Research ArticleA New Model of Ultracapacitors Based on Fractal Fundamentals

Xiaodong Zhang1 Yi Sun2 Jia Song1 Aijun Wu3 and Xiaoyan Cui4

1 School of Electrical Engineering Beijing Jiaotong University Beijing 100044 China2 China Electrotechnical Society Beijing 100823 China3 Beijing Railway Bureau Beijing 100860 China4Automation School Beijing University of Posts and Telecommunications Beijing 100876 China

Correspondence should be addressed to Xiaodong Zhang xdzhangbjtueducn

Received 24 January 2014 Accepted 25 June 2014 Published 23 July 2014

Academic Editor Hongjie Jia

Copyright copy 2014 Xiaodong Zhang 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

An intuitive model is proposed in this paper to describe the electrical behavior of certain ultracapacitors The model is basedon a simple expression that can be fully characterized by five real numbers In this paper the measured impedances of threeultracapacitors as a function of frequency are compared to model results There is good agreement between the model andmeasurements Results presented in a previous study are also reviewed and the paper demonstrates that those results are alsoconsistent with the newly described model

1 Introduction

Smart grids and energy systems always need energy stor-age Ultracapacitors represent an alternative to batteries forstoring electrical energy and can help to compensate forthe limited power density of batteries [1] They resemblerechargeable batteries in terms of their ability to transportand store charge but they employ a very different chargestorage mechanism Ultracapacitors store electric energy byaccumulating and separating opposite charges physically asopposed to batteries which store energy chemically [2]Opposing charges are separated by an electrodeelectrolyteinterface which is referred to as an electrochemical double-layer Compared to batteries ultracapacitors have a muchlonger charge-discharge cycle life [2 3]The power density ofultracapacitors is considerably higher than that of batteriesand the energy density is higher than that of electrolyticcapacitors for power applications Ultracapacitors can storea high level of energy in a small volume and release thisenergy in a powerful burst [4] so they are useful in powerelectronic systems and applications (eg power systemsautomotive telecommunication and military) that need toprovide or absorb sudden current surges [5] The power out-put of ultracapacitors is limited by their internal impedance

The impedance needs to be identified and characterizedin order to develop models for different applications Thedevelopment of these models requires measurements of theirdynamic electrical behavior [2ndash4]

Often ultracapacitors are modeled with simple 119877119862 cir-cuits as shown in Figure 1 Models like this are sufficient forwell-defined and stable electrical signals however they donot accurately describe the electrical behavior of these devicesin dynamic and high-power situations

Practical ultracapacitor models are more complicatedConway [6] described the charge storage mechanism as aFaradic pseudocapacitance involving a redox reaction ofmicroporous transition metal hydrous oxides Some [3 7]have modeled the behavior of ultracapacitors using 119877119862

transmission line equivalent circuits The porous electrodeis described by a line of 119877 and 119862 elements representingthe elemental double layer capacitance and the respectiveelectrolyte resistance at a particular pore depth A morecomplete description of the porous electrode behavior wasgiven by De Levie [8] Gualous et al [4] took into accountthe physics of the ultracapacitor and proposed an equivalentcircuit that described the ultracapacitor electrical behaviorwith two 119877119862 branches as shown in Figure 2 This modelconsiders the nonlinear relationship between the capacitance

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 726108 7 pageshttpdxdoiorg1011552014726108

2 Journal of Applied Mathematics

RpC

Rs

a

b

Figure 1 119877119862 circuit model for ultracapacitors

R1

C1

R2

C2Rp

La

b

Figure 2 Equivalent circuit developed by Gualous et al

of activated carbon particles and their surface area that varieswith the type of activated carbon used and the way it istreated But the model has six parameters to be identified anddoes not accurately describe ultracapacitor behavior at lowfrequencies

Zubieta and Bonert [9] provided amodel for the terminalbehavior of an ultracapacitor based on physical reasoningThis model (see Figure 3) has three distinct 119877119862 time con-stants 119877119891 119862119891 with the voltage-dependent capacitor 1198621198912(119865119881) which reflects the voltage dependence of the capac-itance dominates the behavior of the ultracapacitor in thetime over a period of seconds in response to a charge actionThe other branches 119877119897 with 119862119897 and 119877119898 with 119862119898 separatelydetermine the terminal behavior in the range of minutes andthe behavior for times longer than 10mins The resistor 119877119901is a leakage resistor representing the self discharge propertyof the ultracapacitor Others (eg [10]) have also developedequivalent circuit models based on variable time constantsto fit the measured AC impedance data of ultracapacitorsIn [10] the dependence of the resistance on frequency wasdivided into four distinct frequency zones and this modelincluded a voltage-dependent capacitor as well

However the pores of activated carbon in an ultraca-pacitor have a complex branch pore structure for whichany impedance analysis model should account Furthermorethe parameters in the above models especially the voltagedependent capacitance can be difficult to quantify

Buller et al [11 12] presented a model shown in Figure 4which can be used to describe the behavior of ultracapacitorsover a wide range of frequencies 119885119901 is the complex pore

+

minus

Rf Rm

Cm

Rp

a

b

Cf

VCfCf2 lowast VCf

CL

RL

Figure 3 Equivalent circuit developed by Zubieta and Bonert

RiL

C

2

C

2

n = 1 n = 2

Zp

Zp

C

120591

1205872

2

n2 middot C

120591

1205872

2

n2 middot C

middot middot middot

Figure 4 Equivalent circuit developed by Buller et al

impedance related to the porosity of the ultracapacitor Themathematical expression for 119885119901 is

119885119901 (119895120596) =120591 sdot coth (radic119895120596120591)

119862 sdot radic119895120596120591 (1)

Unfortunately this function is not well suited for circuitsimulation software because of the coth term [13] Moreoverwhen the number 119873 is high the series denominators of theformula in Figure 4

120591

1205872

2

1198992 sdot 119862(2)

become very large and may not be accurately determinedby measurements Buller et al [12] used 10 cells and 20parameters to model a 1400 F ultracapacitor

Qu and Shi [7] proposed an 119877119862-ladder network modelfor ultracapacitors based on the pore structure of activatedcarbon which is shown in Figure 5This model is particularlyintuitive because it illustrates howmore capacitance becomesavailable as the time constant of the charging cycle isincreased 119877119894 and 119862119894 (119894 = 1 119899) can be treated as theresistance and capacitance of the pores with certain pore size119877119894119862119894 gives the unit of time and indicates how fast the pore ofcertain size is

Itagaki et al [14] proposed a model for ultracapacitorsbased on a fractal pore structure with three sizes of cylin-drical pores This model consisted of resistors and compleximpedances connected in a tree-like structure

Journal of Applied Mathematics 3

R2 R3

C1 C2

Rn

Cnminus1

a

b

Cn

R1

C3

Figure 5 Equivalent circuit developed by Qu and Shi

Porouselectrode

Post

Tooth

RC

m teeth

n posts

Figure 6 Pore model of the ultracapacitor electrodes

This paper proposes a new electrical model for ultraca-pacitors that is also based on a fractal interpretation of theirstructure but does not make any assumption about the sizeor shape of the pores The new model is relatively simple andcan be fully described by five parameters

2 Model Description

Ultracapacitors are comprised of two highly porous activatedcarbon electrodes which take surface area and charge sep-aration distance to an extreme The surface areas can begreater than 21500 square feet per gram and the separationbetween the charged surfaces is reduced to distances on theorder of nanometers [15] The electrodes are immersed in asuitable electrolyte to facilitate the charge transfer and storagemechanism Charges accumulate in the pores resulting incapacitance

Consider the sample pore structure illustrated in Figure 6a central conducting structure called a ldquopostrdquo is lined withmany smaller structures referred to as ldquoteethrdquo Half of theposts in this structure are connected to one electrode of anultracapacitor and the other half are connected to the otherelectrodeThe ultracapacitor has 2119899 posts and every post has

Profile

a

d

Figure 7 Capacitance scheme of porosity

119898 teeth In this model each tooth is a smaller version of thepost that it is attached to

If we start by analyzing the capacitance of the structurewithout any teeth the total capacitance is 119899 times to the post-to-post capacitance because all of these capacitances are inparallel The equivalent series resistance of the ultracapacitoris the resistance of one pair of posts divided by 119899 since thepost resistances are also in parallel therefore an equivalentcircuit for an ultracapacitor consisting of interleaved posts ofuniform size would be a simple 119877119862 circuit where the valueof 119877 is 119877post119899 and the value of 119862 is 119862post times 119899 In this case119877post is the resistance associatedwith chargemoving fromoneelectrode into a single post and then to the second electrode119862post is the contribution to the capacitance of a single pair ofposts The time constant associated with charge populatingthe posts (ie the 119877119862 time constant) is

120591 = 119877119862 =

119877post

119899times 119862post times 119899 = 119877post119862post (3)

119877post is proportional to the length of the post and inverselyproportional to the cross-sectional area

119877post = 120590119897

120587 sdot 119860 (4)

where120590 is the electrical conductivity of the post 119897 is the lengthof the post and 119860 is the cross-sectional area of the post

The capacitance 119862post can be expressed as the capac-itance between any two posts times a constant that isdetermined by the number of posts in proximity to a givenpost A uniform post distribution is illustrated in Figure 7119862post is proportional to the length 119897 and inversely proportionalto the natural log of the ratio of the post separation 119889 to thepost radius 119886 as indicated below

119862post = 1198962120587120576119897

ln (119889119886) (5)

where 120576 is the permittivity of the dielectricA circuit model for the post-only ultracapacitor is pro-

vided in Figure 8Now consider the structure of the intermeshed teeth

between posts In fractal geometry each tooth would be ascaled down version of the post to which it was attached Ifwe assume that each post has 119898 teeth and the size of eachtooth is 1119898 times the size of the post then the resistance ofa single tooth is

119877tooth = 120590119897119898

1205871198601198982= 119898 times 119877post (6)

4 Journal of Applied Mathematics

C

R

Figure 8 The post-only model

CC

RR

Figure 9 Equivalent circuit for a capacitor with one set of posts andteeth

The capacitance 119862tooth can be expressed as

119862tooth = 1198962120587120576119897119898

ln ((119889119898) (119886119898))=

119862post

119898 (7)

For 119898 teeth per post in parallel the overall capacitance ismultiplied by119898 and the resistance is divided by119898 yielding

119877teeth per post =(119898 times 119877post)

119898= 119877post

(8)

119862teeth per post = (119862post

119898) times 119898 = 119862post (9)

From (8) and (9) it is clear that the time constantassociated with charge moving from the posts out onto theteeth is the same as the time constant associated with chargemoving out onto the posts and can be calculated with (3)Theequivalent circuit for the capacitor with both posts and teethis shown in Figure 9 Both resistors in this model have thesame value Both capacitors also have the same value

Employing (6) through (9) it is relatively straightforwardto show that adding 119898 smaller teeth to each tooth in thestructure shown in Figure 6 would result in the same amountof additional capacitance provided through the same amountof additional resistance The fractal geometry with manylayers of repeating structures would yield the equivalent

CCCCC C

RRRRa0

b0

Figure 10 Electrical model of a fractal electrode structure

R

CC

R R R

C

a0

b0

CCZa0b0Za0b0

Za0b0

R1

R2

C1a

b

Zab

Figure 11 Ultracapacitor model described by only 5 real numbers

circuit in Figure 10 where all of the resistors would have thesame value and so do all of the capacitors

Although the example in Figure 6 employs cylindricalpore structures the resistance and capacitance will scalesimilarly with nearly any branch-like pore structure Gen-erally the resistance of a branch will be proportional to itslength and inversely proportional to its cross-sectional areaThe capacitance between closely spaced branches will beproportional to its length and independent of cross-sectionalarea if the spacing between branches is also scaledThereforethe model in Figure 10 does not assume a particular porestructure as long as the successively smaller branches of thepores form fractal geometry

As the different number of time between charging anddischarging the deterioration of ultracapacitors is also notthe same Few electrolytes will be decomposed to form thefree insoluble product particles which increase the resistanceTherefore a resistor is paralleled in the model in Figure 10The greater the loss is the higher the resistor value will be

Considering its ohmic resistance characteristics andohmic capacitive characteristics a resistor and a capacitor arein series thus the model in Figure 11 is proposed which has5 independent component values

Although the model in Figure 11 has only 5 independentcomponent values it has an infinite number of elementsIn order to develop a closed-form expression for the inputimpedance it is convenient to use a Fourier transformtechnique For the impedance ladder network in Figure 12119885119899 is the equivalent input complex impedance and 119885119899minus1 is

Journal of Applied Mathematics 5

Zb Zb Zb Zb

Za Za Za Za

Znminus1Zn Z1Z2

Z0

Z0

Figure 12 Complex impedance ladder network

Zb

Za Znminus1Zn

Figure 13 The equivalent circuit of ladder network in Figure 12

the equivalent input complex impedance of the (119899 minus 1)-orderladder network (see Figure 13) So

119885119899 = (119885119887 + 119885119899minus1) 119885119886 (10)

The recurrence formula for 119885119899 is

119885119899 =(119885119886) sdot 119885119899minus1 + (119885119886sdot119885119887)

119885119899minus1 + (119885119886+119885119887) (11)

The solution of (11) can be obtained using amethod presentedin [16 17] In this case the coefficient matrix of 119885119899 which isexpressed by 119885119899minus1 is shown as follows

1198601 = [119885119886 119885119886sdot119885119887

1 119885119886+119885119887] (12)

119860119899 = 1198601198991 (13)

119860119899 in (13) is the coefficient matrix of 119885119899 which is expressedby 1198850 If 1198601 is expressed by its characteristic values andcharacteristic vectors 1198601198991 can be calculated as

1198601 = 119875[1205821 0

0 1205822]119875minus1 (14)

1198601198991 = 119875[

1205821198991 0

0 1205821198992]119875minus1 (15)

The characteristic values of 1198601 and its characteristic vectorsare

1205821 =

2119885119886+119885119887 + radic1198852119887+ 4119885119886sdot119885119887

2

1205822 =

2119885119886+119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2

119875 = [119885119886sdot119885119887 1205822 minus 119885119886minus119885119887

1205821minus119885119886 1]

119875minus1=

[[[[

[

1

119885119887 sdot (1205821+119885119886)minus1205822 minus 119885119886minus119885119887

119885119887 sdot (1205821+119885119886)

minus1205821minus119885119886

119885119887 sdot (1205821+119885119886)

119885119886sdot119885119887

119885119887 sdot (1205821+119885119886)

]]]]

]

(16)

Combining (16) into (15) results in

1198601198991

=

[[[[

[

1198851198861198851198871205821198991 + 1205821119885119886120582

1198992

119885119887 sdot (1205821+119885119886)

119885119886119885119887 (119885119886+119885119887minus1205822) (1205821198991minus1205821198992)

119885119887 sdot (1205821+119885119886)

(1205821minus119885119886) (1205821198991 minus 1205821198992)

119885119887 sdot (1205821+119885119886)

12058211198851198871205821198991 + 119885119886119885119887120582

1198992

119885119887 sdot (1205821+119885119886)

]]]]

]

(17)

119885119899 = ((1198851198861198851198871205821198991 + 1205821119885119886120582

1198992) sdot 1198850

+ 119885119886119885119887 (119885119886+119885119887 minus 1205822) (1205821198991 minus 1205821198992))

times ((1205821minus119885119886) (1205821198991 minus 1205821198992) 1198850 + (1205821119885119887120582

1198991 + 119885119886119885119887120582

1198992))minus1

(18)

where1205821 + 1205822 = 2119885119886 + 119885119887

1205821 sdot 1205822 = 1198852119886

(19)

If 1198850 = 119885119886 (18) can be simplified using (19) and as

119885119899 =(119885119886 minus 1205822) sdot 120582

119899+11 minus (119885119886 minus 1205821) sdot120582

119899+12

120582119899+11 minus 120582

119899+12

(20)

It is assumed that

1205822

1205821

=

2119885119886+119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2119885119886+119885119887 + radic1198852119887+ 4119885119886sdot119885119887

=

10038161003816100381610038161003816100381610038161003816

1205822

1205821

10038161003816100381610038161003816100381610038161003816

119890119895120579 (21)

Therefore

119885119899 =(119885119886 minus 1205822) minus (119885119886 minus 1205821) sdot

1003816100381610038161003816120582212058211003816100381610038161003816

119899+1119890119895(119899+1)120579

1 minus100381610038161003816100381612058221205821

1003816100381610038161003816

119899+1119890119895(119899+1)120579

(22)

If |12058221205821| gt 1 then the lim119899rarrinfin|12058221205821|119899+1

rarr infin

lim119899rarrinfin

119885119899 = 119885119886 minus 1205821 =

minus119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2

(23)

6 Journal of Applied Mathematics

minus05

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0008 01 012 014 016 018 02 022 024 026

Zreal (Ohms)

Zim

ag(O

hms)

MeasuredCalculated

(a) Nyquist plot of a 33 F ultracapacitor (values for the model calculation1198771 = 01151Ohm 1198621 = 3634 F 1198772 = 008732Ohm 119877 = 0004589Ohmand 119862 = 003791 F)

Zreal (Ohms)

Zim

ag(O

hms)

minus14

minus12

minus1

minus08

minus06

minus04

minus02

00 01 02 03 04 05 06 07 08 09 1 11 12

MeasuredCalculated

(b) Nyquist plot of a 2 Fultracapacitor (values for themodel calculation1198771 =007529Ohm 1198621 = 2063 F 1198772 = 3564Ohm 119877 = 002241Ohm and 119862 =003445 F)

Figure 14 The Comparison between measured numbers and calculated numbers in Nyquist plot

Or else if |12058221205821| lt 1 then the lim119899rarrinfin|12058221205821|119899+1

rarr 0

lim119899rarrinfin

119885119899 = 119885119886 minus 1205822 =

minus119885119887 + radic1198852119887+ 4119885119886sdot119885119887

2

(24)

For the porous electrode model 119885119886 = 1119895120596119862 119885119887 = 119877In this case |12058221205821| lt 1 and (24) applies Therefore the

formula for the electrode impedance 11988511988601198870 in Figure 10 is

lim119899rarrinfin

11988511988601198870= minus

119877

2+ radic

1198772

4+

119877

119895120596119862 (25)

and the input impedance of the ultracapacitor model inFigure 11 is

119885119886119887 = 1198771 +1

1198951205961198621

+1

11198772 + 111988511988601198870

= 1198771 +1

1198951205961198621

+1

11198772 + 1 (minus1198772 + radic11987724 + 119877119895120596119862)

(26)

3 Validation of Model

Electrochemical impedance spectroscopy (EIS) is one of themost frequently used analytical tools for the characterizationof ultracapacitors This method was used to determine inputresistance and reactance of the ultracapacitor as a function offrequency at a given excitation voltage

Two ultracapacitors with nominal values of 33 F and 2 Fwere measured at room temperature Potentiostat cyclingtests were performed in the capacitor designing lab of St JudeMedicalrsquos Cardiac Rhythm Manufacture the Liberty SCwith a Gamry potentiostat (Reference 3000) The impedancewas measured by applying a sinusoidal 5mV excitationsuperimposed on a 12 VDC bias to the ultracapacitor andmeasuring the magnitude and phase of the current Thefrequency range was from 100mHz to 1 kHz

In this study a Panasonic 33 F23 V (EEC-HW0D335)ultracapacitor which is a new one and a Taiyo Yuden 2 F23 V(PAS1016LR2R3205) which had been charged and dischargedfor many times were measured Figures 14(a) and 14(b)show the magnitude of the input impedance plotted as afunction of the real part of the input impedance for the twoultracapacitors At frequencies above 100Hz the electricalbehavior of the ultracapacitors is more like a simple resistorthan a capacitor At low frequencies (119891 lt 1Hz) the imagi-nary part of the impedance dominates

The values of 1198771 1198621 1198772 119877 and 119862 for each capacitormodel were obtained by curve-fitting the measured data tothe model input impedance in (26) It seems that the 1198772value of old one (2 F) is larger than the new one (33 F) Theexperimental data curves of the real part and the imaginarypart of119885119886119887 were respectively fit using theMATLABCF Toolsoftware Figure 14 also shows the impedance obtained fromthe model in Figure 11 The plots show excellent agreementbetween the model and the measured values Figure 15 showsthe measured impedance of a new KAMCAP 400 F27 V(HP-2R7-J407UYLL) ultracapacitor with aDCbias voltage of12 V and 10mV disturbance variable from 100mHz to 100Hz

Journal of Applied Mathematics 7

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

00015 0016 0017 0018 0019 002 0021 0022 0023 0024

Zreal (Ohms)

Zim

ag(O

hms)

MeasuredCalculated

times10minus3

Figure 15 Complex-plane representation of a 400 F ultracapacitor(values for the model calculation 1198771 = 001798Ohm 1198621 = 4205 F1198772 = 0007511Ohm 119877 = 00001291Ohm and 119862 = 10 F)

at +25∘CThis experiment was carried out by Zahner IM6eXThe model data is also shown in Figure 15 In this model1198771 = 001798Ohm 1198621 = 4205 F 1198772 = 0007511Ohm119877 = 00001291Ohm and 119862 = 10 F It also shows excellentagreement between the measured and modeled values

4 Conclusion

A new model for describing the electrical behavior of ultra-capacitors is introduced based on an intuitive representationof the electrode pores as a fractal structure The new modelhas an infinite number of elements but is fully representedby only five real numbers A closed-form expression for theinput impedance was derived making it relatively easy to fitmeasured results to the model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014JBZ017)

References

[1] P Barrade ldquoSeries connection of supercapacitors comparativestudy of solutions for the active equalization of the voltagesrdquoin Proceedings of the 7th International Conference on Modeling

and Simulation of Electric Machines Converters and SystemsElectrimacs Montreal Canada August 2002

[2] J Van Mierlo P Van den Bossche and G Maggetto ldquoModelsof energy sources for EV and HEV Fuel cells batteriesultracapacitors flywheels and engine-generatorsrdquo Journal ofPower Sources vol 128 no 1 pp 76ndash89 2004

[3] R Kotz and M Carlen ldquoPrinciples and applications of electro-chemical capacitorsrdquo Electrochimica Acta vol 45 no 15-16 pp2483ndash2498 2000

[4] H Gualous D Bouquain A Berthon and J M KauffmannldquoExperimental study of supercapacitor serial resistance andcapacitance variations with temperaturerdquo Journal of PowerSources vol 123 no 1 pp 86ndash93 2003

[5] P L Taberna P Simon and J F Fauvarque ldquoElectrochemicalcharacteristics and impedance spectroscopy studies of carbon-carbon supercapacitorsrdquo Journal of the Electrochemical Societyvol 150 no 3 pp A292ndashA300 2003

[6] B E Conway Electrochemical Supercapacitors Kluwer Aca-demic PublishersPlenum Press New York NY USA 1999

[7] D Qu and H Shi ldquoStudies of activated carbons used in double-layer capacitorsrdquo Journal of Power Sources vol 74 no 1 pp 99ndash107 1998

[8] R De Levie inAdvancesin Electrochemistry and ElectrochemicalEngineering P Delahay and C T Tobias Eds vol 6 pp 329ndash397 John Wiley amp Sons New York NY USA 1967

[9] L Zubieta and R Bonert ldquoCharacterization of Double-LayerCapacitors (DLCs) for power electronics applicationsrdquo in Pro-ceedings of the IEEE Industry Applications Conference pp 1149ndash1154 October 1998

[10] F Rafik H Gualous R Gallay A Crausaz and A BerthonldquoFrequency thermal and voltage supercapacitor characteriza-tion and modelingrdquo Journal of Power Sources vol 165 no 2 pp928ndash934 2007

[11] E Karden S Buller and R W De Doncker ldquoA frequency-domain approach to dynamical modeling of electrochemicalpower sourcesrdquo Electrochimica Acta vol 47 no 13-14 pp 2347ndash2356 2002

[12] S Buller E Karden D Kok and R W de Doncker ldquoModelingthe dynamic behavior of supercapacitors using impedancespectroscopyrdquo IEEE Transactions on Industry Applications vol38 no 6 pp 1622ndash1626 2002

[13] D Riu N Retiere and D Linzen ldquoHalf-order modelling ofsupercapacitorsrdquo in Proceedings of the 39th IAS Annual Meetingon Record of the Industry Applications Conference vol 4 pp2550ndash2554 IEEE October 2004

[14] M Itagaki S Suzuki I Shitanda K Watanabe and HNakazawa ldquoImpedance analysis on electric double layer capaci-tor with transmission line modelrdquo Journal of Power Sources vol164 no 1 pp 415ndash424 2007

[15] M Spandana and A Jagruthi ldquoEnergy Storage UltracapacitorsrdquoYuva Engineers 2nd BTech EEE 2010

[16] X Yuan ldquoExtraction of the general term formula of fractionalrecursive sequence of numbermdash119909119899 = (119886119909119899minus1 + 119887) (119888119909119899minus1 + 119889)

by applying matrix methodrdquo Journal of China West NormalUniversity (Natural Sciences) vol 26 no 1 pp 102ndash104 2005

[17] Y Jin and Q Wang ldquoThe universal solution of ladder networkequivalent impedancerdquo Journal of Higher Correspondence Edu-cation vol 23 no 4 2010

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

2 Journal of Applied Mathematics

RpC

Rs

a

b

Figure 1 119877119862 circuit model for ultracapacitors

R1

C1

R2

C2Rp

La

b

Figure 2 Equivalent circuit developed by Gualous et al

of activated carbon particles and their surface area that varieswith the type of activated carbon used and the way it istreated But the model has six parameters to be identified anddoes not accurately describe ultracapacitor behavior at lowfrequencies

Zubieta and Bonert [9] provided amodel for the terminalbehavior of an ultracapacitor based on physical reasoningThis model (see Figure 3) has three distinct 119877119862 time con-stants 119877119891 119862119891 with the voltage-dependent capacitor 1198621198912(119865119881) which reflects the voltage dependence of the capac-itance dominates the behavior of the ultracapacitor in thetime over a period of seconds in response to a charge actionThe other branches 119877119897 with 119862119897 and 119877119898 with 119862119898 separatelydetermine the terminal behavior in the range of minutes andthe behavior for times longer than 10mins The resistor 119877119901is a leakage resistor representing the self discharge propertyof the ultracapacitor Others (eg [10]) have also developedequivalent circuit models based on variable time constantsto fit the measured AC impedance data of ultracapacitorsIn [10] the dependence of the resistance on frequency wasdivided into four distinct frequency zones and this modelincluded a voltage-dependent capacitor as well

However the pores of activated carbon in an ultraca-pacitor have a complex branch pore structure for whichany impedance analysis model should account Furthermorethe parameters in the above models especially the voltagedependent capacitance can be difficult to quantify

Buller et al [11 12] presented a model shown in Figure 4which can be used to describe the behavior of ultracapacitorsover a wide range of frequencies 119885119901 is the complex pore

+

minus

Rf Rm

Cm

Rp

a

b

Cf

VCfCf2 lowast VCf

CL

RL

Figure 3 Equivalent circuit developed by Zubieta and Bonert

RiL

C

2

C

2

n = 1 n = 2

Zp

Zp

C

120591

1205872

2

n2 middot C

120591

1205872

2

n2 middot C

middot middot middot

Figure 4 Equivalent circuit developed by Buller et al

impedance related to the porosity of the ultracapacitor Themathematical expression for 119885119901 is

119885119901 (119895120596) =120591 sdot coth (radic119895120596120591)

119862 sdot radic119895120596120591 (1)

Unfortunately this function is not well suited for circuitsimulation software because of the coth term [13] Moreoverwhen the number 119873 is high the series denominators of theformula in Figure 4

120591

1205872

2

1198992 sdot 119862(2)

become very large and may not be accurately determinedby measurements Buller et al [12] used 10 cells and 20parameters to model a 1400 F ultracapacitor

Qu and Shi [7] proposed an 119877119862-ladder network modelfor ultracapacitors based on the pore structure of activatedcarbon which is shown in Figure 5This model is particularlyintuitive because it illustrates howmore capacitance becomesavailable as the time constant of the charging cycle isincreased 119877119894 and 119862119894 (119894 = 1 119899) can be treated as theresistance and capacitance of the pores with certain pore size119877119894119862119894 gives the unit of time and indicates how fast the pore ofcertain size is

Itagaki et al [14] proposed a model for ultracapacitorsbased on a fractal pore structure with three sizes of cylin-drical pores This model consisted of resistors and compleximpedances connected in a tree-like structure

Journal of Applied Mathematics 3

R2 R3

C1 C2

Rn

Cnminus1

a

b

Cn

R1

C3

Figure 5 Equivalent circuit developed by Qu and Shi

Porouselectrode

Post

Tooth

RC

m teeth

n posts

Figure 6 Pore model of the ultracapacitor electrodes

This paper proposes a new electrical model for ultraca-pacitors that is also based on a fractal interpretation of theirstructure but does not make any assumption about the sizeor shape of the pores The new model is relatively simple andcan be fully described by five parameters

2 Model Description

Ultracapacitors are comprised of two highly porous activatedcarbon electrodes which take surface area and charge sep-aration distance to an extreme The surface areas can begreater than 21500 square feet per gram and the separationbetween the charged surfaces is reduced to distances on theorder of nanometers [15] The electrodes are immersed in asuitable electrolyte to facilitate the charge transfer and storagemechanism Charges accumulate in the pores resulting incapacitance

Consider the sample pore structure illustrated in Figure 6a central conducting structure called a ldquopostrdquo is lined withmany smaller structures referred to as ldquoteethrdquo Half of theposts in this structure are connected to one electrode of anultracapacitor and the other half are connected to the otherelectrodeThe ultracapacitor has 2119899 posts and every post has

Profile

a

d

Figure 7 Capacitance scheme of porosity

119898 teeth In this model each tooth is a smaller version of thepost that it is attached to

If we start by analyzing the capacitance of the structurewithout any teeth the total capacitance is 119899 times to the post-to-post capacitance because all of these capacitances are inparallel The equivalent series resistance of the ultracapacitoris the resistance of one pair of posts divided by 119899 since thepost resistances are also in parallel therefore an equivalentcircuit for an ultracapacitor consisting of interleaved posts ofuniform size would be a simple 119877119862 circuit where the valueof 119877 is 119877post119899 and the value of 119862 is 119862post times 119899 In this case119877post is the resistance associatedwith chargemoving fromoneelectrode into a single post and then to the second electrode119862post is the contribution to the capacitance of a single pair ofposts The time constant associated with charge populatingthe posts (ie the 119877119862 time constant) is

120591 = 119877119862 =

119877post

119899times 119862post times 119899 = 119877post119862post (3)

119877post is proportional to the length of the post and inverselyproportional to the cross-sectional area

119877post = 120590119897

120587 sdot 119860 (4)

where120590 is the electrical conductivity of the post 119897 is the lengthof the post and 119860 is the cross-sectional area of the post

The capacitance 119862post can be expressed as the capac-itance between any two posts times a constant that isdetermined by the number of posts in proximity to a givenpost A uniform post distribution is illustrated in Figure 7119862post is proportional to the length 119897 and inversely proportionalto the natural log of the ratio of the post separation 119889 to thepost radius 119886 as indicated below

119862post = 1198962120587120576119897

ln (119889119886) (5)

where 120576 is the permittivity of the dielectricA circuit model for the post-only ultracapacitor is pro-

vided in Figure 8Now consider the structure of the intermeshed teeth

between posts In fractal geometry each tooth would be ascaled down version of the post to which it was attached Ifwe assume that each post has 119898 teeth and the size of eachtooth is 1119898 times the size of the post then the resistance ofa single tooth is

119877tooth = 120590119897119898

1205871198601198982= 119898 times 119877post (6)

4 Journal of Applied Mathematics

C

R

Figure 8 The post-only model

CC

RR

Figure 9 Equivalent circuit for a capacitor with one set of posts andteeth

The capacitance 119862tooth can be expressed as

119862tooth = 1198962120587120576119897119898

ln ((119889119898) (119886119898))=

119862post

119898 (7)

For 119898 teeth per post in parallel the overall capacitance ismultiplied by119898 and the resistance is divided by119898 yielding

119877teeth per post =(119898 times 119877post)

119898= 119877post

(8)

119862teeth per post = (119862post

119898) times 119898 = 119862post (9)

From (8) and (9) it is clear that the time constantassociated with charge moving from the posts out onto theteeth is the same as the time constant associated with chargemoving out onto the posts and can be calculated with (3)Theequivalent circuit for the capacitor with both posts and teethis shown in Figure 9 Both resistors in this model have thesame value Both capacitors also have the same value

Employing (6) through (9) it is relatively straightforwardto show that adding 119898 smaller teeth to each tooth in thestructure shown in Figure 6 would result in the same amountof additional capacitance provided through the same amountof additional resistance The fractal geometry with manylayers of repeating structures would yield the equivalent

CCCCC C

RRRRa0

b0

Figure 10 Electrical model of a fractal electrode structure

R

CC

R R R

C

a0

b0

CCZa0b0Za0b0

Za0b0

R1

R2

C1a

b

Zab

Figure 11 Ultracapacitor model described by only 5 real numbers

circuit in Figure 10 where all of the resistors would have thesame value and so do all of the capacitors

Although the example in Figure 6 employs cylindricalpore structures the resistance and capacitance will scalesimilarly with nearly any branch-like pore structure Gen-erally the resistance of a branch will be proportional to itslength and inversely proportional to its cross-sectional areaThe capacitance between closely spaced branches will beproportional to its length and independent of cross-sectionalarea if the spacing between branches is also scaledThereforethe model in Figure 10 does not assume a particular porestructure as long as the successively smaller branches of thepores form fractal geometry

As the different number of time between charging anddischarging the deterioration of ultracapacitors is also notthe same Few electrolytes will be decomposed to form thefree insoluble product particles which increase the resistanceTherefore a resistor is paralleled in the model in Figure 10The greater the loss is the higher the resistor value will be

Considering its ohmic resistance characteristics andohmic capacitive characteristics a resistor and a capacitor arein series thus the model in Figure 11 is proposed which has5 independent component values

Although the model in Figure 11 has only 5 independentcomponent values it has an infinite number of elementsIn order to develop a closed-form expression for the inputimpedance it is convenient to use a Fourier transformtechnique For the impedance ladder network in Figure 12119885119899 is the equivalent input complex impedance and 119885119899minus1 is

Journal of Applied Mathematics 5

Zb Zb Zb Zb

Za Za Za Za

Znminus1Zn Z1Z2

Z0

Z0

Figure 12 Complex impedance ladder network

Zb

Za Znminus1Zn

Figure 13 The equivalent circuit of ladder network in Figure 12

the equivalent input complex impedance of the (119899 minus 1)-orderladder network (see Figure 13) So

119885119899 = (119885119887 + 119885119899minus1) 119885119886 (10)

The recurrence formula for 119885119899 is

119885119899 =(119885119886) sdot 119885119899minus1 + (119885119886sdot119885119887)

119885119899minus1 + (119885119886+119885119887) (11)

The solution of (11) can be obtained using amethod presentedin [16 17] In this case the coefficient matrix of 119885119899 which isexpressed by 119885119899minus1 is shown as follows

1198601 = [119885119886 119885119886sdot119885119887

1 119885119886+119885119887] (12)

119860119899 = 1198601198991 (13)

119860119899 in (13) is the coefficient matrix of 119885119899 which is expressedby 1198850 If 1198601 is expressed by its characteristic values andcharacteristic vectors 1198601198991 can be calculated as

1198601 = 119875[1205821 0

0 1205822]119875minus1 (14)

1198601198991 = 119875[

1205821198991 0

0 1205821198992]119875minus1 (15)

The characteristic values of 1198601 and its characteristic vectorsare

1205821 =

2119885119886+119885119887 + radic1198852119887+ 4119885119886sdot119885119887

2

1205822 =

2119885119886+119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2

119875 = [119885119886sdot119885119887 1205822 minus 119885119886minus119885119887

1205821minus119885119886 1]

119875minus1=

[[[[

[

1

119885119887 sdot (1205821+119885119886)minus1205822 minus 119885119886minus119885119887

119885119887 sdot (1205821+119885119886)

minus1205821minus119885119886

119885119887 sdot (1205821+119885119886)

119885119886sdot119885119887

119885119887 sdot (1205821+119885119886)

]]]]

]

(16)

Combining (16) into (15) results in

1198601198991

=

[[[[

[

1198851198861198851198871205821198991 + 1205821119885119886120582

1198992

119885119887 sdot (1205821+119885119886)

119885119886119885119887 (119885119886+119885119887minus1205822) (1205821198991minus1205821198992)

119885119887 sdot (1205821+119885119886)

(1205821minus119885119886) (1205821198991 minus 1205821198992)

119885119887 sdot (1205821+119885119886)

12058211198851198871205821198991 + 119885119886119885119887120582

1198992

119885119887 sdot (1205821+119885119886)

]]]]

]

(17)

119885119899 = ((1198851198861198851198871205821198991 + 1205821119885119886120582

1198992) sdot 1198850

+ 119885119886119885119887 (119885119886+119885119887 minus 1205822) (1205821198991 minus 1205821198992))

times ((1205821minus119885119886) (1205821198991 minus 1205821198992) 1198850 + (1205821119885119887120582

1198991 + 119885119886119885119887120582

1198992))minus1

(18)

where1205821 + 1205822 = 2119885119886 + 119885119887

1205821 sdot 1205822 = 1198852119886

(19)

If 1198850 = 119885119886 (18) can be simplified using (19) and as

119885119899 =(119885119886 minus 1205822) sdot 120582

119899+11 minus (119885119886 minus 1205821) sdot120582

119899+12

120582119899+11 minus 120582

119899+12

(20)

It is assumed that

1205822

1205821

=

2119885119886+119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2119885119886+119885119887 + radic1198852119887+ 4119885119886sdot119885119887

=

10038161003816100381610038161003816100381610038161003816

1205822

1205821

10038161003816100381610038161003816100381610038161003816

119890119895120579 (21)

Therefore

119885119899 =(119885119886 minus 1205822) minus (119885119886 minus 1205821) sdot

1003816100381610038161003816120582212058211003816100381610038161003816

119899+1119890119895(119899+1)120579

1 minus100381610038161003816100381612058221205821

1003816100381610038161003816

119899+1119890119895(119899+1)120579

(22)

If |12058221205821| gt 1 then the lim119899rarrinfin|12058221205821|119899+1

rarr infin

lim119899rarrinfin

119885119899 = 119885119886 minus 1205821 =

minus119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2

(23)

6 Journal of Applied Mathematics

minus05

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0008 01 012 014 016 018 02 022 024 026

Zreal (Ohms)

Zim

ag(O

hms)

MeasuredCalculated

(a) Nyquist plot of a 33 F ultracapacitor (values for the model calculation1198771 = 01151Ohm 1198621 = 3634 F 1198772 = 008732Ohm 119877 = 0004589Ohmand 119862 = 003791 F)

Zreal (Ohms)

Zim

ag(O

hms)

minus14

minus12

minus1

minus08

minus06

minus04

minus02

00 01 02 03 04 05 06 07 08 09 1 11 12

MeasuredCalculated

(b) Nyquist plot of a 2 Fultracapacitor (values for themodel calculation1198771 =007529Ohm 1198621 = 2063 F 1198772 = 3564Ohm 119877 = 002241Ohm and 119862 =003445 F)

Figure 14 The Comparison between measured numbers and calculated numbers in Nyquist plot

Or else if |12058221205821| lt 1 then the lim119899rarrinfin|12058221205821|119899+1

rarr 0

lim119899rarrinfin

119885119899 = 119885119886 minus 1205822 =

minus119885119887 + radic1198852119887+ 4119885119886sdot119885119887

2

(24)

For the porous electrode model 119885119886 = 1119895120596119862 119885119887 = 119877In this case |12058221205821| lt 1 and (24) applies Therefore the

formula for the electrode impedance 11988511988601198870 in Figure 10 is

lim119899rarrinfin

11988511988601198870= minus

119877

2+ radic

1198772

4+

119877

119895120596119862 (25)

and the input impedance of the ultracapacitor model inFigure 11 is

119885119886119887 = 1198771 +1

1198951205961198621

+1

11198772 + 111988511988601198870

= 1198771 +1

1198951205961198621

+1

11198772 + 1 (minus1198772 + radic11987724 + 119877119895120596119862)

(26)

3 Validation of Model

Electrochemical impedance spectroscopy (EIS) is one of themost frequently used analytical tools for the characterizationof ultracapacitors This method was used to determine inputresistance and reactance of the ultracapacitor as a function offrequency at a given excitation voltage

Two ultracapacitors with nominal values of 33 F and 2 Fwere measured at room temperature Potentiostat cyclingtests were performed in the capacitor designing lab of St JudeMedicalrsquos Cardiac Rhythm Manufacture the Liberty SCwith a Gamry potentiostat (Reference 3000) The impedancewas measured by applying a sinusoidal 5mV excitationsuperimposed on a 12 VDC bias to the ultracapacitor andmeasuring the magnitude and phase of the current Thefrequency range was from 100mHz to 1 kHz

In this study a Panasonic 33 F23 V (EEC-HW0D335)ultracapacitor which is a new one and a Taiyo Yuden 2 F23 V(PAS1016LR2R3205) which had been charged and dischargedfor many times were measured Figures 14(a) and 14(b)show the magnitude of the input impedance plotted as afunction of the real part of the input impedance for the twoultracapacitors At frequencies above 100Hz the electricalbehavior of the ultracapacitors is more like a simple resistorthan a capacitor At low frequencies (119891 lt 1Hz) the imagi-nary part of the impedance dominates

The values of 1198771 1198621 1198772 119877 and 119862 for each capacitormodel were obtained by curve-fitting the measured data tothe model input impedance in (26) It seems that the 1198772value of old one (2 F) is larger than the new one (33 F) Theexperimental data curves of the real part and the imaginarypart of119885119886119887 were respectively fit using theMATLABCF Toolsoftware Figure 14 also shows the impedance obtained fromthe model in Figure 11 The plots show excellent agreementbetween the model and the measured values Figure 15 showsthe measured impedance of a new KAMCAP 400 F27 V(HP-2R7-J407UYLL) ultracapacitor with aDCbias voltage of12 V and 10mV disturbance variable from 100mHz to 100Hz

Journal of Applied Mathematics 7

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

00015 0016 0017 0018 0019 002 0021 0022 0023 0024

Zreal (Ohms)

Zim

ag(O

hms)

MeasuredCalculated

times10minus3

Figure 15 Complex-plane representation of a 400 F ultracapacitor(values for the model calculation 1198771 = 001798Ohm 1198621 = 4205 F1198772 = 0007511Ohm 119877 = 00001291Ohm and 119862 = 10 F)

at +25∘CThis experiment was carried out by Zahner IM6eXThe model data is also shown in Figure 15 In this model1198771 = 001798Ohm 1198621 = 4205 F 1198772 = 0007511Ohm119877 = 00001291Ohm and 119862 = 10 F It also shows excellentagreement between the measured and modeled values

4 Conclusion

A new model for describing the electrical behavior of ultra-capacitors is introduced based on an intuitive representationof the electrode pores as a fractal structure The new modelhas an infinite number of elements but is fully representedby only five real numbers A closed-form expression for theinput impedance was derived making it relatively easy to fitmeasured results to the model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014JBZ017)

References

[1] P Barrade ldquoSeries connection of supercapacitors comparativestudy of solutions for the active equalization of the voltagesrdquoin Proceedings of the 7th International Conference on Modeling

and Simulation of Electric Machines Converters and SystemsElectrimacs Montreal Canada August 2002

[2] J Van Mierlo P Van den Bossche and G Maggetto ldquoModelsof energy sources for EV and HEV Fuel cells batteriesultracapacitors flywheels and engine-generatorsrdquo Journal ofPower Sources vol 128 no 1 pp 76ndash89 2004

[3] R Kotz and M Carlen ldquoPrinciples and applications of electro-chemical capacitorsrdquo Electrochimica Acta vol 45 no 15-16 pp2483ndash2498 2000

[4] H Gualous D Bouquain A Berthon and J M KauffmannldquoExperimental study of supercapacitor serial resistance andcapacitance variations with temperaturerdquo Journal of PowerSources vol 123 no 1 pp 86ndash93 2003

[5] P L Taberna P Simon and J F Fauvarque ldquoElectrochemicalcharacteristics and impedance spectroscopy studies of carbon-carbon supercapacitorsrdquo Journal of the Electrochemical Societyvol 150 no 3 pp A292ndashA300 2003

[6] B E Conway Electrochemical Supercapacitors Kluwer Aca-demic PublishersPlenum Press New York NY USA 1999

[7] D Qu and H Shi ldquoStudies of activated carbons used in double-layer capacitorsrdquo Journal of Power Sources vol 74 no 1 pp 99ndash107 1998

[8] R De Levie inAdvancesin Electrochemistry and ElectrochemicalEngineering P Delahay and C T Tobias Eds vol 6 pp 329ndash397 John Wiley amp Sons New York NY USA 1967

[9] L Zubieta and R Bonert ldquoCharacterization of Double-LayerCapacitors (DLCs) for power electronics applicationsrdquo in Pro-ceedings of the IEEE Industry Applications Conference pp 1149ndash1154 October 1998

[10] F Rafik H Gualous R Gallay A Crausaz and A BerthonldquoFrequency thermal and voltage supercapacitor characteriza-tion and modelingrdquo Journal of Power Sources vol 165 no 2 pp928ndash934 2007

[11] E Karden S Buller and R W De Doncker ldquoA frequency-domain approach to dynamical modeling of electrochemicalpower sourcesrdquo Electrochimica Acta vol 47 no 13-14 pp 2347ndash2356 2002

[12] S Buller E Karden D Kok and R W de Doncker ldquoModelingthe dynamic behavior of supercapacitors using impedancespectroscopyrdquo IEEE Transactions on Industry Applications vol38 no 6 pp 1622ndash1626 2002

[13] D Riu N Retiere and D Linzen ldquoHalf-order modelling ofsupercapacitorsrdquo in Proceedings of the 39th IAS Annual Meetingon Record of the Industry Applications Conference vol 4 pp2550ndash2554 IEEE October 2004

[14] M Itagaki S Suzuki I Shitanda K Watanabe and HNakazawa ldquoImpedance analysis on electric double layer capaci-tor with transmission line modelrdquo Journal of Power Sources vol164 no 1 pp 415ndash424 2007

[15] M Spandana and A Jagruthi ldquoEnergy Storage UltracapacitorsrdquoYuva Engineers 2nd BTech EEE 2010

[16] X Yuan ldquoExtraction of the general term formula of fractionalrecursive sequence of numbermdash119909119899 = (119886119909119899minus1 + 119887) (119888119909119899minus1 + 119889)

by applying matrix methodrdquo Journal of China West NormalUniversity (Natural Sciences) vol 26 no 1 pp 102ndash104 2005

[17] Y Jin and Q Wang ldquoThe universal solution of ladder networkequivalent impedancerdquo Journal of Higher Correspondence Edu-cation vol 23 no 4 2010

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

Journal of Applied Mathematics 3

R2 R3

C1 C2

Rn

Cnminus1

a

b

Cn

R1

C3

Figure 5 Equivalent circuit developed by Qu and Shi

Porouselectrode

Post

Tooth

RC

m teeth

n posts

Figure 6 Pore model of the ultracapacitor electrodes

This paper proposes a new electrical model for ultraca-pacitors that is also based on a fractal interpretation of theirstructure but does not make any assumption about the sizeor shape of the pores The new model is relatively simple andcan be fully described by five parameters

2 Model Description

Ultracapacitors are comprised of two highly porous activatedcarbon electrodes which take surface area and charge sep-aration distance to an extreme The surface areas can begreater than 21500 square feet per gram and the separationbetween the charged surfaces is reduced to distances on theorder of nanometers [15] The electrodes are immersed in asuitable electrolyte to facilitate the charge transfer and storagemechanism Charges accumulate in the pores resulting incapacitance

Consider the sample pore structure illustrated in Figure 6a central conducting structure called a ldquopostrdquo is lined withmany smaller structures referred to as ldquoteethrdquo Half of theposts in this structure are connected to one electrode of anultracapacitor and the other half are connected to the otherelectrodeThe ultracapacitor has 2119899 posts and every post has

Profile

a

d

Figure 7 Capacitance scheme of porosity

119898 teeth In this model each tooth is a smaller version of thepost that it is attached to

If we start by analyzing the capacitance of the structurewithout any teeth the total capacitance is 119899 times to the post-to-post capacitance because all of these capacitances are inparallel The equivalent series resistance of the ultracapacitoris the resistance of one pair of posts divided by 119899 since thepost resistances are also in parallel therefore an equivalentcircuit for an ultracapacitor consisting of interleaved posts ofuniform size would be a simple 119877119862 circuit where the valueof 119877 is 119877post119899 and the value of 119862 is 119862post times 119899 In this case119877post is the resistance associatedwith chargemoving fromoneelectrode into a single post and then to the second electrode119862post is the contribution to the capacitance of a single pair ofposts The time constant associated with charge populatingthe posts (ie the 119877119862 time constant) is

120591 = 119877119862 =

119877post

119899times 119862post times 119899 = 119877post119862post (3)

119877post is proportional to the length of the post and inverselyproportional to the cross-sectional area

119877post = 120590119897

120587 sdot 119860 (4)

where120590 is the electrical conductivity of the post 119897 is the lengthof the post and 119860 is the cross-sectional area of the post

The capacitance 119862post can be expressed as the capac-itance between any two posts times a constant that isdetermined by the number of posts in proximity to a givenpost A uniform post distribution is illustrated in Figure 7119862post is proportional to the length 119897 and inversely proportionalto the natural log of the ratio of the post separation 119889 to thepost radius 119886 as indicated below

119862post = 1198962120587120576119897

ln (119889119886) (5)

where 120576 is the permittivity of the dielectricA circuit model for the post-only ultracapacitor is pro-

vided in Figure 8Now consider the structure of the intermeshed teeth

between posts In fractal geometry each tooth would be ascaled down version of the post to which it was attached Ifwe assume that each post has 119898 teeth and the size of eachtooth is 1119898 times the size of the post then the resistance ofa single tooth is

119877tooth = 120590119897119898

1205871198601198982= 119898 times 119877post (6)

4 Journal of Applied Mathematics

C

R

Figure 8 The post-only model

CC

RR

Figure 9 Equivalent circuit for a capacitor with one set of posts andteeth

The capacitance 119862tooth can be expressed as

119862tooth = 1198962120587120576119897119898

ln ((119889119898) (119886119898))=

119862post

119898 (7)

For 119898 teeth per post in parallel the overall capacitance ismultiplied by119898 and the resistance is divided by119898 yielding

119877teeth per post =(119898 times 119877post)

119898= 119877post

(8)

119862teeth per post = (119862post

119898) times 119898 = 119862post (9)

From (8) and (9) it is clear that the time constantassociated with charge moving from the posts out onto theteeth is the same as the time constant associated with chargemoving out onto the posts and can be calculated with (3)Theequivalent circuit for the capacitor with both posts and teethis shown in Figure 9 Both resistors in this model have thesame value Both capacitors also have the same value

Employing (6) through (9) it is relatively straightforwardto show that adding 119898 smaller teeth to each tooth in thestructure shown in Figure 6 would result in the same amountof additional capacitance provided through the same amountof additional resistance The fractal geometry with manylayers of repeating structures would yield the equivalent

CCCCC C

RRRRa0

b0

Figure 10 Electrical model of a fractal electrode structure

R

CC

R R R

C

a0

b0

CCZa0b0Za0b0

Za0b0

R1

R2

C1a

b

Zab

Figure 11 Ultracapacitor model described by only 5 real numbers

circuit in Figure 10 where all of the resistors would have thesame value and so do all of the capacitors

Although the example in Figure 6 employs cylindricalpore structures the resistance and capacitance will scalesimilarly with nearly any branch-like pore structure Gen-erally the resistance of a branch will be proportional to itslength and inversely proportional to its cross-sectional areaThe capacitance between closely spaced branches will beproportional to its length and independent of cross-sectionalarea if the spacing between branches is also scaledThereforethe model in Figure 10 does not assume a particular porestructure as long as the successively smaller branches of thepores form fractal geometry

As the different number of time between charging anddischarging the deterioration of ultracapacitors is also notthe same Few electrolytes will be decomposed to form thefree insoluble product particles which increase the resistanceTherefore a resistor is paralleled in the model in Figure 10The greater the loss is the higher the resistor value will be

Considering its ohmic resistance characteristics andohmic capacitive characteristics a resistor and a capacitor arein series thus the model in Figure 11 is proposed which has5 independent component values

Although the model in Figure 11 has only 5 independentcomponent values it has an infinite number of elementsIn order to develop a closed-form expression for the inputimpedance it is convenient to use a Fourier transformtechnique For the impedance ladder network in Figure 12119885119899 is the equivalent input complex impedance and 119885119899minus1 is

Journal of Applied Mathematics 5

Zb Zb Zb Zb

Za Za Za Za

Znminus1Zn Z1Z2

Z0

Z0

Figure 12 Complex impedance ladder network

Zb

Za Znminus1Zn

Figure 13 The equivalent circuit of ladder network in Figure 12

the equivalent input complex impedance of the (119899 minus 1)-orderladder network (see Figure 13) So

119885119899 = (119885119887 + 119885119899minus1) 119885119886 (10)

The recurrence formula for 119885119899 is

119885119899 =(119885119886) sdot 119885119899minus1 + (119885119886sdot119885119887)

119885119899minus1 + (119885119886+119885119887) (11)

The solution of (11) can be obtained using amethod presentedin [16 17] In this case the coefficient matrix of 119885119899 which isexpressed by 119885119899minus1 is shown as follows

1198601 = [119885119886 119885119886sdot119885119887

1 119885119886+119885119887] (12)

119860119899 = 1198601198991 (13)

119860119899 in (13) is the coefficient matrix of 119885119899 which is expressedby 1198850 If 1198601 is expressed by its characteristic values andcharacteristic vectors 1198601198991 can be calculated as

1198601 = 119875[1205821 0

0 1205822]119875minus1 (14)

1198601198991 = 119875[

1205821198991 0

0 1205821198992]119875minus1 (15)

The characteristic values of 1198601 and its characteristic vectorsare

1205821 =

2119885119886+119885119887 + radic1198852119887+ 4119885119886sdot119885119887

2

1205822 =

2119885119886+119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2

119875 = [119885119886sdot119885119887 1205822 minus 119885119886minus119885119887

1205821minus119885119886 1]

119875minus1=

[[[[

[

1

119885119887 sdot (1205821+119885119886)minus1205822 minus 119885119886minus119885119887

119885119887 sdot (1205821+119885119886)

minus1205821minus119885119886

119885119887 sdot (1205821+119885119886)

119885119886sdot119885119887

119885119887 sdot (1205821+119885119886)

]]]]

]

(16)

Combining (16) into (15) results in

1198601198991

=

[[[[

[

1198851198861198851198871205821198991 + 1205821119885119886120582

1198992

119885119887 sdot (1205821+119885119886)

119885119886119885119887 (119885119886+119885119887minus1205822) (1205821198991minus1205821198992)

119885119887 sdot (1205821+119885119886)

(1205821minus119885119886) (1205821198991 minus 1205821198992)

119885119887 sdot (1205821+119885119886)

12058211198851198871205821198991 + 119885119886119885119887120582

1198992

119885119887 sdot (1205821+119885119886)

]]]]

]

(17)

119885119899 = ((1198851198861198851198871205821198991 + 1205821119885119886120582

1198992) sdot 1198850

+ 119885119886119885119887 (119885119886+119885119887 minus 1205822) (1205821198991 minus 1205821198992))

times ((1205821minus119885119886) (1205821198991 minus 1205821198992) 1198850 + (1205821119885119887120582

1198991 + 119885119886119885119887120582

1198992))minus1

(18)

where1205821 + 1205822 = 2119885119886 + 119885119887

1205821 sdot 1205822 = 1198852119886

(19)

If 1198850 = 119885119886 (18) can be simplified using (19) and as

119885119899 =(119885119886 minus 1205822) sdot 120582

119899+11 minus (119885119886 minus 1205821) sdot120582

119899+12

120582119899+11 minus 120582

119899+12

(20)

It is assumed that

1205822

1205821

=

2119885119886+119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2119885119886+119885119887 + radic1198852119887+ 4119885119886sdot119885119887

=

10038161003816100381610038161003816100381610038161003816

1205822

1205821

10038161003816100381610038161003816100381610038161003816

119890119895120579 (21)

Therefore

119885119899 =(119885119886 minus 1205822) minus (119885119886 minus 1205821) sdot

1003816100381610038161003816120582212058211003816100381610038161003816

119899+1119890119895(119899+1)120579

1 minus100381610038161003816100381612058221205821

1003816100381610038161003816

119899+1119890119895(119899+1)120579

(22)

If |12058221205821| gt 1 then the lim119899rarrinfin|12058221205821|119899+1

rarr infin

lim119899rarrinfin

119885119899 = 119885119886 minus 1205821 =

minus119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2

(23)

6 Journal of Applied Mathematics

minus05

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0008 01 012 014 016 018 02 022 024 026

Zreal (Ohms)

Zim

ag(O

hms)

MeasuredCalculated

(a) Nyquist plot of a 33 F ultracapacitor (values for the model calculation1198771 = 01151Ohm 1198621 = 3634 F 1198772 = 008732Ohm 119877 = 0004589Ohmand 119862 = 003791 F)

Zreal (Ohms)

Zim

ag(O

hms)

minus14

minus12

minus1

minus08

minus06

minus04

minus02

00 01 02 03 04 05 06 07 08 09 1 11 12

MeasuredCalculated

(b) Nyquist plot of a 2 Fultracapacitor (values for themodel calculation1198771 =007529Ohm 1198621 = 2063 F 1198772 = 3564Ohm 119877 = 002241Ohm and 119862 =003445 F)

Figure 14 The Comparison between measured numbers and calculated numbers in Nyquist plot

Or else if |12058221205821| lt 1 then the lim119899rarrinfin|12058221205821|119899+1

rarr 0

lim119899rarrinfin

119885119899 = 119885119886 minus 1205822 =

minus119885119887 + radic1198852119887+ 4119885119886sdot119885119887

2

(24)

For the porous electrode model 119885119886 = 1119895120596119862 119885119887 = 119877In this case |12058221205821| lt 1 and (24) applies Therefore the

formula for the electrode impedance 11988511988601198870 in Figure 10 is

lim119899rarrinfin

11988511988601198870= minus

119877

2+ radic

1198772

4+

119877

119895120596119862 (25)

and the input impedance of the ultracapacitor model inFigure 11 is

119885119886119887 = 1198771 +1

1198951205961198621

+1

11198772 + 111988511988601198870

= 1198771 +1

1198951205961198621

+1

11198772 + 1 (minus1198772 + radic11987724 + 119877119895120596119862)

(26)

3 Validation of Model

Electrochemical impedance spectroscopy (EIS) is one of themost frequently used analytical tools for the characterizationof ultracapacitors This method was used to determine inputresistance and reactance of the ultracapacitor as a function offrequency at a given excitation voltage

Two ultracapacitors with nominal values of 33 F and 2 Fwere measured at room temperature Potentiostat cyclingtests were performed in the capacitor designing lab of St JudeMedicalrsquos Cardiac Rhythm Manufacture the Liberty SCwith a Gamry potentiostat (Reference 3000) The impedancewas measured by applying a sinusoidal 5mV excitationsuperimposed on a 12 VDC bias to the ultracapacitor andmeasuring the magnitude and phase of the current Thefrequency range was from 100mHz to 1 kHz

In this study a Panasonic 33 F23 V (EEC-HW0D335)ultracapacitor which is a new one and a Taiyo Yuden 2 F23 V(PAS1016LR2R3205) which had been charged and dischargedfor many times were measured Figures 14(a) and 14(b)show the magnitude of the input impedance plotted as afunction of the real part of the input impedance for the twoultracapacitors At frequencies above 100Hz the electricalbehavior of the ultracapacitors is more like a simple resistorthan a capacitor At low frequencies (119891 lt 1Hz) the imagi-nary part of the impedance dominates

The values of 1198771 1198621 1198772 119877 and 119862 for each capacitormodel were obtained by curve-fitting the measured data tothe model input impedance in (26) It seems that the 1198772value of old one (2 F) is larger than the new one (33 F) Theexperimental data curves of the real part and the imaginarypart of119885119886119887 were respectively fit using theMATLABCF Toolsoftware Figure 14 also shows the impedance obtained fromthe model in Figure 11 The plots show excellent agreementbetween the model and the measured values Figure 15 showsthe measured impedance of a new KAMCAP 400 F27 V(HP-2R7-J407UYLL) ultracapacitor with aDCbias voltage of12 V and 10mV disturbance variable from 100mHz to 100Hz

Journal of Applied Mathematics 7

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

00015 0016 0017 0018 0019 002 0021 0022 0023 0024

Zreal (Ohms)

Zim

ag(O

hms)

MeasuredCalculated

times10minus3

Figure 15 Complex-plane representation of a 400 F ultracapacitor(values for the model calculation 1198771 = 001798Ohm 1198621 = 4205 F1198772 = 0007511Ohm 119877 = 00001291Ohm and 119862 = 10 F)

at +25∘CThis experiment was carried out by Zahner IM6eXThe model data is also shown in Figure 15 In this model1198771 = 001798Ohm 1198621 = 4205 F 1198772 = 0007511Ohm119877 = 00001291Ohm and 119862 = 10 F It also shows excellentagreement between the measured and modeled values

4 Conclusion

A new model for describing the electrical behavior of ultra-capacitors is introduced based on an intuitive representationof the electrode pores as a fractal structure The new modelhas an infinite number of elements but is fully representedby only five real numbers A closed-form expression for theinput impedance was derived making it relatively easy to fitmeasured results to the model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014JBZ017)

References

[1] P Barrade ldquoSeries connection of supercapacitors comparativestudy of solutions for the active equalization of the voltagesrdquoin Proceedings of the 7th International Conference on Modeling

and Simulation of Electric Machines Converters and SystemsElectrimacs Montreal Canada August 2002

[2] J Van Mierlo P Van den Bossche and G Maggetto ldquoModelsof energy sources for EV and HEV Fuel cells batteriesultracapacitors flywheels and engine-generatorsrdquo Journal ofPower Sources vol 128 no 1 pp 76ndash89 2004

[3] R Kotz and M Carlen ldquoPrinciples and applications of electro-chemical capacitorsrdquo Electrochimica Acta vol 45 no 15-16 pp2483ndash2498 2000

[4] H Gualous D Bouquain A Berthon and J M KauffmannldquoExperimental study of supercapacitor serial resistance andcapacitance variations with temperaturerdquo Journal of PowerSources vol 123 no 1 pp 86ndash93 2003

[5] P L Taberna P Simon and J F Fauvarque ldquoElectrochemicalcharacteristics and impedance spectroscopy studies of carbon-carbon supercapacitorsrdquo Journal of the Electrochemical Societyvol 150 no 3 pp A292ndashA300 2003

[6] B E Conway Electrochemical Supercapacitors Kluwer Aca-demic PublishersPlenum Press New York NY USA 1999

[7] D Qu and H Shi ldquoStudies of activated carbons used in double-layer capacitorsrdquo Journal of Power Sources vol 74 no 1 pp 99ndash107 1998

[8] R De Levie inAdvancesin Electrochemistry and ElectrochemicalEngineering P Delahay and C T Tobias Eds vol 6 pp 329ndash397 John Wiley amp Sons New York NY USA 1967

[9] L Zubieta and R Bonert ldquoCharacterization of Double-LayerCapacitors (DLCs) for power electronics applicationsrdquo in Pro-ceedings of the IEEE Industry Applications Conference pp 1149ndash1154 October 1998

[10] F Rafik H Gualous R Gallay A Crausaz and A BerthonldquoFrequency thermal and voltage supercapacitor characteriza-tion and modelingrdquo Journal of Power Sources vol 165 no 2 pp928ndash934 2007

[11] E Karden S Buller and R W De Doncker ldquoA frequency-domain approach to dynamical modeling of electrochemicalpower sourcesrdquo Electrochimica Acta vol 47 no 13-14 pp 2347ndash2356 2002

[12] S Buller E Karden D Kok and R W de Doncker ldquoModelingthe dynamic behavior of supercapacitors using impedancespectroscopyrdquo IEEE Transactions on Industry Applications vol38 no 6 pp 1622ndash1626 2002

[13] D Riu N Retiere and D Linzen ldquoHalf-order modelling ofsupercapacitorsrdquo in Proceedings of the 39th IAS Annual Meetingon Record of the Industry Applications Conference vol 4 pp2550ndash2554 IEEE October 2004

[14] M Itagaki S Suzuki I Shitanda K Watanabe and HNakazawa ldquoImpedance analysis on electric double layer capaci-tor with transmission line modelrdquo Journal of Power Sources vol164 no 1 pp 415ndash424 2007

[15] M Spandana and A Jagruthi ldquoEnergy Storage UltracapacitorsrdquoYuva Engineers 2nd BTech EEE 2010

[16] X Yuan ldquoExtraction of the general term formula of fractionalrecursive sequence of numbermdash119909119899 = (119886119909119899minus1 + 119887) (119888119909119899minus1 + 119889)

by applying matrix methodrdquo Journal of China West NormalUniversity (Natural Sciences) vol 26 no 1 pp 102ndash104 2005

[17] Y Jin and Q Wang ldquoThe universal solution of ladder networkequivalent impedancerdquo Journal of Higher Correspondence Edu-cation vol 23 no 4 2010

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

4 Journal of Applied Mathematics

C

R

Figure 8 The post-only model

CC

RR

Figure 9 Equivalent circuit for a capacitor with one set of posts andteeth

The capacitance 119862tooth can be expressed as

119862tooth = 1198962120587120576119897119898

ln ((119889119898) (119886119898))=

119862post

119898 (7)

For 119898 teeth per post in parallel the overall capacitance ismultiplied by119898 and the resistance is divided by119898 yielding

119877teeth per post =(119898 times 119877post)

119898= 119877post

(8)

119862teeth per post = (119862post

119898) times 119898 = 119862post (9)

From (8) and (9) it is clear that the time constantassociated with charge moving from the posts out onto theteeth is the same as the time constant associated with chargemoving out onto the posts and can be calculated with (3)Theequivalent circuit for the capacitor with both posts and teethis shown in Figure 9 Both resistors in this model have thesame value Both capacitors also have the same value

Employing (6) through (9) it is relatively straightforwardto show that adding 119898 smaller teeth to each tooth in thestructure shown in Figure 6 would result in the same amountof additional capacitance provided through the same amountof additional resistance The fractal geometry with manylayers of repeating structures would yield the equivalent

CCCCC C

RRRRa0

b0

Figure 10 Electrical model of a fractal electrode structure

R

CC

R R R

C

a0

b0

CCZa0b0Za0b0

Za0b0

R1

R2

C1a

b

Zab

Figure 11 Ultracapacitor model described by only 5 real numbers

circuit in Figure 10 where all of the resistors would have thesame value and so do all of the capacitors

Although the example in Figure 6 employs cylindricalpore structures the resistance and capacitance will scalesimilarly with nearly any branch-like pore structure Gen-erally the resistance of a branch will be proportional to itslength and inversely proportional to its cross-sectional areaThe capacitance between closely spaced branches will beproportional to its length and independent of cross-sectionalarea if the spacing between branches is also scaledThereforethe model in Figure 10 does not assume a particular porestructure as long as the successively smaller branches of thepores form fractal geometry

As the different number of time between charging anddischarging the deterioration of ultracapacitors is also notthe same Few electrolytes will be decomposed to form thefree insoluble product particles which increase the resistanceTherefore a resistor is paralleled in the model in Figure 10The greater the loss is the higher the resistor value will be

Considering its ohmic resistance characteristics andohmic capacitive characteristics a resistor and a capacitor arein series thus the model in Figure 11 is proposed which has5 independent component values

Although the model in Figure 11 has only 5 independentcomponent values it has an infinite number of elementsIn order to develop a closed-form expression for the inputimpedance it is convenient to use a Fourier transformtechnique For the impedance ladder network in Figure 12119885119899 is the equivalent input complex impedance and 119885119899minus1 is

Journal of Applied Mathematics 5

Zb Zb Zb Zb

Za Za Za Za

Znminus1Zn Z1Z2

Z0

Z0

Figure 12 Complex impedance ladder network

Zb

Za Znminus1Zn

Figure 13 The equivalent circuit of ladder network in Figure 12

the equivalent input complex impedance of the (119899 minus 1)-orderladder network (see Figure 13) So

119885119899 = (119885119887 + 119885119899minus1) 119885119886 (10)

The recurrence formula for 119885119899 is

119885119899 =(119885119886) sdot 119885119899minus1 + (119885119886sdot119885119887)

119885119899minus1 + (119885119886+119885119887) (11)

The solution of (11) can be obtained using amethod presentedin [16 17] In this case the coefficient matrix of 119885119899 which isexpressed by 119885119899minus1 is shown as follows

1198601 = [119885119886 119885119886sdot119885119887

1 119885119886+119885119887] (12)

119860119899 = 1198601198991 (13)

119860119899 in (13) is the coefficient matrix of 119885119899 which is expressedby 1198850 If 1198601 is expressed by its characteristic values andcharacteristic vectors 1198601198991 can be calculated as

1198601 = 119875[1205821 0

0 1205822]119875minus1 (14)

1198601198991 = 119875[

1205821198991 0

0 1205821198992]119875minus1 (15)

The characteristic values of 1198601 and its characteristic vectorsare

1205821 =

2119885119886+119885119887 + radic1198852119887+ 4119885119886sdot119885119887

2

1205822 =

2119885119886+119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2

119875 = [119885119886sdot119885119887 1205822 minus 119885119886minus119885119887

1205821minus119885119886 1]

119875minus1=

[[[[

[

1

119885119887 sdot (1205821+119885119886)minus1205822 minus 119885119886minus119885119887

119885119887 sdot (1205821+119885119886)

minus1205821minus119885119886

119885119887 sdot (1205821+119885119886)

119885119886sdot119885119887

119885119887 sdot (1205821+119885119886)

]]]]

]

(16)

Combining (16) into (15) results in

1198601198991

=

[[[[

[

1198851198861198851198871205821198991 + 1205821119885119886120582

1198992

119885119887 sdot (1205821+119885119886)

119885119886119885119887 (119885119886+119885119887minus1205822) (1205821198991minus1205821198992)

119885119887 sdot (1205821+119885119886)

(1205821minus119885119886) (1205821198991 minus 1205821198992)

119885119887 sdot (1205821+119885119886)

12058211198851198871205821198991 + 119885119886119885119887120582

1198992

119885119887 sdot (1205821+119885119886)

]]]]

]

(17)

119885119899 = ((1198851198861198851198871205821198991 + 1205821119885119886120582

1198992) sdot 1198850

+ 119885119886119885119887 (119885119886+119885119887 minus 1205822) (1205821198991 minus 1205821198992))

times ((1205821minus119885119886) (1205821198991 minus 1205821198992) 1198850 + (1205821119885119887120582

1198991 + 119885119886119885119887120582

1198992))minus1

(18)

where1205821 + 1205822 = 2119885119886 + 119885119887

1205821 sdot 1205822 = 1198852119886

(19)

If 1198850 = 119885119886 (18) can be simplified using (19) and as

119885119899 =(119885119886 minus 1205822) sdot 120582

119899+11 minus (119885119886 minus 1205821) sdot120582

119899+12

120582119899+11 minus 120582

119899+12

(20)

It is assumed that

1205822

1205821

=

2119885119886+119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2119885119886+119885119887 + radic1198852119887+ 4119885119886sdot119885119887

=

10038161003816100381610038161003816100381610038161003816

1205822

1205821

10038161003816100381610038161003816100381610038161003816

119890119895120579 (21)

Therefore

119885119899 =(119885119886 minus 1205822) minus (119885119886 minus 1205821) sdot

1003816100381610038161003816120582212058211003816100381610038161003816

119899+1119890119895(119899+1)120579

1 minus100381610038161003816100381612058221205821

1003816100381610038161003816

119899+1119890119895(119899+1)120579

(22)

If |12058221205821| gt 1 then the lim119899rarrinfin|12058221205821|119899+1

rarr infin

lim119899rarrinfin

119885119899 = 119885119886 minus 1205821 =

minus119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2

(23)

6 Journal of Applied Mathematics

minus05

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0008 01 012 014 016 018 02 022 024 026

Zreal (Ohms)

Zim

ag(O

hms)

MeasuredCalculated

(a) Nyquist plot of a 33 F ultracapacitor (values for the model calculation1198771 = 01151Ohm 1198621 = 3634 F 1198772 = 008732Ohm 119877 = 0004589Ohmand 119862 = 003791 F)

Zreal (Ohms)

Zim

ag(O

hms)

minus14

minus12

minus1

minus08

minus06

minus04

minus02

00 01 02 03 04 05 06 07 08 09 1 11 12

MeasuredCalculated

(b) Nyquist plot of a 2 Fultracapacitor (values for themodel calculation1198771 =007529Ohm 1198621 = 2063 F 1198772 = 3564Ohm 119877 = 002241Ohm and 119862 =003445 F)

Figure 14 The Comparison between measured numbers and calculated numbers in Nyquist plot

Or else if |12058221205821| lt 1 then the lim119899rarrinfin|12058221205821|119899+1

rarr 0

lim119899rarrinfin

119885119899 = 119885119886 minus 1205822 =

minus119885119887 + radic1198852119887+ 4119885119886sdot119885119887

2

(24)

For the porous electrode model 119885119886 = 1119895120596119862 119885119887 = 119877In this case |12058221205821| lt 1 and (24) applies Therefore the

formula for the electrode impedance 11988511988601198870 in Figure 10 is

lim119899rarrinfin

11988511988601198870= minus

119877

2+ radic

1198772

4+

119877

119895120596119862 (25)

and the input impedance of the ultracapacitor model inFigure 11 is

119885119886119887 = 1198771 +1

1198951205961198621

+1

11198772 + 111988511988601198870

= 1198771 +1

1198951205961198621

+1

11198772 + 1 (minus1198772 + radic11987724 + 119877119895120596119862)

(26)

3 Validation of Model

Electrochemical impedance spectroscopy (EIS) is one of themost frequently used analytical tools for the characterizationof ultracapacitors This method was used to determine inputresistance and reactance of the ultracapacitor as a function offrequency at a given excitation voltage

Two ultracapacitors with nominal values of 33 F and 2 Fwere measured at room temperature Potentiostat cyclingtests were performed in the capacitor designing lab of St JudeMedicalrsquos Cardiac Rhythm Manufacture the Liberty SCwith a Gamry potentiostat (Reference 3000) The impedancewas measured by applying a sinusoidal 5mV excitationsuperimposed on a 12 VDC bias to the ultracapacitor andmeasuring the magnitude and phase of the current Thefrequency range was from 100mHz to 1 kHz

In this study a Panasonic 33 F23 V (EEC-HW0D335)ultracapacitor which is a new one and a Taiyo Yuden 2 F23 V(PAS1016LR2R3205) which had been charged and dischargedfor many times were measured Figures 14(a) and 14(b)show the magnitude of the input impedance plotted as afunction of the real part of the input impedance for the twoultracapacitors At frequencies above 100Hz the electricalbehavior of the ultracapacitors is more like a simple resistorthan a capacitor At low frequencies (119891 lt 1Hz) the imagi-nary part of the impedance dominates

The values of 1198771 1198621 1198772 119877 and 119862 for each capacitormodel were obtained by curve-fitting the measured data tothe model input impedance in (26) It seems that the 1198772value of old one (2 F) is larger than the new one (33 F) Theexperimental data curves of the real part and the imaginarypart of119885119886119887 were respectively fit using theMATLABCF Toolsoftware Figure 14 also shows the impedance obtained fromthe model in Figure 11 The plots show excellent agreementbetween the model and the measured values Figure 15 showsthe measured impedance of a new KAMCAP 400 F27 V(HP-2R7-J407UYLL) ultracapacitor with aDCbias voltage of12 V and 10mV disturbance variable from 100mHz to 100Hz

Journal of Applied Mathematics 7

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

00015 0016 0017 0018 0019 002 0021 0022 0023 0024

Zreal (Ohms)

Zim

ag(O

hms)

MeasuredCalculated

times10minus3

Figure 15 Complex-plane representation of a 400 F ultracapacitor(values for the model calculation 1198771 = 001798Ohm 1198621 = 4205 F1198772 = 0007511Ohm 119877 = 00001291Ohm and 119862 = 10 F)

at +25∘CThis experiment was carried out by Zahner IM6eXThe model data is also shown in Figure 15 In this model1198771 = 001798Ohm 1198621 = 4205 F 1198772 = 0007511Ohm119877 = 00001291Ohm and 119862 = 10 F It also shows excellentagreement between the measured and modeled values

4 Conclusion

A new model for describing the electrical behavior of ultra-capacitors is introduced based on an intuitive representationof the electrode pores as a fractal structure The new modelhas an infinite number of elements but is fully representedby only five real numbers A closed-form expression for theinput impedance was derived making it relatively easy to fitmeasured results to the model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014JBZ017)

References

[1] P Barrade ldquoSeries connection of supercapacitors comparativestudy of solutions for the active equalization of the voltagesrdquoin Proceedings of the 7th International Conference on Modeling

and Simulation of Electric Machines Converters and SystemsElectrimacs Montreal Canada August 2002

[2] J Van Mierlo P Van den Bossche and G Maggetto ldquoModelsof energy sources for EV and HEV Fuel cells batteriesultracapacitors flywheels and engine-generatorsrdquo Journal ofPower Sources vol 128 no 1 pp 76ndash89 2004

[3] R Kotz and M Carlen ldquoPrinciples and applications of electro-chemical capacitorsrdquo Electrochimica Acta vol 45 no 15-16 pp2483ndash2498 2000

[4] H Gualous D Bouquain A Berthon and J M KauffmannldquoExperimental study of supercapacitor serial resistance andcapacitance variations with temperaturerdquo Journal of PowerSources vol 123 no 1 pp 86ndash93 2003

[5] P L Taberna P Simon and J F Fauvarque ldquoElectrochemicalcharacteristics and impedance spectroscopy studies of carbon-carbon supercapacitorsrdquo Journal of the Electrochemical Societyvol 150 no 3 pp A292ndashA300 2003

[6] B E Conway Electrochemical Supercapacitors Kluwer Aca-demic PublishersPlenum Press New York NY USA 1999

[7] D Qu and H Shi ldquoStudies of activated carbons used in double-layer capacitorsrdquo Journal of Power Sources vol 74 no 1 pp 99ndash107 1998

[8] R De Levie inAdvancesin Electrochemistry and ElectrochemicalEngineering P Delahay and C T Tobias Eds vol 6 pp 329ndash397 John Wiley amp Sons New York NY USA 1967

[9] L Zubieta and R Bonert ldquoCharacterization of Double-LayerCapacitors (DLCs) for power electronics applicationsrdquo in Pro-ceedings of the IEEE Industry Applications Conference pp 1149ndash1154 October 1998

[10] F Rafik H Gualous R Gallay A Crausaz and A BerthonldquoFrequency thermal and voltage supercapacitor characteriza-tion and modelingrdquo Journal of Power Sources vol 165 no 2 pp928ndash934 2007

[11] E Karden S Buller and R W De Doncker ldquoA frequency-domain approach to dynamical modeling of electrochemicalpower sourcesrdquo Electrochimica Acta vol 47 no 13-14 pp 2347ndash2356 2002

[12] S Buller E Karden D Kok and R W de Doncker ldquoModelingthe dynamic behavior of supercapacitors using impedancespectroscopyrdquo IEEE Transactions on Industry Applications vol38 no 6 pp 1622ndash1626 2002

[13] D Riu N Retiere and D Linzen ldquoHalf-order modelling ofsupercapacitorsrdquo in Proceedings of the 39th IAS Annual Meetingon Record of the Industry Applications Conference vol 4 pp2550ndash2554 IEEE October 2004

[14] M Itagaki S Suzuki I Shitanda K Watanabe and HNakazawa ldquoImpedance analysis on electric double layer capaci-tor with transmission line modelrdquo Journal of Power Sources vol164 no 1 pp 415ndash424 2007

[15] M Spandana and A Jagruthi ldquoEnergy Storage UltracapacitorsrdquoYuva Engineers 2nd BTech EEE 2010

[16] X Yuan ldquoExtraction of the general term formula of fractionalrecursive sequence of numbermdash119909119899 = (119886119909119899minus1 + 119887) (119888119909119899minus1 + 119889)

by applying matrix methodrdquo Journal of China West NormalUniversity (Natural Sciences) vol 26 no 1 pp 102ndash104 2005

[17] Y Jin and Q Wang ldquoThe universal solution of ladder networkequivalent impedancerdquo Journal of Higher Correspondence Edu-cation vol 23 no 4 2010

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

Journal of Applied Mathematics 5

Zb Zb Zb Zb

Za Za Za Za

Znminus1Zn Z1Z2

Z0

Z0

Figure 12 Complex impedance ladder network

Zb

Za Znminus1Zn

Figure 13 The equivalent circuit of ladder network in Figure 12

the equivalent input complex impedance of the (119899 minus 1)-orderladder network (see Figure 13) So

119885119899 = (119885119887 + 119885119899minus1) 119885119886 (10)

The recurrence formula for 119885119899 is

119885119899 =(119885119886) sdot 119885119899minus1 + (119885119886sdot119885119887)

119885119899minus1 + (119885119886+119885119887) (11)

The solution of (11) can be obtained using amethod presentedin [16 17] In this case the coefficient matrix of 119885119899 which isexpressed by 119885119899minus1 is shown as follows

1198601 = [119885119886 119885119886sdot119885119887

1 119885119886+119885119887] (12)

119860119899 = 1198601198991 (13)

119860119899 in (13) is the coefficient matrix of 119885119899 which is expressedby 1198850 If 1198601 is expressed by its characteristic values andcharacteristic vectors 1198601198991 can be calculated as

1198601 = 119875[1205821 0

0 1205822]119875minus1 (14)

1198601198991 = 119875[

1205821198991 0

0 1205821198992]119875minus1 (15)

The characteristic values of 1198601 and its characteristic vectorsare

1205821 =

2119885119886+119885119887 + radic1198852119887+ 4119885119886sdot119885119887

2

1205822 =

2119885119886+119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2

119875 = [119885119886sdot119885119887 1205822 minus 119885119886minus119885119887

1205821minus119885119886 1]

119875minus1=

[[[[

[

1

119885119887 sdot (1205821+119885119886)minus1205822 minus 119885119886minus119885119887

119885119887 sdot (1205821+119885119886)

minus1205821minus119885119886

119885119887 sdot (1205821+119885119886)

119885119886sdot119885119887

119885119887 sdot (1205821+119885119886)

]]]]

]

(16)

Combining (16) into (15) results in

1198601198991

=

[[[[

[

1198851198861198851198871205821198991 + 1205821119885119886120582

1198992

119885119887 sdot (1205821+119885119886)

119885119886119885119887 (119885119886+119885119887minus1205822) (1205821198991minus1205821198992)

119885119887 sdot (1205821+119885119886)

(1205821minus119885119886) (1205821198991 minus 1205821198992)

119885119887 sdot (1205821+119885119886)

12058211198851198871205821198991 + 119885119886119885119887120582

1198992

119885119887 sdot (1205821+119885119886)

]]]]

]

(17)

119885119899 = ((1198851198861198851198871205821198991 + 1205821119885119886120582

1198992) sdot 1198850

+ 119885119886119885119887 (119885119886+119885119887 minus 1205822) (1205821198991 minus 1205821198992))

times ((1205821minus119885119886) (1205821198991 minus 1205821198992) 1198850 + (1205821119885119887120582

1198991 + 119885119886119885119887120582

1198992))minus1

(18)

where1205821 + 1205822 = 2119885119886 + 119885119887

1205821 sdot 1205822 = 1198852119886

(19)

If 1198850 = 119885119886 (18) can be simplified using (19) and as

119885119899 =(119885119886 minus 1205822) sdot 120582

119899+11 minus (119885119886 minus 1205821) sdot120582

119899+12

120582119899+11 minus 120582

119899+12

(20)

It is assumed that

1205822

1205821

=

2119885119886+119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2119885119886+119885119887 + radic1198852119887+ 4119885119886sdot119885119887

=

10038161003816100381610038161003816100381610038161003816

1205822

1205821

10038161003816100381610038161003816100381610038161003816

119890119895120579 (21)

Therefore

119885119899 =(119885119886 minus 1205822) minus (119885119886 minus 1205821) sdot

1003816100381610038161003816120582212058211003816100381610038161003816

119899+1119890119895(119899+1)120579

1 minus100381610038161003816100381612058221205821

1003816100381610038161003816

119899+1119890119895(119899+1)120579

(22)

If |12058221205821| gt 1 then the lim119899rarrinfin|12058221205821|119899+1

rarr infin

lim119899rarrinfin

119885119899 = 119885119886 minus 1205821 =

minus119885119887 minus radic1198852119887+ 4119885119886sdot119885119887

2

(23)

6 Journal of Applied Mathematics

minus05

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0008 01 012 014 016 018 02 022 024 026

Zreal (Ohms)

Zim

ag(O

hms)

MeasuredCalculated

(a) Nyquist plot of a 33 F ultracapacitor (values for the model calculation1198771 = 01151Ohm 1198621 = 3634 F 1198772 = 008732Ohm 119877 = 0004589Ohmand 119862 = 003791 F)

Zreal (Ohms)

Zim

ag(O

hms)

minus14

minus12

minus1

minus08

minus06

minus04

minus02

00 01 02 03 04 05 06 07 08 09 1 11 12

MeasuredCalculated

(b) Nyquist plot of a 2 Fultracapacitor (values for themodel calculation1198771 =007529Ohm 1198621 = 2063 F 1198772 = 3564Ohm 119877 = 002241Ohm and 119862 =003445 F)

Figure 14 The Comparison between measured numbers and calculated numbers in Nyquist plot

Or else if |12058221205821| lt 1 then the lim119899rarrinfin|12058221205821|119899+1

rarr 0

lim119899rarrinfin

119885119899 = 119885119886 minus 1205822 =

minus119885119887 + radic1198852119887+ 4119885119886sdot119885119887

2

(24)

For the porous electrode model 119885119886 = 1119895120596119862 119885119887 = 119877In this case |12058221205821| lt 1 and (24) applies Therefore the

formula for the electrode impedance 11988511988601198870 in Figure 10 is

lim119899rarrinfin

11988511988601198870= minus

119877

2+ radic

1198772

4+

119877

119895120596119862 (25)

and the input impedance of the ultracapacitor model inFigure 11 is

119885119886119887 = 1198771 +1

1198951205961198621

+1

11198772 + 111988511988601198870

= 1198771 +1

1198951205961198621

+1

11198772 + 1 (minus1198772 + radic11987724 + 119877119895120596119862)

(26)

3 Validation of Model

Electrochemical impedance spectroscopy (EIS) is one of themost frequently used analytical tools for the characterizationof ultracapacitors This method was used to determine inputresistance and reactance of the ultracapacitor as a function offrequency at a given excitation voltage

Two ultracapacitors with nominal values of 33 F and 2 Fwere measured at room temperature Potentiostat cyclingtests were performed in the capacitor designing lab of St JudeMedicalrsquos Cardiac Rhythm Manufacture the Liberty SCwith a Gamry potentiostat (Reference 3000) The impedancewas measured by applying a sinusoidal 5mV excitationsuperimposed on a 12 VDC bias to the ultracapacitor andmeasuring the magnitude and phase of the current Thefrequency range was from 100mHz to 1 kHz

In this study a Panasonic 33 F23 V (EEC-HW0D335)ultracapacitor which is a new one and a Taiyo Yuden 2 F23 V(PAS1016LR2R3205) which had been charged and dischargedfor many times were measured Figures 14(a) and 14(b)show the magnitude of the input impedance plotted as afunction of the real part of the input impedance for the twoultracapacitors At frequencies above 100Hz the electricalbehavior of the ultracapacitors is more like a simple resistorthan a capacitor At low frequencies (119891 lt 1Hz) the imagi-nary part of the impedance dominates

The values of 1198771 1198621 1198772 119877 and 119862 for each capacitormodel were obtained by curve-fitting the measured data tothe model input impedance in (26) It seems that the 1198772value of old one (2 F) is larger than the new one (33 F) Theexperimental data curves of the real part and the imaginarypart of119885119886119887 were respectively fit using theMATLABCF Toolsoftware Figure 14 also shows the impedance obtained fromthe model in Figure 11 The plots show excellent agreementbetween the model and the measured values Figure 15 showsthe measured impedance of a new KAMCAP 400 F27 V(HP-2R7-J407UYLL) ultracapacitor with aDCbias voltage of12 V and 10mV disturbance variable from 100mHz to 100Hz

Journal of Applied Mathematics 7

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

00015 0016 0017 0018 0019 002 0021 0022 0023 0024

Zreal (Ohms)

Zim

ag(O

hms)

MeasuredCalculated

times10minus3

Figure 15 Complex-plane representation of a 400 F ultracapacitor(values for the model calculation 1198771 = 001798Ohm 1198621 = 4205 F1198772 = 0007511Ohm 119877 = 00001291Ohm and 119862 = 10 F)

at +25∘CThis experiment was carried out by Zahner IM6eXThe model data is also shown in Figure 15 In this model1198771 = 001798Ohm 1198621 = 4205 F 1198772 = 0007511Ohm119877 = 00001291Ohm and 119862 = 10 F It also shows excellentagreement between the measured and modeled values

4 Conclusion

A new model for describing the electrical behavior of ultra-capacitors is introduced based on an intuitive representationof the electrode pores as a fractal structure The new modelhas an infinite number of elements but is fully representedby only five real numbers A closed-form expression for theinput impedance was derived making it relatively easy to fitmeasured results to the model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014JBZ017)

References

[1] P Barrade ldquoSeries connection of supercapacitors comparativestudy of solutions for the active equalization of the voltagesrdquoin Proceedings of the 7th International Conference on Modeling

and Simulation of Electric Machines Converters and SystemsElectrimacs Montreal Canada August 2002

[2] J Van Mierlo P Van den Bossche and G Maggetto ldquoModelsof energy sources for EV and HEV Fuel cells batteriesultracapacitors flywheels and engine-generatorsrdquo Journal ofPower Sources vol 128 no 1 pp 76ndash89 2004

[3] R Kotz and M Carlen ldquoPrinciples and applications of electro-chemical capacitorsrdquo Electrochimica Acta vol 45 no 15-16 pp2483ndash2498 2000

[4] H Gualous D Bouquain A Berthon and J M KauffmannldquoExperimental study of supercapacitor serial resistance andcapacitance variations with temperaturerdquo Journal of PowerSources vol 123 no 1 pp 86ndash93 2003

[5] P L Taberna P Simon and J F Fauvarque ldquoElectrochemicalcharacteristics and impedance spectroscopy studies of carbon-carbon supercapacitorsrdquo Journal of the Electrochemical Societyvol 150 no 3 pp A292ndashA300 2003

[6] B E Conway Electrochemical Supercapacitors Kluwer Aca-demic PublishersPlenum Press New York NY USA 1999

[7] D Qu and H Shi ldquoStudies of activated carbons used in double-layer capacitorsrdquo Journal of Power Sources vol 74 no 1 pp 99ndash107 1998

[8] R De Levie inAdvancesin Electrochemistry and ElectrochemicalEngineering P Delahay and C T Tobias Eds vol 6 pp 329ndash397 John Wiley amp Sons New York NY USA 1967

[9] L Zubieta and R Bonert ldquoCharacterization of Double-LayerCapacitors (DLCs) for power electronics applicationsrdquo in Pro-ceedings of the IEEE Industry Applications Conference pp 1149ndash1154 October 1998

[10] F Rafik H Gualous R Gallay A Crausaz and A BerthonldquoFrequency thermal and voltage supercapacitor characteriza-tion and modelingrdquo Journal of Power Sources vol 165 no 2 pp928ndash934 2007

[11] E Karden S Buller and R W De Doncker ldquoA frequency-domain approach to dynamical modeling of electrochemicalpower sourcesrdquo Electrochimica Acta vol 47 no 13-14 pp 2347ndash2356 2002

[12] S Buller E Karden D Kok and R W de Doncker ldquoModelingthe dynamic behavior of supercapacitors using impedancespectroscopyrdquo IEEE Transactions on Industry Applications vol38 no 6 pp 1622ndash1626 2002

[13] D Riu N Retiere and D Linzen ldquoHalf-order modelling ofsupercapacitorsrdquo in Proceedings of the 39th IAS Annual Meetingon Record of the Industry Applications Conference vol 4 pp2550ndash2554 IEEE October 2004

[14] M Itagaki S Suzuki I Shitanda K Watanabe and HNakazawa ldquoImpedance analysis on electric double layer capaci-tor with transmission line modelrdquo Journal of Power Sources vol164 no 1 pp 415ndash424 2007

[15] M Spandana and A Jagruthi ldquoEnergy Storage UltracapacitorsrdquoYuva Engineers 2nd BTech EEE 2010

[16] X Yuan ldquoExtraction of the general term formula of fractionalrecursive sequence of numbermdash119909119899 = (119886119909119899minus1 + 119887) (119888119909119899minus1 + 119889)

by applying matrix methodrdquo Journal of China West NormalUniversity (Natural Sciences) vol 26 no 1 pp 102ndash104 2005

[17] Y Jin and Q Wang ldquoThe universal solution of ladder networkequivalent impedancerdquo Journal of Higher Correspondence Edu-cation vol 23 no 4 2010

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 Journal of Applied Mathematics

minus05

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0008 01 012 014 016 018 02 022 024 026

Zreal (Ohms)

Zim

ag(O

hms)

MeasuredCalculated

(a) Nyquist plot of a 33 F ultracapacitor (values for the model calculation1198771 = 01151Ohm 1198621 = 3634 F 1198772 = 008732Ohm 119877 = 0004589Ohmand 119862 = 003791 F)

Zreal (Ohms)

Zim

ag(O

hms)

minus14

minus12

minus1

minus08

minus06

minus04

minus02

00 01 02 03 04 05 06 07 08 09 1 11 12

MeasuredCalculated

(b) Nyquist plot of a 2 Fultracapacitor (values for themodel calculation1198771 =007529Ohm 1198621 = 2063 F 1198772 = 3564Ohm 119877 = 002241Ohm and 119862 =003445 F)

Figure 14 The Comparison between measured numbers and calculated numbers in Nyquist plot

Or else if |12058221205821| lt 1 then the lim119899rarrinfin|12058221205821|119899+1

rarr 0

lim119899rarrinfin

119885119899 = 119885119886 minus 1205822 =

minus119885119887 + radic1198852119887+ 4119885119886sdot119885119887

2

(24)

For the porous electrode model 119885119886 = 1119895120596119862 119885119887 = 119877In this case |12058221205821| lt 1 and (24) applies Therefore the

formula for the electrode impedance 11988511988601198870 in Figure 10 is

lim119899rarrinfin

11988511988601198870= minus

119877

2+ radic

1198772

4+

119877

119895120596119862 (25)

and the input impedance of the ultracapacitor model inFigure 11 is

119885119886119887 = 1198771 +1

1198951205961198621

+1

11198772 + 111988511988601198870

= 1198771 +1

1198951205961198621

+1

11198772 + 1 (minus1198772 + radic11987724 + 119877119895120596119862)

(26)

3 Validation of Model

Electrochemical impedance spectroscopy (EIS) is one of themost frequently used analytical tools for the characterizationof ultracapacitors This method was used to determine inputresistance and reactance of the ultracapacitor as a function offrequency at a given excitation voltage

Two ultracapacitors with nominal values of 33 F and 2 Fwere measured at room temperature Potentiostat cyclingtests were performed in the capacitor designing lab of St JudeMedicalrsquos Cardiac Rhythm Manufacture the Liberty SCwith a Gamry potentiostat (Reference 3000) The impedancewas measured by applying a sinusoidal 5mV excitationsuperimposed on a 12 VDC bias to the ultracapacitor andmeasuring the magnitude and phase of the current Thefrequency range was from 100mHz to 1 kHz

In this study a Panasonic 33 F23 V (EEC-HW0D335)ultracapacitor which is a new one and a Taiyo Yuden 2 F23 V(PAS1016LR2R3205) which had been charged and dischargedfor many times were measured Figures 14(a) and 14(b)show the magnitude of the input impedance plotted as afunction of the real part of the input impedance for the twoultracapacitors At frequencies above 100Hz the electricalbehavior of the ultracapacitors is more like a simple resistorthan a capacitor At low frequencies (119891 lt 1Hz) the imagi-nary part of the impedance dominates

The values of 1198771 1198621 1198772 119877 and 119862 for each capacitormodel were obtained by curve-fitting the measured data tothe model input impedance in (26) It seems that the 1198772value of old one (2 F) is larger than the new one (33 F) Theexperimental data curves of the real part and the imaginarypart of119885119886119887 were respectively fit using theMATLABCF Toolsoftware Figure 14 also shows the impedance obtained fromthe model in Figure 11 The plots show excellent agreementbetween the model and the measured values Figure 15 showsthe measured impedance of a new KAMCAP 400 F27 V(HP-2R7-J407UYLL) ultracapacitor with aDCbias voltage of12 V and 10mV disturbance variable from 100mHz to 100Hz

Journal of Applied Mathematics 7

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

00015 0016 0017 0018 0019 002 0021 0022 0023 0024

Zreal (Ohms)

Zim

ag(O

hms)

MeasuredCalculated

times10minus3

Figure 15 Complex-plane representation of a 400 F ultracapacitor(values for the model calculation 1198771 = 001798Ohm 1198621 = 4205 F1198772 = 0007511Ohm 119877 = 00001291Ohm and 119862 = 10 F)

at +25∘CThis experiment was carried out by Zahner IM6eXThe model data is also shown in Figure 15 In this model1198771 = 001798Ohm 1198621 = 4205 F 1198772 = 0007511Ohm119877 = 00001291Ohm and 119862 = 10 F It also shows excellentagreement between the measured and modeled values

4 Conclusion

A new model for describing the electrical behavior of ultra-capacitors is introduced based on an intuitive representationof the electrode pores as a fractal structure The new modelhas an infinite number of elements but is fully representedby only five real numbers A closed-form expression for theinput impedance was derived making it relatively easy to fitmeasured results to the model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014JBZ017)

References

[1] P Barrade ldquoSeries connection of supercapacitors comparativestudy of solutions for the active equalization of the voltagesrdquoin Proceedings of the 7th International Conference on Modeling

and Simulation of Electric Machines Converters and SystemsElectrimacs Montreal Canada August 2002

[2] J Van Mierlo P Van den Bossche and G Maggetto ldquoModelsof energy sources for EV and HEV Fuel cells batteriesultracapacitors flywheels and engine-generatorsrdquo Journal ofPower Sources vol 128 no 1 pp 76ndash89 2004

[3] R Kotz and M Carlen ldquoPrinciples and applications of electro-chemical capacitorsrdquo Electrochimica Acta vol 45 no 15-16 pp2483ndash2498 2000

[4] H Gualous D Bouquain A Berthon and J M KauffmannldquoExperimental study of supercapacitor serial resistance andcapacitance variations with temperaturerdquo Journal of PowerSources vol 123 no 1 pp 86ndash93 2003

[5] P L Taberna P Simon and J F Fauvarque ldquoElectrochemicalcharacteristics and impedance spectroscopy studies of carbon-carbon supercapacitorsrdquo Journal of the Electrochemical Societyvol 150 no 3 pp A292ndashA300 2003

[6] B E Conway Electrochemical Supercapacitors Kluwer Aca-demic PublishersPlenum Press New York NY USA 1999

[7] D Qu and H Shi ldquoStudies of activated carbons used in double-layer capacitorsrdquo Journal of Power Sources vol 74 no 1 pp 99ndash107 1998

[8] R De Levie inAdvancesin Electrochemistry and ElectrochemicalEngineering P Delahay and C T Tobias Eds vol 6 pp 329ndash397 John Wiley amp Sons New York NY USA 1967

[9] L Zubieta and R Bonert ldquoCharacterization of Double-LayerCapacitors (DLCs) for power electronics applicationsrdquo in Pro-ceedings of the IEEE Industry Applications Conference pp 1149ndash1154 October 1998

[10] F Rafik H Gualous R Gallay A Crausaz and A BerthonldquoFrequency thermal and voltage supercapacitor characteriza-tion and modelingrdquo Journal of Power Sources vol 165 no 2 pp928ndash934 2007

[11] E Karden S Buller and R W De Doncker ldquoA frequency-domain approach to dynamical modeling of electrochemicalpower sourcesrdquo Electrochimica Acta vol 47 no 13-14 pp 2347ndash2356 2002

[12] S Buller E Karden D Kok and R W de Doncker ldquoModelingthe dynamic behavior of supercapacitors using impedancespectroscopyrdquo IEEE Transactions on Industry Applications vol38 no 6 pp 1622ndash1626 2002

[13] D Riu N Retiere and D Linzen ldquoHalf-order modelling ofsupercapacitorsrdquo in Proceedings of the 39th IAS Annual Meetingon Record of the Industry Applications Conference vol 4 pp2550ndash2554 IEEE October 2004

[14] M Itagaki S Suzuki I Shitanda K Watanabe and HNakazawa ldquoImpedance analysis on electric double layer capaci-tor with transmission line modelrdquo Journal of Power Sources vol164 no 1 pp 415ndash424 2007

[15] M Spandana and A Jagruthi ldquoEnergy Storage UltracapacitorsrdquoYuva Engineers 2nd BTech EEE 2010

[16] X Yuan ldquoExtraction of the general term formula of fractionalrecursive sequence of numbermdash119909119899 = (119886119909119899minus1 + 119887) (119888119909119899minus1 + 119889)

by applying matrix methodrdquo Journal of China West NormalUniversity (Natural Sciences) vol 26 no 1 pp 102ndash104 2005

[17] Y Jin and Q Wang ldquoThe universal solution of ladder networkequivalent impedancerdquo Journal of Higher Correspondence Edu-cation vol 23 no 4 2010

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

Journal of Applied Mathematics 7

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

00015 0016 0017 0018 0019 002 0021 0022 0023 0024

Zreal (Ohms)

Zim

ag(O

hms)

MeasuredCalculated

times10minus3

Figure 15 Complex-plane representation of a 400 F ultracapacitor(values for the model calculation 1198771 = 001798Ohm 1198621 = 4205 F1198772 = 0007511Ohm 119877 = 00001291Ohm and 119862 = 10 F)

at +25∘CThis experiment was carried out by Zahner IM6eXThe model data is also shown in Figure 15 In this model1198771 = 001798Ohm 1198621 = 4205 F 1198772 = 0007511Ohm119877 = 00001291Ohm and 119862 = 10 F It also shows excellentagreement between the measured and modeled values

4 Conclusion

A new model for describing the electrical behavior of ultra-capacitors is introduced based on an intuitive representationof the electrode pores as a fractal structure The new modelhas an infinite number of elements but is fully representedby only five real numbers A closed-form expression for theinput impedance was derived making it relatively easy to fitmeasured results to the model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Fundamental Research Fundsfor the Central Universities (no 2014JBZ017)

References

[1] P Barrade ldquoSeries connection of supercapacitors comparativestudy of solutions for the active equalization of the voltagesrdquoin Proceedings of the 7th International Conference on Modeling

and Simulation of Electric Machines Converters and SystemsElectrimacs Montreal Canada August 2002

[2] J Van Mierlo P Van den Bossche and G Maggetto ldquoModelsof energy sources for EV and HEV Fuel cells batteriesultracapacitors flywheels and engine-generatorsrdquo Journal ofPower Sources vol 128 no 1 pp 76ndash89 2004

[3] R Kotz and M Carlen ldquoPrinciples and applications of electro-chemical capacitorsrdquo Electrochimica Acta vol 45 no 15-16 pp2483ndash2498 2000

[4] H Gualous D Bouquain A Berthon and J M KauffmannldquoExperimental study of supercapacitor serial resistance andcapacitance variations with temperaturerdquo Journal of PowerSources vol 123 no 1 pp 86ndash93 2003

[5] P L Taberna P Simon and J F Fauvarque ldquoElectrochemicalcharacteristics and impedance spectroscopy studies of carbon-carbon supercapacitorsrdquo Journal of the Electrochemical Societyvol 150 no 3 pp A292ndashA300 2003

[6] B E Conway Electrochemical Supercapacitors Kluwer Aca-demic PublishersPlenum Press New York NY USA 1999

[7] D Qu and H Shi ldquoStudies of activated carbons used in double-layer capacitorsrdquo Journal of Power Sources vol 74 no 1 pp 99ndash107 1998

[8] R De Levie inAdvancesin Electrochemistry and ElectrochemicalEngineering P Delahay and C T Tobias Eds vol 6 pp 329ndash397 John Wiley amp Sons New York NY USA 1967

[9] L Zubieta and R Bonert ldquoCharacterization of Double-LayerCapacitors (DLCs) for power electronics applicationsrdquo in Pro-ceedings of the IEEE Industry Applications Conference pp 1149ndash1154 October 1998

[10] F Rafik H Gualous R Gallay A Crausaz and A BerthonldquoFrequency thermal and voltage supercapacitor characteriza-tion and modelingrdquo Journal of Power Sources vol 165 no 2 pp928ndash934 2007

[11] E Karden S Buller and R W De Doncker ldquoA frequency-domain approach to dynamical modeling of electrochemicalpower sourcesrdquo Electrochimica Acta vol 47 no 13-14 pp 2347ndash2356 2002

[12] S Buller E Karden D Kok and R W de Doncker ldquoModelingthe dynamic behavior of supercapacitors using impedancespectroscopyrdquo IEEE Transactions on Industry Applications vol38 no 6 pp 1622ndash1626 2002

[13] D Riu N Retiere and D Linzen ldquoHalf-order modelling ofsupercapacitorsrdquo in Proceedings of the 39th IAS Annual Meetingon Record of the Industry Applications Conference vol 4 pp2550ndash2554 IEEE October 2004

[14] M Itagaki S Suzuki I Shitanda K Watanabe and HNakazawa ldquoImpedance analysis on electric double layer capaci-tor with transmission line modelrdquo Journal of Power Sources vol164 no 1 pp 415ndash424 2007

[15] M Spandana and A Jagruthi ldquoEnergy Storage UltracapacitorsrdquoYuva Engineers 2nd BTech EEE 2010

[16] X Yuan ldquoExtraction of the general term formula of fractionalrecursive sequence of numbermdash119909119899 = (119886119909119899minus1 + 119887) (119888119909119899minus1 + 119889)

by applying matrix methodrdquo Journal of China West NormalUniversity (Natural Sciences) vol 26 no 1 pp 102ndash104 2005

[17] Y Jin and Q Wang ldquoThe universal solution of ladder networkequivalent impedancerdquo Journal of Higher Correspondence Edu-cation vol 23 no 4 2010

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