pp 2645ndash2657 2006

[5] J D Gabano and T Poinot ldquoFractional modelling and identifi-cation of thermal systemsrdquo Signal Processing vol 91 no 3 pp531ndash541 2011

[6] A G Radwan A S Elwakil and A M Soliman ldquoFractional-order sinusoidal oscillators design procedure and practicalexamplesrdquo IEEE Transactions on Circuits and Systems vol 55no 7 pp 2051ndash2063 2008

[7] A G Radwan A S Elwakil and A M Soliman ldquoOn thegeneralization of second-order filters to the fractional-orderdomainrdquo Journal of Circuits Systems and Computers vol 18 no2 pp 361ndash386 2009

[8] A Soltan A G Radwan and A M Soliman ldquoButterworthpassive filter in the fractional orderrdquo in Proceedings of theInternational Conference on Microelectronics (ICM rsquo11) pp 1ndash5Hammamet Tunisia 2011

[9] P Ahmadi BMaundy A S Elwakil and L Belostostski ldquoHigh-quality factor asymmetric-slope band-pass filters a fractional-order capacitor approachrdquo IET Circuits Devices amp Systems vol6 no 3 pp 187ndash197 2012

[10] A Lahiri and T K Rawat ldquoNoise analysis of single stagefractional-order low pass filter using stochastic and fractionalcalculusrdquo ECTI Transactions on Electronics and Communica-tions vol 7 no 2 pp 136ndash143 2009

[11] A Radwan ldquoStability analysis of the fractional-order 119877119871120573119862120572circuitrdquo Journal of Fractional Calculus and Applications vol 3no 1 pp 1ndash15 2012

[12] T J Freeborn B Maundy and A S Elwakil ldquoField pro-grammable analogue array implementations of fractional stepfiltersrdquo IET Circuits Devices amp Systems vol 4 no 6 pp 514ndash524 2010

[13] B Maundy A S Elwakil and T J Freeborn ldquoOn the practicalrealization of higher-order filters with fractional steppingrdquoSignal Processing vol 91 no 3 pp 484ndash491 2011

[14] T J Freeborn B Maundy and A S Elwakil ldquoFractional-step Tow-Thomas biquad filtersrdquo Nonlinear Theory and ItsApplications IEICE (NOLTA) vol 3 no 3 pp 357ndash374 2012

[15] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[16] M Nakagawa and K Sorimachi ldquoBasic characteristics of afractance devicerdquo IEICEmdashTransactions on Fundamentals ofElectronics Communications and Computer Sciences E vol 75pp 1814ndash1819 1992

[17] M Sivarama Krishna S Das K Biswas and B GoswamildquoFabrication of a fractional order capacitor with desired spec-ifications a study on process identification and characteriza-tionrdquo IEEE Transactions on Electron Devices vol 58 no 11 pp4067ndash4073 2011

[18] T Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons Fractals vol 24no 2 pp 479ndash490 2005

[19] J J Quintana A Ramos and I Nuez ldquoIdentification of thefractional impedance of ultracapacitorsrdquo in Proceedings of the2nd Workshop on Fractional Differentiation and Its Applications(IFAC rsquo06) pp 289ndash293 2006

[20] A Dzielinski G Sarwas and D Sierociuk ldquoComparison andvalidation of integer and fractional order ultracapacitor mod-elsrdquo Advances in Difference Equations vol 2011 article 11 2011

[21] I Podlubny I Petras B M Vinagre P OrsquoLeary and Lrsquo DorcakldquoAnalogue realizations of fractional-order controllersrdquo Nonlin-ear Dynamics An International Journal of Nonlinear Dynamicsand Chaos in Engineering Systems vol 29 no 1ndash4 pp 281ndash2962002

[22] B Krishna and K Reddy ldquoActive and passive realization offractance device of order 12rdquo Active and Passive ElectronicComponents vol 2008 Article ID 369421 5 pages 2008

[23] R Schaumann and M E Van Valkenburg Design of AnalogFilters Oxford University Press 2001

[24] L T Bruton ldquoNetwork transfer functions using the conceptof frequency-dependent negative resistancerdquo IEEE Transactionson Circuit Theory vol CT-16 no 3 pp 406ndash408 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 4 Component values to realize 4th order approximations of fractional order capacitors with a center frequency of 1 kHz

ComponentValues

119862 = 1 120583F 119862 = 1 120583F 119862 = 126 120583F 119862 = 0382 120583F 119862 = 1 120583F120572 = 01 120572 = 05 120572 = 05 120572 = 09 120572 = 09

119877119886 (Ω) 2747 k 1402 k 1111 68 26119877119887 (Ω) 819 k 317 k 2517 433 165119877119888 (Ω) 561 k 478 k 3787 1307 499119877119889 (Ω) 663 k 112 k 8889 6704 2559119877119890 (Ω) 1541 k 929 k 73697 1461897 557898119862119887 (nF) 0165 664 689 705 1846119862119888 (120583F) 00015 0023 0296 113 297119862119889 (120583F) 00052 0043 0537 103 269119862119890 (120583F) 0015 0055 0695 0207 0544

0

Mag

nitu

de (d

B)

Frequency (Hz)

minus7

minus6

minus5

minus4

minus3

minus2

minus1

101

102

103

104

105

120573 = 09

120573 = 05

Figure 16 PSPICE simulations using Figure 14 compared to idealsimulations of (14) as dashed and solid lines respectively to realizeapproximated FBRFs of orders 120572 + 120573 = 10 and 14 when 120572 = 05

that the PSPICE simulations deviate from the MATLABsimulated transfer function at low and high frequencieswhich is a result of using the 4th order approximation ofthe fractional order capacitors We highlight that this filterpresents two characteristics not possible using integer orderfilters (a) bandreject responses have not been possible frominteger order filters with order less than 2 and (b) asymmetricresponses with independent control of stopband attenuationhave never been presented

31 Fractional Bruton Transformation The Bruton transfor-mation applies an impedance transformation to each elementof a passive ladder circuit to create an active circuit realizationusing the concept of frequency-dependent negative resistancethat does not require the use of inductors [24] This methodcan be expanded to the fractional-order domain to removethe fractional order inductor from the 119877119871120573119862120572 circuit Wherethe integer order transformation scales each element by 1119904the fractional Bruton transformation scales each element by1119904120573 Applying this scaling to a resistor transforms it to afractional order capacitor a fractional order inductor to aresistor and a fractional order capacitor to a new fractionalelement of order 0 le 120572 + 120573 le 1 This new fractional elementcan be a resistor capacitor or frequency-dependent negative

119881in (s)

1

119904120573119862119887 119877119887

119881119900 (s)+

1

119904120572+120573119863119887

Figure 17 FLPF of Figure 1(a) after applying the fractional Brutontransformation

minus

+

minus

+

1198772

1198775

1

1199041205731198621

1198774

asymp1

119904120572+1205731198631198871

1199041205721198623

Figure 18 GIC topology to simulate a grounded fractional elementof order 0 le 120572 + 120573 le 2 using two fractional order capacitorscomponents

resistor (FDNR) when 120572 + 120573 = 0 1 and 2 respectivelyNote that we will use the units of [Ω] when referring tothis fractional element in order to remain consistent withthe FDNR whose symbol we are also using to represent thiselementThe FLPF of Figure 1(a) after applying the fractionalBruton transformation is shown in Figure 17 The transferfunction of this transformed circuit is given by

119879FLPF (119904) =119881119900 (119904)

119881in (119904)=

1119877119887119863119887119904120572+120573 + 119904120572119877119887119862119887 + 1119877119887119863119887

(17)

where119862119887 = 1119877119877119887 = 119871 and119863119887 = 119862 for the transfer functionto be equivalent to (5)

Mathematical Problems in Engineering 9

119877119887

119877119888

119877119889

119877119890119862119890

119862119889

119862119887

119877119886

minus

minus

+

+

+

119862119888

1198772

1198775

1198774

1

1199041198621

1

119904119862119887 119877119887

11199041+120572

119863119887

119881119900 (s)

119881in (s)1119872Ω

1

1199041205721198623

Figure 19 Approximated FLPF of Figure 17 realized with RC ladder approximations of fractional order capacitors and GIC approximationof a fractional element of order (120572 + 120573)

Table 5 Component values to realize fractional elements with GICtopology of Figure 18

ComponentValues

119863119887 = 6638 nΩ 119863119887 = 201 nΩ 119863119887 = 6074 pΩOrder = 11 Order = 15 Order = 19

1198621 (120583F) 120573 1 1 1 1 1 1

1198623 (120583F) 120572 1 01 1 05 1 091198772 (Ω) 6638 k 201 k 6074

1198774 (Ω) 1000 1000 1000

1198775 (Ω) 1000 1000 1000

This new fractional element can be realized using a GICwith two fractional order capacitors this topology is shownin Figure 18 and has an impedance

119885119863 =1198775

119904120572+1205731198621119862311987721198774(18)

simulating a fractional element with impedance 119863119887 =11986211198623119877211987741198775 A GIC has also previously been employed torealize a fractional order capacitor of 36 120583F and 120572 = 16 in[7] Using the topology of Figure 17 to simulate the FLPFsof Figure 2 magnitude scaled by 1000 and frequency shiftedto 1 kHz requires 119862119887 = 01592 120583F 119877119887 = 1000Ω and 119863119887 =(6638 n 201 n 6074 p)Ω for the (120572 + 120573) = 11 15 and19 order filters respectively when 120573 = 1 The componentsrequired to simulate the impedance of the fractional elementusing theGIC topology are given in Table 5 with the values ofthe fractional order capacitor 1198623 approximated with the RCladder of Figure 12 given in Table 4

The approximated FLPF shown in Figure 19 was simu-lated in PSPICE usingMC1458 op amps to realize a (120572+120573) =19 order filter when120573 = 1The PSPICE simulatedmagnituderesponses (dashed lines) compared to the ideal responses(solid lines) are shown in Figure 20 Note that a 1MΩ resistorhas been added to bypass the capacitor and provide aDCpathto the noninverting input terminal of the upper op amp in thefractional element realization for its bias current [23]

The PSPICE simulatedmagnitude responses of the FLPFsshowvery good agreementwith theMATLAB simulated ideal

minus60

minus50

minus40

minus30

minus20

minus10

0

Mag

nitu

de (d

B)

101 10

2 103

104

Frequency (Hz)

120573 = 1

120572 = 05

120572 = 09

120572 = 01

Figure 20 PSPICE simulation using Figure 19 compared to idealsimulations of (17) as dashed and solid lines respectively to realizeapproximated FLPF of order 120572 + 120573 = 19 when 120573 = 1

response Verifying the fractional Bruton transformed FLPFsas well as the GIC realizations of a fractional element of order(120572 + 120573) using approximated fractional order capacitors Thedeviations above 20 kHz can be attributed to the approxima-tions of the fractional order capacitors and nonidealities ofthe op amps used to realize the fractional order inductors

4 Conclusion

We have proposed modifying the traditional series RLCcircuit to use a fractional order capacitor and fractionalorder inductor to realize a fractional 119877119871120573119862120572 circuit that iscapable of realizing fractional lowpass highpass bandpassand bandreject filter responses of order 0 lt 120572 + 120573 le 2requiring onlymodification of the element arrangementThistopology can realize bandpass or bandreject responses withorder less than 2 which are not possible using an integerorder circuit In addition these proposed bandpass and ban-dreject filters show asymmetric bandpass characteristics withindependent control of the stopband attenuations throughmanipulation of the fractional elements orders which is noteasily accomplished using integer order filters We have alsoshown how to realize a new fractional element of order (120572 +

10 Mathematical Problems in Engineering

120573) le 2 and a fractional order inductor using fractional ordercapacitors in a GIC circuit PSPICE simulations of FBRFs andFLPFs verify the fractional characteristics of these circuitsas well as the proposed realizations using approximatedfractional elements

Acknowledgments

T J Freeborn would like to acknowledge Canadarsquos NationalSciences and Engineering Research Council (NSERC) Al-berta Innovates-Technology Futures and Alberta AdvancedEducation and Technology for their financial support of thisresearch through their graduate student scholarships

References

[1] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[2] J A Tenreiro Machado I S Jesus A Galhano and JBoaventura Cunha ldquoFractional order electromagneticsrdquo SignalProcessing vol 86 no 10 pp 2637ndash2644 2006

[3] N Sebaa Z E A Fellah W Lauriks and C Depollier ldquoAppli-cation of fractional calculus to ultrasonic wave propagation inhuman cancellous bonerdquo Signal Processing vol 86 no 10 pp2668ndash2677 2006

[4] J Sabatier M Aoun A Oustaloup G Gregoire F Ragot and PRoy ldquoFractional system identification for lead acid battery stateof charge estimationrdquo Signal Processing vol 86 no 10 pp 2645ndash2657 2006

[5] J D Gabano and T Poinot ldquoFractional modelling and identifi-cation of thermal systemsrdquo Signal Processing vol 91 no 3 pp531ndash541 2011

[6] A G Radwan A S Elwakil and A M Soliman ldquoFractional-order sinusoidal oscillators design procedure and practicalexamplesrdquo IEEE Transactions on Circuits and Systems vol 55no 7 pp 2051ndash2063 2008

[7] A G Radwan A S Elwakil and A M Soliman ldquoOn thegeneralization of second-order filters to the fractional-orderdomainrdquo Journal of Circuits Systems and Computers vol 18 no2 pp 361ndash386 2009

[8] A Soltan A G Radwan and A M Soliman ldquoButterworthpassive filter in the fractional orderrdquo in Proceedings of theInternational Conference on Microelectronics (ICM rsquo11) pp 1ndash5Hammamet Tunisia 2011

[9] P Ahmadi BMaundy A S Elwakil and L Belostostski ldquoHigh-quality factor asymmetric-slope band-pass filters a fractional-order capacitor approachrdquo IET Circuits Devices amp Systems vol6 no 3 pp 187ndash197 2012

[10] A Lahiri and T K Rawat ldquoNoise analysis of single stagefractional-order low pass filter using stochastic and fractionalcalculusrdquo ECTI Transactions on Electronics and Communica-tions vol 7 no 2 pp 136ndash143 2009

[11] A Radwan ldquoStability analysis of the fractional-order 119877119871120573119862120572circuitrdquo Journal of Fractional Calculus and Applications vol 3no 1 pp 1ndash15 2012

[12] T J Freeborn B Maundy and A S Elwakil ldquoField pro-grammable analogue array implementations of fractional stepfiltersrdquo IET Circuits Devices amp Systems vol 4 no 6 pp 514ndash524 2010

[13] B Maundy A S Elwakil and T J Freeborn ldquoOn the practicalrealization of higher-order filters with fractional steppingrdquoSignal Processing vol 91 no 3 pp 484ndash491 2011

[14] T J Freeborn B Maundy and A S Elwakil ldquoFractional-step Tow-Thomas biquad filtersrdquo Nonlinear Theory and ItsApplications IEICE (NOLTA) vol 3 no 3 pp 357ndash374 2012

[15] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[16] M Nakagawa and K Sorimachi ldquoBasic characteristics of afractance devicerdquo IEICEmdashTransactions on Fundamentals ofElectronics Communications and Computer Sciences E vol 75pp 1814ndash1819 1992

[17] M Sivarama Krishna S Das K Biswas and B GoswamildquoFabrication of a fractional order capacitor with desired spec-ifications a study on process identification and characteriza-tionrdquo IEEE Transactions on Electron Devices vol 58 no 11 pp4067ndash4073 2011

[18] T Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons Fractals vol 24no 2 pp 479ndash490 2005

[19] J J Quintana A Ramos and I Nuez ldquoIdentification of thefractional impedance of ultracapacitorsrdquo in Proceedings of the2nd Workshop on Fractional Differentiation and Its Applications(IFAC rsquo06) pp 289ndash293 2006

[20] A Dzielinski G Sarwas and D Sierociuk ldquoComparison andvalidation of integer and fractional order ultracapacitor mod-elsrdquo Advances in Difference Equations vol 2011 article 11 2011

[21] I Podlubny I Petras B M Vinagre P OrsquoLeary and Lrsquo DorcakldquoAnalogue realizations of fractional-order controllersrdquo Nonlin-ear Dynamics An International Journal of Nonlinear Dynamicsand Chaos in Engineering Systems vol 29 no 1ndash4 pp 281ndash2962002

[22] B Krishna and K Reddy ldquoActive and passive realization offractance device of order 12rdquo Active and Passive ElectronicComponents vol 2008 Article ID 369421 5 pages 2008

[23] R Schaumann and M E Van Valkenburg Design of AnalogFilters Oxford University Press 2001

[24] L T Bruton ldquoNetwork transfer functions using the conceptof frequency-dependent negative resistancerdquo IEEE Transactionson Circuit Theory vol CT-16 no 3 pp 406ndash408 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

119877119887

119877119888

119877119889

119877119890119862119890

119862119889

119862119887

119877119886

minus

minus

+

+

+

119862119888

1198772

1198775

1198774

1

1199041198621

1

119904119862119887 119877119887

11199041+120572

119863119887

119881119900 (s)

119881in (s)1119872Ω

1

1199041205721198623

Figure 19 Approximated FLPF of Figure 17 realized with RC ladder approximations of fractional order capacitors and GIC approximationof a fractional element of order (120572 + 120573)

Table 5 Component values to realize fractional elements with GICtopology of Figure 18

ComponentValues

119863119887 = 6638 nΩ 119863119887 = 201 nΩ 119863119887 = 6074 pΩOrder = 11 Order = 15 Order = 19

1198621 (120583F) 120573 1 1 1 1 1 1

1198623 (120583F) 120572 1 01 1 05 1 091198772 (Ω) 6638 k 201 k 6074

1198774 (Ω) 1000 1000 1000

1198775 (Ω) 1000 1000 1000

This new fractional element can be realized using a GICwith two fractional order capacitors this topology is shownin Figure 18 and has an impedance

119885119863 =1198775

119904120572+1205731198621119862311987721198774(18)

simulating a fractional element with impedance 119863119887 =11986211198623119877211987741198775 A GIC has also previously been employed torealize a fractional order capacitor of 36 120583F and 120572 = 16 in[7] Using the topology of Figure 17 to simulate the FLPFsof Figure 2 magnitude scaled by 1000 and frequency shiftedto 1 kHz requires 119862119887 = 01592 120583F 119877119887 = 1000Ω and 119863119887 =(6638 n 201 n 6074 p)Ω for the (120572 + 120573) = 11 15 and19 order filters respectively when 120573 = 1 The componentsrequired to simulate the impedance of the fractional elementusing theGIC topology are given in Table 5 with the values ofthe fractional order capacitor 1198623 approximated with the RCladder of Figure 12 given in Table 4

The approximated FLPF shown in Figure 19 was simu-lated in PSPICE usingMC1458 op amps to realize a (120572+120573) =19 order filter when120573 = 1The PSPICE simulatedmagnituderesponses (dashed lines) compared to the ideal responses(solid lines) are shown in Figure 20 Note that a 1MΩ resistorhas been added to bypass the capacitor and provide aDCpathto the noninverting input terminal of the upper op amp in thefractional element realization for its bias current [23]

The PSPICE simulatedmagnitude responses of the FLPFsshowvery good agreementwith theMATLAB simulated ideal

minus60

minus50

minus40

minus30

minus20

minus10

0

Mag

nitu

de (d

B)

101 10

2 103

104

Frequency (Hz)

120573 = 1

120572 = 05

120572 = 09

120572 = 01

Figure 20 PSPICE simulation using Figure 19 compared to idealsimulations of (17) as dashed and solid lines respectively to realizeapproximated FLPF of order 120572 + 120573 = 19 when 120573 = 1

response Verifying the fractional Bruton transformed FLPFsas well as the GIC realizations of a fractional element of order(120572 + 120573) using approximated fractional order capacitors Thedeviations above 20 kHz can be attributed to the approxima-tions of the fractional order capacitors and nonidealities ofthe op amps used to realize the fractional order inductors

4 Conclusion

We have proposed modifying the traditional series RLCcircuit to use a fractional order capacitor and fractionalorder inductor to realize a fractional 119877119871120573119862120572 circuit that iscapable of realizing fractional lowpass highpass bandpassand bandreject filter responses of order 0 lt 120572 + 120573 le 2requiring onlymodification of the element arrangementThistopology can realize bandpass or bandreject responses withorder less than 2 which are not possible using an integerorder circuit In addition these proposed bandpass and ban-dreject filters show asymmetric bandpass characteristics withindependent control of the stopband attenuations throughmanipulation of the fractional elements orders which is noteasily accomplished using integer order filters We have alsoshown how to realize a new fractional element of order (120572 +

10 Mathematical Problems in Engineering

120573) le 2 and a fractional order inductor using fractional ordercapacitors in a GIC circuit PSPICE simulations of FBRFs andFLPFs verify the fractional characteristics of these circuitsas well as the proposed realizations using approximatedfractional elements

Acknowledgments

T J Freeborn would like to acknowledge Canadarsquos NationalSciences and Engineering Research Council (NSERC) Al-berta Innovates-Technology Futures and Alberta AdvancedEducation and Technology for their financial support of thisresearch through their graduate student scholarships

References

[1] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[2] J A Tenreiro Machado I S Jesus A Galhano and JBoaventura Cunha ldquoFractional order electromagneticsrdquo SignalProcessing vol 86 no 10 pp 2637ndash2644 2006

[3] N Sebaa Z E A Fellah W Lauriks and C Depollier ldquoAppli-cation of fractional calculus to ultrasonic wave propagation inhuman cancellous bonerdquo Signal Processing vol 86 no 10 pp2668ndash2677 2006

[4] J Sabatier M Aoun A Oustaloup G Gregoire F Ragot and PRoy ldquoFractional system identification for lead acid battery stateof charge estimationrdquo Signal Processing vol 86 no 10 pp 2645ndash2657 2006

[5] J D Gabano and T Poinot ldquoFractional modelling and identifi-cation of thermal systemsrdquo Signal Processing vol 91 no 3 pp531ndash541 2011

[6] A G Radwan A S Elwakil and A M Soliman ldquoFractional-order sinusoidal oscillators design procedure and practicalexamplesrdquo IEEE Transactions on Circuits and Systems vol 55no 7 pp 2051ndash2063 2008

[7] A G Radwan A S Elwakil and A M Soliman ldquoOn thegeneralization of second-order filters to the fractional-orderdomainrdquo Journal of Circuits Systems and Computers vol 18 no2 pp 361ndash386 2009

[8] A Soltan A G Radwan and A M Soliman ldquoButterworthpassive filter in the fractional orderrdquo in Proceedings of theInternational Conference on Microelectronics (ICM rsquo11) pp 1ndash5Hammamet Tunisia 2011

[9] P Ahmadi BMaundy A S Elwakil and L Belostostski ldquoHigh-quality factor asymmetric-slope band-pass filters a fractional-order capacitor approachrdquo IET Circuits Devices amp Systems vol6 no 3 pp 187ndash197 2012

[10] A Lahiri and T K Rawat ldquoNoise analysis of single stagefractional-order low pass filter using stochastic and fractionalcalculusrdquo ECTI Transactions on Electronics and Communica-tions vol 7 no 2 pp 136ndash143 2009

[11] A Radwan ldquoStability analysis of the fractional-order 119877119871120573119862120572circuitrdquo Journal of Fractional Calculus and Applications vol 3no 1 pp 1ndash15 2012

[12] T J Freeborn B Maundy and A S Elwakil ldquoField pro-grammable analogue array implementations of fractional stepfiltersrdquo IET Circuits Devices amp Systems vol 4 no 6 pp 514ndash524 2010

[13] B Maundy A S Elwakil and T J Freeborn ldquoOn the practicalrealization of higher-order filters with fractional steppingrdquoSignal Processing vol 91 no 3 pp 484ndash491 2011

[14] T J Freeborn B Maundy and A S Elwakil ldquoFractional-step Tow-Thomas biquad filtersrdquo Nonlinear Theory and ItsApplications IEICE (NOLTA) vol 3 no 3 pp 357ndash374 2012

[15] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[16] M Nakagawa and K Sorimachi ldquoBasic characteristics of afractance devicerdquo IEICEmdashTransactions on Fundamentals ofElectronics Communications and Computer Sciences E vol 75pp 1814ndash1819 1992

[17] M Sivarama Krishna S Das K Biswas and B GoswamildquoFabrication of a fractional order capacitor with desired spec-ifications a study on process identification and characteriza-tionrdquo IEEE Transactions on Electron Devices vol 58 no 11 pp4067ndash4073 2011

[18] T Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons Fractals vol 24no 2 pp 479ndash490 2005

[19] J J Quintana A Ramos and I Nuez ldquoIdentification of thefractional impedance of ultracapacitorsrdquo in Proceedings of the2nd Workshop on Fractional Differentiation and Its Applications(IFAC rsquo06) pp 289ndash293 2006

[20] A Dzielinski G Sarwas and D Sierociuk ldquoComparison andvalidation of integer and fractional order ultracapacitor mod-elsrdquo Advances in Difference Equations vol 2011 article 11 2011

[21] I Podlubny I Petras B M Vinagre P OrsquoLeary and Lrsquo DorcakldquoAnalogue realizations of fractional-order controllersrdquo Nonlin-ear Dynamics An International Journal of Nonlinear Dynamicsand Chaos in Engineering Systems vol 29 no 1ndash4 pp 281ndash2962002

[22] B Krishna and K Reddy ldquoActive and passive realization offractance device of order 12rdquo Active and Passive ElectronicComponents vol 2008 Article ID 369421 5 pages 2008

[23] R Schaumann and M E Van Valkenburg Design of AnalogFilters Oxford University Press 2001

[24] L T Bruton ldquoNetwork transfer functions using the conceptof frequency-dependent negative resistancerdquo IEEE Transactionson Circuit Theory vol CT-16 no 3 pp 406ndash408 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

120573) le 2 and a fractional order inductor using fractional ordercapacitors in a GIC circuit PSPICE simulations of FBRFs andFLPFs verify the fractional characteristics of these circuitsas well as the proposed realizations using approximatedfractional elements

Acknowledgments

T J Freeborn would like to acknowledge Canadarsquos NationalSciences and Engineering Research Council (NSERC) Al-berta Innovates-Technology Futures and Alberta AdvancedEducation and Technology for their financial support of thisresearch through their graduate student scholarships

References

[1] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[2] J A Tenreiro Machado I S Jesus A Galhano and JBoaventura Cunha ldquoFractional order electromagneticsrdquo SignalProcessing vol 86 no 10 pp 2637ndash2644 2006

[3] N Sebaa Z E A Fellah W Lauriks and C Depollier ldquoAppli-cation of fractional calculus to ultrasonic wave propagation inhuman cancellous bonerdquo Signal Processing vol 86 no 10 pp2668ndash2677 2006

[4] J Sabatier M Aoun A Oustaloup G Gregoire F Ragot and PRoy ldquoFractional system identification for lead acid battery stateof charge estimationrdquo Signal Processing vol 86 no 10 pp 2645ndash2657 2006

[5] J D Gabano and T Poinot ldquoFractional modelling and identifi-cation of thermal systemsrdquo Signal Processing vol 91 no 3 pp531ndash541 2011

[6] A G Radwan A S Elwakil and A M Soliman ldquoFractional-order sinusoidal oscillators design procedure and practicalexamplesrdquo IEEE Transactions on Circuits and Systems vol 55no 7 pp 2051ndash2063 2008

[7] A G Radwan A S Elwakil and A M Soliman ldquoOn thegeneralization of second-order filters to the fractional-orderdomainrdquo Journal of Circuits Systems and Computers vol 18 no2 pp 361ndash386 2009

[8] A Soltan A G Radwan and A M Soliman ldquoButterworthpassive filter in the fractional orderrdquo in Proceedings of theInternational Conference on Microelectronics (ICM rsquo11) pp 1ndash5Hammamet Tunisia 2011

[9] P Ahmadi BMaundy A S Elwakil and L Belostostski ldquoHigh-quality factor asymmetric-slope band-pass filters a fractional-order capacitor approachrdquo IET Circuits Devices amp Systems vol6 no 3 pp 187ndash197 2012

[10] A Lahiri and T K Rawat ldquoNoise analysis of single stagefractional-order low pass filter using stochastic and fractionalcalculusrdquo ECTI Transactions on Electronics and Communica-tions vol 7 no 2 pp 136ndash143 2009

[11] A Radwan ldquoStability analysis of the fractional-order 119877119871120573119862120572circuitrdquo Journal of Fractional Calculus and Applications vol 3no 1 pp 1ndash15 2012

[12] T J Freeborn B Maundy and A S Elwakil ldquoField pro-grammable analogue array implementations of fractional stepfiltersrdquo IET Circuits Devices amp Systems vol 4 no 6 pp 514ndash524 2010

[13] B Maundy A S Elwakil and T J Freeborn ldquoOn the practicalrealization of higher-order filters with fractional steppingrdquoSignal Processing vol 91 no 3 pp 484ndash491 2011

[14] T J Freeborn B Maundy and A S Elwakil ldquoFractional-step Tow-Thomas biquad filtersrdquo Nonlinear Theory and ItsApplications IEICE (NOLTA) vol 3 no 3 pp 357ndash374 2012

[15] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and some of Their Applications vol198 Academic Press San Diego Calif USA 1999

[16] M Nakagawa and K Sorimachi ldquoBasic characteristics of afractance devicerdquo IEICEmdashTransactions on Fundamentals ofElectronics Communications and Computer Sciences E vol 75pp 1814ndash1819 1992

[17] M Sivarama Krishna S Das K Biswas and B GoswamildquoFabrication of a fractional order capacitor with desired spec-ifications a study on process identification and characteriza-tionrdquo IEEE Transactions on Electron Devices vol 58 no 11 pp4067ndash4073 2011

[18] T Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons Fractals vol 24no 2 pp 479ndash490 2005

[19] J J Quintana A Ramos and I Nuez ldquoIdentification of thefractional impedance of ultracapacitorsrdquo in Proceedings of the2nd Workshop on Fractional Differentiation and Its Applications(IFAC rsquo06) pp 289ndash293 2006

[20] A Dzielinski G Sarwas and D Sierociuk ldquoComparison andvalidation of integer and fractional order ultracapacitor mod-elsrdquo Advances in Difference Equations vol 2011 article 11 2011

[21] I Podlubny I Petras B M Vinagre P OrsquoLeary and Lrsquo DorcakldquoAnalogue realizations of fractional-order controllersrdquo Nonlin-ear Dynamics An International Journal of Nonlinear Dynamicsand Chaos in Engineering Systems vol 29 no 1ndash4 pp 281ndash2962002

[22] B Krishna and K Reddy ldquoActive and passive realization offractance device of order 12rdquo Active and Passive ElectronicComponents vol 2008 Article ID 369421 5 pages 2008

[23] R Schaumann and M E Van Valkenburg Design of AnalogFilters Oxford University Press 2001

[24] L T Bruton ldquoNetwork transfer functions using the conceptof frequency-dependent negative resistancerdquo IEEE Transactionson Circuit Theory vol CT-16 no 3 pp 406ndash408 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of