Derivation of a Distribution Function of Relaxation Times for the (fractal) Finite Length Warburg.

Derivation

of

a

Distribution

Function

of

Relaxation

Times

for

the

(fractal)

Finite

Length

Warburg.

Bernard

A.

Boukamp

UniversityofTwente,Fac.ofScienceandTechnology&MESA+InstituteforNanotechnology,P.O.Box217,7500AE,Enschede,TheNetherlands

ARTICLE INFO

Articlehistory: Received9June2017

Receivedinrevisedform22August2017 Accepted27August2017

Availableonline1September2017

Keywords:

ImpedanceSpectroscopy

DistributionFunctionofRelaxationTimes FiniteLengthWarburg

Simulation

ABSTRACT

AnanalyticDistributionFunctionofRelaxationTimes(DFRT)isderivedforthefractalFiniteLength Warburg (f-FLW, also called ‘Generalized FLW’) with impedance expression: Zf
FLW(v)=Z0
tanh

(vt0)n
(vt0)n.t0isthecharacteristictimeconstantofthef-FLW.Analysisshowsthatforn!0.5(i.e.

theidealFLW)theDFRTtransformsintoaninfiniteseriesof_d-functionsthatappearinthe_t-domainat positionsgivenby_tk=t0/[p2(k1/2)2]withk=1,2,3, ... 1.Themathematicalsurfaceareasofthese

d-functionsareproportionaltotk.ItisfoundthattheFLWimpedancecanbesimulatedbyaninfinite

seriescombinationofparallel(RkC0)-circuits,withRk=C0tk1andtkasdefinedabove.Rk=2tkZ0and

C0=0.5Z01.Z0isthedc-resistancevalueoftheFLW.

A fullanalysis of theseDFRTexpressionsis presented and comparedwithimpedance inversion techniquesbasedonTikhonovregularizationandmulti-(RQ)CNLS-fits(m(RQ)fit).Transformationof simplem(RQ)fitsprovideareasonablyclosepresentationint-spaceofthef-FLW,clearlyshowingthefirst twomajorpeaks.ImpedancereconstructionsfromboththeTikhonovandm(RQ)fitderivedDFRT’sshowa closematchtotheoriginaldata.

©2017TheAuthor.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

OverthelastfiftyyearsElectrochemicalImpedance Spectros-copy(EIS)hasbeendevelopedintoanimportantresearchtoolfor studying electrochemically active systems [1]. The principles, practisesand applicationsofEIS havenowbeenlaiddownin a number ofimportant textbooks [2–5]. Impedancespectroscopy hasalsobecomeamajorresearchmethodforcharacterisingSolid Oxide Fuel Cells (SOFC) [6] and Solid Oxide Electrolyser Cells (SOEC)[7]atafundamentallevelforwell-de_finedhalfcells,aswell asonasystemslevelforcompletecells.Duetothenatureofthe complex,porouselectrodes,wheretheadsorption,chargetransfer andtransportprocessesaredistributedoverasignificantpartof themicrostructure,theimpedancegraphsarenoteasytomodel withso-called equivalentcircuitsusingcomplexnonlinearleast squares(CNLS)fitting routines[8–12]. Recentlythere hasbeen renewed interest into the transformation of the frequency dispersionintoadistributionfunctionofrelaxationtimes(DFRT)

[13–29].EspeciallythegroupofIvers-Tifféehaspioneeredtheuse of DFRT analysis in the study of SOFC’s [15,17,18] and Li-ion batteries[19,21].TheDFRTprovidesamodel-freerepresentationof essential relaxation times that are directly connected to the

physicaltransportand(charge)transferprocesses.ForSOFC,SOEC andotherfuelcells,thestudyofthepositionandthepeakheightas functionoftemperatureand/orpartialpressureofthegasphase components provides insight in the transport processes in the electrodes.InbatteryresearchthechangeoftheDFRTwiththe ‘StateofCharge’providesessentialinformation,whileforSolarCell research the relation between illumination level and the correspondingDFRTcouldbeabletorevealfundamental process-es.Furthermore,itcanbeaninterestingapproachtouseDFRTto studyaginginavarietyofelectrochemicalsystems.

Thetransformationtothe

_t

-domainisdefinedby: Zð

_v

iÞ ¼R1þRp Zþ1 1 Gð

t

Þ 1þj

v

i

t

dln

_t

ð1Þ

Z(

v

)isthemeasuredimpedance,R1isthehighfrequency

cut-off resistance, Rp is thepolarization resistanceand G(

t

) is the

soughtdistributionfunctionofrelaxationtimes(DFRT).G(

_t

)isa normalizedrealfunctionwith:RG(

_t

)dln

_t

=1.The

t

-domainisthe inverseofthefrequencydomainwith

t

=(2

p

f)1.Unfortunately, Eq.(1)isknownasanill-posedinverseproblemforwhichmany solutionsarepossible.Imposingthe(logical)restrictionthatG(

t

) must always be positive reduces the number of possibilities. SeveralmethodsexisttoarriveataviableDFRT:FourierTransform (FT) [15,17,28,29], Tikhonov Regularization (TR) [16,18–27], E-mailaddress:b.a.boukamp@utwente.nl(B.A. Boukamp).

http://dx.doi.org/10.1016/j.electacta.2017.08.154

0013-4686/©2017TheAuthor.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

ElectrochimicaActa252(2017)154–163

ContentslistsavailableatScienceDirect

Electrochimica

Acta

Maximum Entropy (ME) [13,14] and multiple-(RQ) CNLS-fit, abbreviated‘m(RQ)fit’[28,29].In theFourier Transformmethod awindowfunctionmustbeapplied,whichwilldampenunwanted oscillations [28]. The width and shape of this window has significant influence on the shape of the DFRT. But also the TikhonovRegularizationand theMaximumEntropyrequire the adjustment of a ‘smoothing’ parameter in order to obtain an acceptableDFRT.Recentlyacriticalcomparison betweentheFT andm(RQ)fithasbeenpublishedbythisauthor[28].Inasecond paper [29] a comparison was made between the FT and TR methodsandthem(RQ)fit.FortheTR-transformationsapublicly availableMatLabapplicationwasused[30].

AquitedifferentapproachistakenbythegroupofTsur[31–33]. Theyuseevolutionaryprogramming(dubbed‘ISGP’,ref[33])for constructingadistributionfunctionbasedonaselectedsetfroma largelibraryofplausiblefunctions.Intheevolutionary program-mingthenumberandtypesoffunctions,andtheirparameters,are automatically adjusted until the impedance conversion of the DFRTmatches,withinpredefinedcriteria,theoriginal measure-mentdata.Theclearadvantageisthatnosmoothingparameteris neededinordertoarriveatanoptimalresult.Thedrawbackcould bethelackofaphysicalinterpretationofcertainfunctionsthat havenoclearcounterpartintheimpedancerepresentation.There areafewimpedancefunctionsthatdo haveanexactDFRT:the (RQ)orZARC[2],whichisaparallelcombinationofaresistance andaconstantphaseelement(symbol:Q),withZQð

v

Þ¼Z0ðj

v

Þn;

theHavriliak-Negami(H-N)response[29,34,35]andtheGerischer

[29,36]orchemicalimpedance[37],whichisaspecialcaseofthe H-Nrelation.Itisimportanttonotethattheparallelcombination of a resistance with a capacitance, (RC), is represented in the

t

-domainbya

d

-function.

TheareaunderaDFRTcurve,definedasRRG(

_t

)dln(

_t

),isequal tothedc-resistanceoftheimpedancecounterpart(i.e.Rforthe ‘(RQ)’orZARC),whichfollowsfromthenormalizationofG(

t

)and Eq.(1).Fora(RC)circuit,however,theresistanceinformationis inaccessibleinthe

_t

-domainastheDFRTbecomesa

_d

-function, althoughthemathematicalsurfaceareaisequaltoRp1.Hence,a

purecapacitanceinparallelwitha resistancein theimpedance complicatestheconstructionofaDFRT.Asimpleapproachisto approximatethe

d

-functionbyasharpGaussfunction[28,29]: R
G_ðRCÞð

t

Þ R Wp effiffiffiffi

_p

lnðt0 =tÞ W  2 ð2Þ where

t

0=R

C.W was set at 0.15 as a good compromise for

visibility(width)andpeakheight(nottoolargewithrespectto other peaks) and small error with respect to the impedance reconstructionfrom theDFRT [28]. A smallervaluefor Wwill decreasethewidthandincreasethepeakheight,whilethearea underthecurveremainsequaltoR.

BesidestheGerischerimpedance,theFinite LengthWarburg (FLW,[38])isalsoacommondiffusionbasedimpedancefunction. Itpresentsthefrequencydispersionforone-dimensionaldiffusion througha layer witha fixed activity for the mobileion atone interface.Stoynov[39]has, basedonaboundedconstantphase element(BCP),derivedafractalformforthefinitelengthWarburg, indicatedherebyf-FLW(seethenextsection).Itseemsthatfor thesetwoboundeddiffusiondispersionsnoexactDFRThasbeen publishedsofar.InthispublicationageneralDFRTforthef-FLWis derivedusingaspecialtransformationmethod[40].Forthisanew frequencydependentvariableisintroduced:

z¼lnð

_v

t

0Þ¼ln

t

0

t

 

ð3Þ with

_t

0thecharacteristictimeconstantforthedispersionrelation

thatistobetransformed.TheDFRT,G(

t

),isthenobtainedfromthe

imaginary part of the impedance expression by the following transformprocess: Z0
Gð

t

Þ¼ 1

p

Zimag zþj

p

2   þZimag zj

p

2   h i ð4Þ Z0 represents the dc-resistance of the impedance function.

Using thisapproachitis quitesimpletoderivethewell-known DFRT for a (RQ) circuit [41] from the imaginary part Zimagð

v

Þ¼Z0

v

nsinðn

p

=2Þ:

R
GðRQÞð

t

Þ¼₂R

p

sinðn

p

Þ

coshðnlnð

_t

0=

t

ÞÞþcosðn

p

Þ ð5Þ

Thecharacteristictimeconstantisgivenby:

t

0=(R/Z0)1/n,Z0is

theCPEconstantdefinedabove.Eq.(5) representsasymmetric peak centredaround

t

0 ona log(

t

)scale.For n=1, i.e.theCPE

becomesa purecapacitance,a

d

-functionisobtained.Forlower valuesofnthespreadofthepeakincreasesrapidly[28].

In the following section the impedance expression for one-dimensional diffusion through a mixed conducting layer is presented.FromthisthefractalFLW,alsoknownas_{‘generalized} FLW’,isderived.Thecompactimpedanceexpressionisdividedinto aseparaterealandimaginarypart,neededforthetransformtoa DFRTusingEq.(4).ItisnotpossibletodirectlyderivetheDFRTfor theidealFLWusingthetransformmethodofEq.(4),asisindicated in Section 4. Therefore first the general DFRT of a f-FLW is developedandthelimitingcasesfor

t

!0and

t

!1areexplored. The DFRT for the ideal FLW is then obtained using a special approachforn!0.5.FinallytheDFRT’sobtainedfromTikhonov Regularization and multi-(RQ) CNLS-fits (i.e. ‘m(RQ)fit’) are comparedtotheexactDFRTrepresentations.

2.FLWdispersionrelation

The FiniteLength Warburg is derived for a linear diffusion problem with a special boundary condition, see Fig. 1. The electrodeisa mixedionicandelectronicconductor(MIEC).It is assumed thattheelectronicconductivityis sufficientlylarge,so that it can be ignored in the derivation. In the presented 3-electrodesetuptheactivityofthemobileionismeasuredatthe electrode/electrolyte interface with respect to the (standard) activity of the reference electrode. The current through the

Fig.1.ModelfortheFiniteLengthWarburg,basedon1-dimensionalbounded diffusion.Asanexamplethediffusionofamobilecation,M+

,inamixedionicand electronicconductor(MIEC)ispresented.Butthismodelappliesequallywellto oxygeniondiffusion.Theelectrolyte/MIECinterfaceisatx=0.Thebacksideofthe MIECiscoveredbyafixedactivity(concentration)ofthemobileion,aM,eq.The

currentpassesbetweentheworkingandcounterelectrode.Theactivity(potential) ofthemobileionismeasuredbetweenthereferenceandworkingelectrode.

working electrode is provided by the counter electrode. The electrodeareaisA,thethicknessisland ~Disthechemicaldiffusion coefficient.Atthebackplaneatx=l,theconcentration(oractivity) ofthemobileionisfixedtotheequilibriumconcentration,Ceq.(or

fixedactivity,aeq.)Thispresentsanideallyfasttransferofmobile

ionsatthebackplane.Fortheelectrode/electrolyteinterfacethe particlefluxisdescribedbyFick’sfirstlaw:

J¼~DdC_dðx;_xtÞjx¼0 ð6Þ

Thetimedependentconcentrationintheelectrodeisgivenby Fick’ssecondlaw:

dCðx;tÞ dt ¼ ~D

d2Cðx;tÞ

dx2 ð7Þ

Theboundaryconditionforx=lisexpressedby:

Cðx;tÞjx¼l¼Ceq: ð8Þ

Applyingasmallpotentialperturbation,

_D

E,willcauseasmall changeintheconcentration,withC(x,t)=Ceq.+

D

c(x,t).Thechange

inactivity,a(t),attheelectrode/electrolyteinterfaceisgivenbythe Nernstrelation,which becomeslinear fora smallperturbation, withzthechargeofthemobileion:

D

E_¼RT zFln aðtÞ aeq: ¼ RT zFln aeq:þ

D

aðtÞ aeq:  RT zFaeq:

D

aðtÞ ð9Þ

Therelationbetweenactivityandconcentrationisgivenbythe ThermodynamicFactor,

G

:

D

a

D

c da dc¼ a C dlna dlnc¼ a C

G

ð10Þ

Thethermodynamicfactor,

_G

,canbeobtainedfromaso-called titrationexperimentwheretheNernstpotential(oroxygenpartial pressure)ismeasuredasfunctionofaddedconcentrationofthe mobile ion. The solution of the set of diffusion and boundary equationsiseasilyobtainedbyLaplacetransform,seeAppendixA. TheLaplacetransformleadsdirectlytotheimpedanceexpression fortheFLW(Eq.(A7)):

ZFLWð

v

Þ ¼ Z0 ffiffiffiffiffiffiffiffiffiffiffi j

_vt

0 p tanh ffiffiffiffiffiffiffiffiffiffiffij

_vt

0 p   with: Z0¼ RTl z2_F2 ACeq~D d ln a d ln c   and:

t

0¼ l2 ~D ð11Þ

wherezisthechargeofthemobileion.F,RandThavetheirusual meaning. An example of the dispersion in the impedance representation,forZ0=1

V

and

t

0=1s,isgiveninFig.2.Forhigh

frequencies a semi-infinite diffusion (or Warburg: Z(

v

)=

Z0

(j

vt

0)0.5) is obtained, i.e. the perturbation does not reach

thebackplaneatx=l.Atlowfrequenciesthedispersionbecomes veryclosetoa semi-circle,i.e. aR(RC)circuit.Here the‘Circuit DescriptionCode’isused(CDC,[42]).Itisimportanttonotethatin thisderivationitisassumedthattheelectronicconductivityofthe electrode is significantlylarger thanthe ionicconductivity and hence can be ignored. When the electronic conductivity is relativelylow then thespecial derivation by Jamnikand Maier

[43,44]mustbeused,whichleadstoamorecomplexdispersion relation.

3.FractalFiniteLengthWarburg

Thefractal-FLWisaphenomenologicalexpression,inanalogy withtheparallelcombinationofaresistanceandaconstantphase element(CPE)or‘ZARC’element[39].Eq.(11)isthenrephrasedby: Zf
FLWð

v

Þ¼ _j Z0

vt

0

ð Þntanhðj

vt

0Þn; n0:5 ð12Þ

InAppendixBtheseparationintoarealandimaginarypartis presented,whichresultsin(Eq.(B4)):

Zf
FLWð

v

Þ¼ Z0

vt

0

ð Þn

C
sinh 2

a

þS
sin 2

b

jðS
sinh 2

a

C
sin 2

b

Þ

cosh2

a

þcos2

b

ð13Þ

With C=cos(n

_p

/2), S=sin(n

_p

/2),

_a

=(

_vt

0)nC and

b

=(

vt

0)nS.

Fig.2 alsoshows thedispersions fora f-FLWwithn=0.45 and n=0.4.

4.DerivationoftheDFRT

ThetransformationprocedurepresentedbyEqs.(3)and(4)in theintroduction,willbeusedforderivingtheDFRTforthef-FLW. Insertingthevariablezj1

2

p

,withz=ln(

t

0/

t

),intheimaginarypart

ofEq.(13)resultsin: Z0
Gf
FLWðzÞ¼ Z0

p

Zf
DLW;im zþj

p

2   þZf
DLW;im zj

p

₂   h i ¼Z0 Gþf
FLWð

t

ÞþGf
FLWð

t

Þ h i ð14Þ FromthecompleteanalysisofEq.(14)itwasfoundthatonlythe Gþf
FLWð

t

Þtermneedstobecalculated.ThetransformtoGf
FLWð

t

Þ

yieldsexactlythesame realpartwithan oppositesignfor the imaginarypart.ThisisnotsurprizingastheDFRTshouldbeareal function. Hence only the real part of Gþ_f
FLWð

_t

_Þ needs to be consideredwithGf
FLWð

t

Þ¼2< Gþf
FLWð

t

Þ

 

Replacing

_v

by(z+j

_p

/2)inthefirstpartofEq.(13)leadstothe followingtransformation: Z0

vt

0 ð Þn¼>Z0 ezþj p 2  n ¼Z0
enz cos n

_p

2   jsin n

p

2   h i ¼Z0 Q½CjS ð15Þ

withQ=(

_t

0/

t

)n.Here‘=>’signifiesthetransformationstep.The

transformofthedenominatorofEq.(13)yields,seeAppendixCfor thederivation,Eq.(C3):

cosh2

a

þcos2

b

¼>cosh2QC2þcosh2QS2
cosB

þjsinh2QC2sinh2QS2
sinB ð16Þ

Fig.2.ImpedancedispersionsofatrueFLW(n=0.5)andtwofractalFLW’swith n=0.45andn=0.4.Frequencyrange:1mHz100kHz.

withB=2QCS=Qsin(n

_p

).Thetransformoftheimaginarypartof the numerator of Eq. (13) is derived in the same manner, see

AppendixD,Eq.(D3):

S sinh2

a

C sin2

b

¼>S
sinh2QC2
cosBC
cosh2QS2
sinBþ þjnS
cosh2QC2
sinBC
sinh2QS2
cosBo

ð17Þ Inthefollowingsectionsthetermsnumeratoranddenominator willbeabbreviatedinequationsbyNum.andDenom.respectively. MultiplyingthisresultwiththeZ0

Q1

(CjS)fromEq.(15)gives:

Num:¼Z0 Q CS sinh2QC2 sinh2QS2   cosB þ S2_cosh2QC2 C2_cosh2QS2   sinB n o þ þj  S2_sinh2QC2 þC2_sinh2QS2   cosB þCS cosh2QC2 þcosh2QS2   sinB n o 2 4 3 5 ð18Þ It is interesting to note that for n=0.5, i.e. C=S=p1_/

2, the

denominatorchangesto:2Q
coshðQÞcosðQÞ,whiletherealpartof thenumerator becomes zero!Eqs. (16) and (18) can easily be combinedinaspreadsheettocalculatethedistributionfunction foralimitedrangeof

t

-values.TheDFRT’sforn=0.45,0.48and 0.49arepresentedinFig.3.ItshowsforeachDFRTonemajorpeak, situatedatalowervaluethanthecharacteristictimeconstant,

_t

0,

followedbysmallerpeaksfordecreasing

t

.Theappearanceofthe DFRT’s are similar to those presented by Leonide et al. [17], although no mathematical expression is presented in that publication. For too small values of

t

, the arguments of the hyperbolicfunctionsbecometoolargetobeevaluatedinaspread sheet.Henceadifferentapproachisneededforsmall

_t

.

4.1.Analysisforsmall

_t

Forsmall

_t

-values theargumentQbecomes large,hencethe sinh() and the cosh() functions can be approximated by the positiveexponentials,e.g.:

sinh 2QC2cosh 2QC21₂e2QC2

ð19Þ ThusEqs.(16),(18)canbereducedbyreplacingthehyperbolic functionsbytheirrespectiveexponentialfunctions.Furthermore, boththenumeratoranddenominatorcanbedividedbytheright handsideofEq.(19).NowarelativelysimplefunctionfortheDFRT isfound(inwhichthefactor2hasbeentakencareof):

Z0
Gð

t

Þ¼ Z0

p

t

t

0 n sinðn

p

Þ1Y2_2cosðn

p

Þ
Y
sinð2Qsinðn

p

ÞÞ 1þ2Y
cosð2Qsinðn

p

ÞÞþY2 ð20Þ withY¼exp½2Qcosðn

p

Þ.Writtenoutinfull,Eq.(20)becomes:

Z0
Gð

t

Þ¼ Z0

p

t

t

0 n sinðnpÞ 1e4ðtt0Þ n cosðnpÞ   2cosðnpÞ
e2ðtt0Þ n cosðnpÞ
_{sin 2}t0 t  n_sinðn pÞ   1þ2e2ðtt0Þ n cosðnpÞ
_{cos 2} t0 t  n_sinðn pÞ   þe4ðtt0Þ n cosðnpÞ ð21Þ In Fig. 4 the DFRTof the simplifiedfunction of Eq. (21) is comparedtotheDFRTobtainedfromasimulationusingthefull expressions defined by thecombination of Eqs. (16), (18).It is surprizingthat,withintheaccuracyofExcel,theresultsareexactly identicalovertheentire

t

-rangeforwhichEqs.(16),(18)canbe evaluated.HenceEq.(21)canbetakenasasimpli_fiedexpression thatexactlyrepresentstheDFRTofaf-FLW.Forverysmallvaluesof

t

(e.g.

_t

<

t

0

103forn=0.45)Eq.(21)canbefurtherreducedtoa

simplepowerrelation,presentedasthedashedlinesinFig.5: Z0
Gð

t

Þ¼ Z0

p

t

t

0 n
sinðn

p

Þ ð22Þ

WhereastheDFRTshowsforn=0.45afewdecreasingpeaksfor

t

<

t

0,thenumberofpeaksincreaserapidlyfornapproaching0.5.

ThisisclearlydemonstratedbytheDFRTforn=0.48inFig.5.The limiting

t

-valuebelowwhichEq.(22)canbeappliedshiftsrapidly forn_!0.5to

_t

=0,orln(

_t

)=_1.AlthoughitseemsthatEq.(22)

givesafiniteresultforn!0.5,theupperlimitin

_t

forwhichthis equationcanbeusedgoestozero.

4.2.Analysisforlarge

_t

Forlarge

t

(i.e.

t

>>

t

0)theargumentsoftheexponentialand

goniometricfunctionsofQapproachzero,hencethesefunctions mustbeexpandedintheirseries.Itisessentialtoincludehigher ordertermscontainingQ3.WithwritingEq.(21)as:

Z0
Gð

t

Þ¼

Z0

p

Num:ANum:B

Denom: ð23Þ

Fig.3.CalculatedDFRT’sforn=0.45,n=0.48andn=0.49.Themajorpeaksfall belowthecharacteristictimeconstantt0.

Fig.4.ComparisonbetweentheexactDFRT,eqs(23,25)andthesimplifiedsolution, eq.(28)forn=0.45.

theseparatecontributionscanbeevaluated: Num:A¼2CS 1e4QcosðnpÞ

 



¼4Qcosðn

p

Þsinðn

p

Þ8Q2cos2_ðn

_p

_Þsinðn

_p

_Þþ32

3 Q 3 cos3_ðn

p

Þsinðn

p

Þ ð24Þ and:

Num_:B ¼2cosðn

p

Þ
e2QcosðnpÞ
sinð2Qsinðn

p

ÞÞ 

¼4Qcos_ðn

_p

_Þsinðn

_p

_Þ8Q2_cos2_ðn

_p

_Þsinðn

_p

_Þþ

þ8Q3

cos3_ðn

p

Þsinðn

p

Þ8₃Q3cosðn

p

Þsin3

ðn

p

Þ

ð25Þ ThedenominatorofEq.(21)canbeexpandedto:

Denom:¼Qh1þ2e2QcosðnpÞ
_{cosð2Qsinðn}

_p

_ÞÞþe4QcosðnpÞi

4Q ð26Þ

CombinationofEqs.(24)–(26)leadstothelimitingexpression for

t

>>

t

0: Z0
Gð

t

Þ¼ Z0

p

8 3Q 3

cosðn

p

Þsinðn

p

Þcos2_ðn

p

Þþsin2ðn

p

Þ h i 4Q ¼Z0 3

_p

t

0

t

 2n sinð2n

p

Þ ð27Þ

Forn=0.5Eq.(27)becomeszero.

5.DerivationoftheDFRTfortheidealFLW Analysisforlimitn!0.5

Insertingn=0.5directlyintoEq.(18)doesnotyieldaresultas thehyperbolicfunctionswillreducetocosh(Q)andsinh(Q),which willcanceloutinthenumeratorofEq.(18).Inordertoanalysethe DFRTforn=0.5,oneshouldapproachthisvaluebyreplacingnby: n¼0:5

d

p

ð28Þ

Asaresultfor

_d

_!0thefollowingrelationssimplifyto: sinðn

p

Þ¼sin

p

2

d

  ¼cosð

d

Þ1 cosðn

p

Þ¼cos

p

2

d

  ¼sinð

d

Þ

d

ð29Þ

Usingthefirsttermsofthegoniometricandexponentialseries expansions,Eq.(21)canberewrittenas:

Z0Gð

t

Þ¼

Z0

p

2

d

þ2

d

2

d

Q1
sinð2QÞ

1þð12Q

_d

Þ
cosð2QÞ2Q

_d

ð30Þ

Forn=0.5,i.e.

d

=0,thedenominatorofEq.(30)reducesto:

Denom:¼1þcosð2QÞ ð31Þ

ForQ=

_p

(k-0.5),withk=1,2, ...,thedenominatorbecomes zero,whichresultsina

d

-functionfortheDFRT.HenceG(

t

)will becomeasumof

_d

-functions.The

_t

kpositionsaregivenby:

t

k¼

t

0

p

2
_ðk0:5Þ2; k¼1; 2; ::: ð32Þ

Thefirst

d

-functionoccursat0.4053

t

0.Forsmallvaluesof

d

(i.e.

d

<103_{)itispossibletocalculatethemaximumvaluesforG}

(

t

k).Fig.6showstheresultsforthefirst10maximainG(

t

)for

decreasingvalues of

_d

. Fromthis a clear relationbetween the maxima,

t

and

d

isobtained:

ð33Þ Thesurfaceareaunderthepeakrepresentsthecorresponding dc-resistance value of the related impedance expression. The problemhereisthatthereisnoclearwaytoseparatetheareas undertheconsecutivepeaks(seeFig.5forthecasen=0.48)andto relatethesetocorrespondingresistancevalues.When

d

!0,the peakscollapseto

_d

-functionswithnoobservablesurfacearea.In the impedance representation, however, a

d

-function is repre-sentedbyacapacitanceandresistanceinparallel,e.g.:(RC).Hence itseemsplausibletosimulatetheDFRTforaFLWwithaninfinite seriesof

d

-functionsthatarerepresentedinthefrequencydomain byaninfiniteseriesof(RC)circuitswithtimeconstantsdefinedby Eq.(32).Inasimulationeffortinaspreadsheetitwasobservedthat the following relation leads to the Finite Length Warburg dispersion: ZFLWð

v

Þ¼Z0 X1 k¼1 2

_t

k 1j

_vt

k 1þ

v

2

_t

2 k ð34Þ

Fig.7showsthesimulationofaFLWwithZ0=1

V

and

t

0=1s,

togetherwiththeapproximationusingEq.(34)forthefirstten

t

k

values.Uptoafrequencyoff20

t

10 Hztheerrorislessthan1%

fora10-(RC)simulation.Fora100-(RC)simulationthe1%errorlies abovef2
103

t

10 Hz.

6.Discussion

AremarkableresultistheidentitybetweenthecomplexDFRT, formedbythequotientofEq.(18)andEq.(16),andthesimplified Eq.(21),whichis obtainedthroughan approximationfor small

t

-values.Noefforthasbeenundertakentoprovidearigorousproof Fig.5. ComparisonbetweenthetwoDFRT’sforn=0.45andn=0.48,derivedfrom

eq.(21).Thesmalltapproximation,eq.(22),ispresentedbythedashedlines.

Fig.6.RelationbetweenG(tk)maxandd.Thefirsttenmaximaareshown.

for this identity, as this would be outside the scope of this contribution.

ThederivedDFRTequations providea usefuladditiontothe libraryoffunctionsusedintheevolutionaryalgorithm method. Thisreversedtransformmethod,developedbythegroupofTsur

[31–33],hasbeendescribedbrieflyintheintroduction.Butthese equationsarealsousefulforthecomparisonofregularinversions ofimpedancedatatothe

_t

-domain.Theorderofmismatchshows how reliable the DFRT transform methods are. First we will considerthef-FLWwithn=0.45.Thesimulation,withZ0=1

V

and

t

0=1s,isperformedoverthefrequencyrange0.1mHz1MHz.

FortheDFRTanalysiswithTikhonovregularizationthepublically availableMatLabapplication‘DRTtools’[30]isused.Thisprogram hasmanyadjustableparameters,but herethedefault settingis used except for theregularization parameter, RP.In a previous study[29]itwasobservedthatanincreaseinRPdiminishedthe (unwanted)oscillationsintheDFRT,butalsowidenedandlowered thepeakssignificantly.

The m(RQ)fit was performed with the CNLS-fit program ‘EqCWin’[46].Inthisprocedure_firstapartofthelowfrequency end isfitted toa R(RAQA) circuit.Next the(RAQA)dispersionis

subtractedfromtheoveralldispersion.AgainaR(RBQB)circuitis

fitted to an appropriate low frequency part of the altered dispersion. Next the combination of a R(RBQB)(RAQA) circuit is

re-fittedtoalowfrequencysectionoftheoriginaldata,resultingin anew,improvedsetofcircuitparameters.The(RQ)(RQ)-partofthe fitted R(RQ)(RQ) circuit is again subtracted from the original dispersionand anewR(RCQC)circuitis fittedtoanappropriate

sectionoftheremainderofthefrequencydispersion.Thisprocess ofiterativeaddinga(RQ)tothesubtractedmulti-(RQ)iscontinued untilanacceptablematchbetweendataand modelisobtained. Each(RQ)circuithasacounterpartinthe

t

-domainaccordingto Eq. (5), hence adding the separate Ri

Gi(

t

) functions yields a

(close)approximationof theDFRT.In someinstances theCNLS procedureresultedintheshiftofa(RQ)intoa(RC)combination (i.e.: n=1). In those cases the corresponding Ri

Gi(

t

) was

calculatedusingtheGaussapproximationofEq.(2).Thenumber of(RQ)and(RC)combinationsinthem(RQ)fitareindicatedasx (RQ)-y(RC),withxandythenumberofrespective(RQ)and(RC) sub-circuits.Thehighfrequencycut-offresistance,R1,whichis

partoftheCNLS-fit,hasbeenomittedfromthedescriptionofthe circuitsusedforthetransformation.

Fig.8showstheexactDFRT(Eq.(21))andtwoTR-DFRT’swith regularization parameters, RP=10-6 _{and RP=10}4_{. Also a DFRT}

derivedfromam(RQ)fitwitha6(RQ)circuit(CNLS-

_x

2_=3.4

₁₀9₎

is presented. The m(RQ)fit follows the two major peaks quite closely inshape andposition.BothTRresults showlowerpeak height and a somewhat increased FWHM (full width at half maximum)forthemainpeak.WithRP=106thesecondpeakis poorlyreproducedwhileRP=104givesabetterresult,butwith lowerpeakheights.

A good checkof thevalidity of theDFRTis thecomparison betweenthereversedtransform,i.e.applicationofEq.(1)toobtain theimpedancefromG(

_t

),andtheoriginalimpedancedata[28,29]. Thereconstructionprocedurehasbeendescribedinref.[28].Both theTR-DFRT,withRP=106,andthem(RQ)fit-DFRTwitha6(RQ) circuit show a very good match with the simulated f-FLW impedance data. This can clearly be seen from the so-called differencesgraphsofFig.9A.Heretherelativerealandimaginary Fig.7.DispersionoftheFLWandthesimulationwithaseriesof10(RC)-circuits.Intheinsertthefirst7(RC)’sareshown.Thebluelineistheenvelopeoftheseparate10-(RC) contributions.

Fig.8.ComparisonofDFRT’sforaf-FLWwithn=0.45.RedlineistheexactDFRT. GreenandpurplelinesarefortheTikhonovregularizationswithresp.RP=106_and

104.Bluelineisderivedfromthem(RQ)fitwitha6(RQ)circuit.Dashedlinesare scaledwithafactor5forbettervisibility.

differences,

D

real and

D

imag, are plotted against log f. The

differencesaredefinedas[1]:

D

real;i¼

Zreal;iZreconreal;i:

jZij ;

D

imag;i¼

Z_imag;iZrecon:_imag;i

jZij ð35Þ

Zrecon:i represents the reconstructed impedance. The

recon-structedm(RQ)fitdatashowasignificantdeviationin

D

realabove

1kHz.Thisisduetothe‘limited’rangeofthecalculatedDFRTinthe

t

-domain (from 107 to103_{s). The areaunder the DFRTcurve}

representsthetotal dc-resistance(polarisation),whichfallsjust shortof1

V

,i.e.Z0.Addingacorrectionof0.176m

V

totherealpart

ofthereconstructeddatashowsasignificantimprovementinthe residualsplot,seeFig.9B.Butoneshouldalsorealizethatabove 2kHz(i.e.2

103_/

_t

0)theimpedancemagnitudeislessthan1%ofZ0.

Thefitted6(RQ)impedanceshowsanexcellentfittothesimulated datawitha

_x

2_=3.4

₁₀9_{,wellbelowthegeneralnoiselevel.The}

highfrequencydeviation(Fig.9A)ofthereconstructedimpedance showsthatthe

t

-rangeoftheDFRTshouldbeextendedfurtherin ordertocoverthetotalpolarizationresistance(orZ0).

ThecaseoftheexactFLW(n=0.5)ismorecomplicatedasthe frequencydispersioninvolvescapacitivebehaviour,seeEq.(34). TheTikhonovregularizationshowsratherbroadpeaks,seeFig.10. Thepositionsofthefirstten

_d

-functions,i.e.therootsofEq.(32), arepresentedbytheverticalgreylines.Theheightsrepresentthe relativedifferencesin_{‘strength’,}whichareproportionalto

_t

k.The

m(RQ)fityieldeda4(RQ)-2(RC)circuit(CNLS-

_x

2_=1.8

₁₀8_).Inthe

transformtothe

t

-domainthetwo(RC)circuitswere approximat-edwithGaussfunctions,Eq.(2),withwidthparameterW=0.15. Thesetwomajorpeakscoincidequitewellwiththepositionsofthe firsttwo

d

-functions,ascanbeseeninFig.10.Therelativeheights alsocorrespondquitewelltotheratio

t

1/

t

2.TheTRandthem(RQ)

fit transforms show quite some differences, however, the reconstructedimpedancesagree remarkablywellwiththeFLW simulation. Fig. 11 shows the residuals graph for the TR with RP=106andthem(RQ)_fitwitha4(RQ)-2(RC)circuit.Thisonce again shows that it is difficult to define a unique DFRT from invertedimpedancedata.

Inref.[28]ithasbeenarguedthattheDFRTisonlycapableof showingmajorrelaxationpeaks.Itisnotaprocedurethatallowsto

detectminorcontributionstothefrequencydispersion.Themain poweroftheDFRTisshowingtheshiftin

t

-positionand/orchange inheightofthemajorpeakswithtemperatureorpartialpressure ofoneofthegascomponents[15].

Diffusionisanimportantprocessinfuelcellelectrodesandcan bepartoftheratecontrollingprocess.Thefinitelengthdiffusionor FLWimpedancehassomesimilarityinshapewiththeGerischer impedance, see inset in Fig. 12. The latter is a semi-infinite diffusionwithacoupledsidereaction,whichresultsinafinite dc-resistance. The exact DFRT’s are, however, quite different. The Gerischer DFRT is characterized by an asymptotic function for

t

!

t

0andisnon-existing(zero)for

t

>

t

0[29],whiletheDFRTfor

aFLWisaninfiniteseriesof

d

-functions,asshowninFig.12.The TikhonovregularizationsratherpoorlyreproducetheexactDFRT’s, although the reconstructed impedances closely match the simulatedimpedances.Them(RQ)fitsfor theFLWshowa good approximationofthepositionand heightratiofor thefirsttwo peaks.ButfortheGerischerapoorapproximationofitsanalytical DFRTisobtainedwithtwopeaksthatlieclosertogetherthanfor the FLW DFRT [28,29]. This couldbe used as a discrimination between a FLW and a Gerischer, but interference of other relaxationprocessescouldobscurethisobservation.

For the Gerischer impedance the asymptote in the DFRT coincideswiththecharacteristictimeconstant,

_t

0.FortheFLWitis

foundfromEq.(32)thatthepositionofthefirst

d

-functionat

t

1isa

factor2.47smallerthanthecharacteristictimeconstant,

t

0.The

timeconstantassociatedwiththeminimumintheimaginarypart Fig.9.A)Relativedifferencegraphfortheimpedancereconstructionsfromthe

TikhonovregularizationofFig.8(RP=10-6_{)andthem(RQ)fitwitha6(RQ)circuit.}

Opensymbolsrefertotherighthandaxis.B)Relativedifferencesfortheadjustedm (RQ)fitreconstruction,seetextforexplanation.

Fig.10.DFRT’sfortheexactFLW.RedandgreenlinesrepresenttheTikhonov regularizationswithresp.RP=106_and104_{.Bluelineisforam(RQ)fitwitha4}

(RQ)-2(RC)circuit.Vertical grey linesrepresentthepositions ofthe first ten d-functions of the exact DFRT. The height differencesrepresent the relative magnitudesofthed-functions.

Fig.11. Difference graphfor thereconstructed impedancesfor the Tikhonov regularization(RP=106)andthem(RQ)fitderivedDFRT’s,ascomparedtothe simulatedFLWdispersion.Them(RQ)fitwasbasedona4(RQ)-2(RC)circuit.

oftheFLW,

_t

min,isafactor2.53smallerthan

t

0.This

t

minisfound

fromthederivativeofZFLW,im.withrespectto

v

.TherootfordZFLW, im/d

v

=0canbefoundthroughaNewton-Raphsoniteration[47].

Thedifferencebetween

_t

minand

t

1isduetothehigherorder

(RC)-circuits(seeFig.7)thatareincludedin

t

min.Thepositionof

t

minfor

the Gerischer impedance can be obtained directly from the derivative:

_t

0/

t

min=p3[47].

ItappearsthattheTikhonovregularization,TR,alwaysshowsa veryrapiddecayoutsidetheinversefrequencyrange.Thislimited

t

-rangeeasesthereconstructionoftheimpedancefromtheDFRT. Inthem(RQ)fitmethodn-valuesfortheCPEcanbeobtainedthat aresignificantlylessthan0.7.Fig.16inref.[28]clearlyshowsthat thesigni_ficant

_t

-rangeoftheDFRTofsucha(RQ)rapidlyincreases withdecreasingn.Forn=0.5(Warburgdiffusion)theFWHMofthe curveextendsalreadyover2.3decades,whilethe10%peakheight valuesoftheDFRTcurvespanmorethan 5decades.Hencethe reconstructionrequires theintegration over a significant larger

t

-range, or the application of a resistance correction for the incompletecoverageoftheareaundertheDFRTcurve,ashasbeen demonstratedabove.

7.Conclusions

For thefractal FiniteLength Warburg (bounded one-dimen-sionaldiffusion)arelativelycomplexanalyticDFRTexpressionis derivedusingthetransformprocedureintroducedbyFuossand Kirkwood[40]. Aremarkablepoint is thepositionof themajor peakat

t

1,whichlieswellbelowthecharacteristictimeconstant

t

0.FromEq.(32)aratioof2.47isobtained,slightlybelowtheratio

of2.53obtainedfromthefrequencyoftheminimuminZFLW,im.,

t

min=(2

p

fmin)1,and the characteristic time constant

t

0. The

DFRTofthef-FLWischaracterizedbyamajorpeakfollowedbya seriesofminorpeaks,thenumberandsharpnessstronglydepends onthevalueofn.Forn=0.5theidealFLWisobtained,butadirect derivationoftheDFRTisnotpossible.Thoroughanalysisof the casewherenapproaches0.5showsthattheDFRTofanidealFLW existsofaninfiniteseriesof

d

-functions.AsaresulttheidealFLW canbedescribedasaninfinitesumof(RC)’swithR=2Z0

t

0/[

p

2(k

1_/

2)2], k=1, 2, ... 1, and C0=(2Z0)1. Although the Tikhonov

regularization presents a DFRT withrather broad peaks in the

t

-domain,thereconstructedimpedanceshowsaclosematchwith theoriginaldata.Usingthem(RQ)fitapproachagood approxima-tionoftheDFRTcanbeobtained.

Asstatedbeforeinref.[29],scientistsshouldbeawareofthe inherentlimitsoftheDistributionFunctionof RelaxationTimes analysis method, although with prudent use it can help in understandingtheessentialelectrochemicalprocessesincomplex systems like fuel- and electrolyser cells, complex electrodes, batteriesandsolarcells.

Acknowledgement

TheauthorwantstoexpresshisgratitudetoDr.Wanforsharing onthe Internet theuserfriendly MatLabapplication ‘DRTtools’

[30].

AppendixA.LaplacetransformofFiniteLengthDiffusionModel The Laplace transforms of Eqs. (6)–(9), with p as Laplace variable,leadto:

J¼~Dd

D

cðx;pÞ dx jx¼0 p

_D

c_ðx;p_{Þ¼ ~D}d 2

D

cðx;pÞ dx2

D

c_ðx;p_Þjx¼l¼0 ðA1Þ

ThegeneralsolutionfortheLaplacetransformofFick-2isgiven by:

D

cðx;pÞ¼

j

sinh x ffiffiffiffiffiffiffiffiffi p=~D q þ

z

cosh x ffiffiffiffiffiffiffiffiffi p=~D q ðA2Þ Fromtheboundaryconditionatx=l(Eq.(A1))itfollowsthat:

z

¼

j

tanh l ffiffiffiffi p ~D r ðA3Þ ThisleadstoanewexpressionforFick-1inLaplacespace: J¼~D

j

ffiffiffiffi p ~D r cosh x ffiffiffiffi p ~D r þ

z

ffiffiffiffi p ~D r sinh x ffiffiffiffi p ~D r   jx¼0 ¼~D

j

ffiffiffiffi p ~D r ðA4Þ Fig.12.ComparisonbetweentheexactDFRT’sfortheFLW(heavygraylines)andtheGerischer(heavyblackline).TheapproximateDFRT’sforthem(RQ)fitmethod(FLW: blue,Gerischer:red)andtheTikhonovregularization(dashedlines)arealsodisplayed.InsertshowstheimpedancedispersionoftheFLWandtheGerischerwithZ0=1Vand

Thepotentialattheelectrode/electrolyteinterfacecanthenbe expressed by (with z the charge of the mobile ion and

G

the thermodynamicfactor):

D

EðpÞ¼RT_zF
Caeq:dlna

eq:dlnC¼

RT

G

zFCeq:

D

cðx;pÞjx¼0 ðA5Þ

TheimpedanceinLaplacespacecanbedefinedbydividingthe voltagebythecurrent,I(p)=zFAJ(p),whereAisthesurfacearea:

ZðpÞ¼_z

D

_FAEðpÞ_J ðpÞ¼ RT

_G

z2_F2 ACeq

j

sinh x ffiffiffiffi_p ~D r 

j

cosh x ffiffiffiffi_p ~D r tanh l ffiffiffiffi p ~D r 

j

ffiffiffiffiffiffiffip ~D q jx¼0 ¼ RT

G

z2_F2 A Ceq
ffiffiffiffiffiffiffi p~D q tanh l ffiffiffiffi p ~D r ðA6Þ TheadvantageoftheLaplacespaceisthatthevariablepcanbe writtenasp=s+j

v

,wheresrepresentsthetransienttoasteady statesituationand

v

represents the frequencydependent part (with

_v

=2

_p

f). Hence, the FLW transfer function follows from Eq.(A6): ZFLWð

v

Þ FLW ¼ RT

G

z2_F2_AC eq
ffiffiffiffiffiffiffiffiffi j

_v

~D q tanh l ffiffiffiffiffiffi j

_v

~D s ! ¼ Z0 ffiffiffiffiffiffiffiffiffiffiffi j

_vt

0 p tanh ffiffiffiffiffiffiffiffiffiffiffij

_vt

0 p   ðA7Þ AppendixB.Derivationofthefullimpedanceexpression foraf-FLW

Theterm(j

_vt

0)ninEq.(12)canbewrittenas:

ðj

vt

0Þn¼ð

vt

0Þncos

n

_p

2 þjð

vt

0Þ

n_sinn

p

2 ¼

a

þj

b

ðB1Þ

Substitution ofEq.(B1)inEq.(12)andexpandingthetanh() functionyields: Zf
FLWð

v

Þ¼ _j Z0

vt

0 ð Þn:tanhðj

vt

0Þn   ¼ Z0

a

þj

_b

:tanhð

a

þj

b

Þ ¼Z0ð

a

j

b

Þ

vt

0 ð Þ2n : eaþjb_eajb eaþjb_þeajb ðB2Þ

ApplicationoftheEulerformula,ejx_{=cosx+jsinx,leadstothe}

followingexpression: ZfFLWð

v

Þ¼Z0ð

a

j

b

Þ

vt

0

ð Þ2n :

ea_cos

_b

_ea_cos

_b

_þjðea_sin

_b

_þea_sin

_b

_Þ

ea_cos

_b

_þea_cos

_b

_þjðea_sin

_b

_ea_sin

_b

_Þ

ðB3Þ Multiplying by the complex conjugate of the denominator resultsinaseparaterealandimaginarypart:

ZfFLWð

v

Þ¼Z0ð

a

j

b

Þ

vt

0 ð Þ2n : sinh2

_a

þjsin2

_b

cosh2

_a

þcos2

_b

  ¼ ¼ Z0

vt

0 ð Þn: cos n

p

2   sinh2

_a

þsin n

p

2   sin2

_b

cosh2

a

þcos2

b

2 6 4 jsin n

_p

2   sinh2

_a

cos n

p

2   sin2

_b

cosh2

_a

_þcos2

_b

 ðB4Þ

For n=0.5 this equation reduces to the well-known FLW-function: ZFLWð

v

Þ¼ Z0 ffiffiffiffiffiffiffiffiffiffiffiffi 2

_vt

0 p
sinh ffiffiffiffiffiffiffiffiffiffiffiffi 2

_vt

0 p þsin ffiffiffiffiffiffiffiffiffiffiffiffi2

_vt

0 p cosh ffiffiffiffiffiffiffiffiffiffiffiffi2

vt

0 p þcos ffiffiffiffiffiffiffiffiffiffiffiffi2

vt

0 p jsinh ffiffiffiffiffiffiffiffiffiffiffiffi 2

_vt

0 p sin ffiffiffiffiffiffiffiffiffiffiffiffi2

_vt

0 p cosh ffiffiffiffiffiffiffiffiffiffiffiffi2

vt

0 p þcos ffiffiffiffiffiffiffiffiffiffiffiffi2

vt

0 p " # ðB5Þ AppendixC.Transformationofthedenominator,Eq.(16)

ThetransformofthedenominatorofEq.(B4)willbedonein twoparts.Firstthecosh()functionistransformed,followedbythe transformofthecosine:

cosh2

_a

¼>1₂ e2

t

0

t

 n CþjS ½ C þe2

t

0

t

 n CþjS ½ C 2 4 3 5 ¼ ¼1₂ e2QC2 cos2QCS_þjsin2QCS ð Þ h þe2QC2 ðcos2QCSjsin2QCSÞ¼

¼cosh2QC2
cosBþjsinh2QC2
sinB ðC1Þ Transformofthecosineyields:

cos2

b

¼>1₂ e2j

t

0

t

 n CþjS ð ÞS þe2j

t

0

t

 n CþjS ð ÞS 2 4 3 5 ¼1₂ ej2QCS2QS2 þej2QCSþ2QS2 h i ¼ ¼1₂ e2QS2ðcosBþjsinBÞþe2QS2 ðcosBjsinBÞ h i ¼ ¼cosh2QS2
cosBjsinh2QS2
sinB ðC2Þ CombiningEqs.(C1),(C2)yieldsthefollowingexpressionfor thedenominator:

Denom:¼cosh2QC2þcosh2QS2
cosB

þjsinh2QC2_sinh2QS2
_{sinB ðC3Þ}

AppendixD.Transformationofthenumerator,Eq.(18)

Developmentofthenumeratoroftheimaginarypart,excluding thetermbeforethebracketsinEq.(B4),isalsodoneintwoparts:

Ssinh2

_a

¼>S 2 e 2

t

0

t

 n ðCþjSÞC e2

t

0

t

 n ðCþjSÞC 2 4 3 5 ¼S₂ e2QC2_þj2QCS e2QC2j2QCS h i ¼₂S e2QC2_{ðcosBþjsinBÞe}2QC2 cosBjsinB ð Þ h i ¼ ¼Shsinh2QC2
cosBþjcosh2QC2
sinBi ðD1Þ And: Csin2

_b

¼>_2jChejð2QCSþj2QS2Þejð2QCSþj2QS2Þi ¼_2jC ej2QCS2QS2_Þ ej2QCSþ2QS2Þ h i ¼ ¼_2jC e2QS2ðcosBþjsinBÞe2QS2 ðcosBjsinBÞ h i ¼ ¼Chcosh2QS2
sinBþjsinh2QS2
cosBi ðD2Þ

CombinationofEqs.(D1),(D2)yields:

Ssin2

_a

þCsin2

_b

¼>hSsinh2QC2
cosBCcosh2QS2
sinBiþ þjhScosh2QC2
_{sinBCsinh2QS}2
_cosBi

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