Numerical Differentiation Methods for the Logarithmic Derivative Technique

Processing and Application of Ceramics 4 [2] (2010) 87–93

Numerical differentiation methods for the logarithmic derivative technique used in dielectric spectroscopy# Henrik Haspel, Ákos Kukovecz*, Zoltán Kónya, Imre Kiricsi Department of Applied and Environmental Chemistry, University of Szeged, Rerrich 1, 6720 Szeged, Hungary Received 15 January 2010; received in revised form 28 June 2010; accepted 30 June 2010

Abstract In dielectric relaxation spectroscopy the conduction contribution often hampers the evaluation of dielectric spectra, especially in the low-frequency regime. In order to overcome this the logarithmic derivative technique could be used, where the calculation of the logarithmic derivative of the real part of the complex permittivity function is needed. Since broadband dielectric measurement provides discrete permittivity function, numerical differentiation has to be used. Applicability of the Savitzky-Golay convolution method in the derivative analysis is examined, and a detailed investigation of the influential parameters (frequency, spectrum resolution, peak shape) is presented on synthetic dielectric data. Keywords: dielectric spectroscopy, logarithmic derivative, Savitzky-Golay method

I. Introduction ential parameters are considered, and the capabilities Dielectric relaxation spectroscopy (DRS) has be- and limitations of the approach are discussed. come a popular and powerful technique for studying II. Theoretical Background the dielectric properties of almost any kind of mate- rial. Modern measurement systems allow the acquisi- In DRS the frequency-dependent complex dielec- tion of relaxation spectra over a wide range in frequen- tric function is used to determine the electrical/dielec- cy and temperature with a high accuracy. Hence we tric properties of materials [1]: ∗ are able to obtain information about molecular relax- ()= ′ ()− ′′ ()ωεωεωε ation dynamics and conduction processes, leading to a where ε’(ω) is the real and ε”(ω) is the imaginary part deeper understanding of the materials’ nature. of the complex dielectric function. In order to extract most of the information car- If the measured sample contains mobile charge ried by the measurement data, mathematical meth- carriers, i.e. it exhibits conduction, a considerable ods may be applied. In broadband dielectric spec- increase shows up in the low frequency part of the troscopy the so-called harmonic analysis is used. imaginary permittivity spectrum (loss spectrum). In This means that the value of the permittivity func- moderately to highly conducting materials this “con- tion is only known at given measurement frequen- ductivity tail” hampers the proper analysis of slow cies. Since the spectra are discrete datasets, only nu- dipolar relaxations. Furthermore, if the charge car- merical approximations can be used. Therefore, the riers are mobile ions, an undesirable effect, the so- result of the evaluation depends strongly on the pre- called electrode polarization takes place (EP) [2]. cision of these methods. This is due to the partial blocking of the charge ex- In this paper we will present numerical derivative change at the sample/electrode interfaces which re- calculations on synthetic dielectric data. All the influ- sults in the formation of two double layers. These double layers give rise to large capacitances in series #Paper presented at 8th Students’ Meeting, SM-2009, Processing and Application of Ceramics, Novi Sad, Serbia, 2009 to the conducting bulk of the sample. This manifests * Corresponding author: tel: +380 44 424 3364, itself in a high apparent dielectric constant typically fax: +380 44 424 2131, e-mail: [email protected] in the low frequency range. There are several meth-

87 H. Haspel et al. / Processing and Application of Ceramics 4 [2] (2010) 87–93

E ½ 2 ° 'H ° ª D § SD · 2D º H cc EH )' ZW cos21sin ¨ ¸ ZW ® D E ¾ « » ¯° 1 iZW ¿° ¬ © 2 ¹ ¼ where ª § SD · º « sin¨ ¸ » © 2 ¹ ) arctan« » « D § SD ·» H. Haspel et al. / Processing and Application of Ceramics 4 [2] (2010)« ZW 87–93 cos¨ ¸» ¬ © 2 ¹¼

H. Haspelǻİ et= al.İs /– Processing İ is the relaxationand Application strength, of Ceramics İs, and 4 İ[2] is (2010) the permittivity 87–93 as the frequency Ȧĺ0, and Ȧĺ . ods which help reduce the coveringThe effect logarithmic of the con derivative- The of thislogarithmic function derivativeis: of this function is: duction and the electrode polarization. One of them which is now solely based on relaxation phenomena. It which approximately equalsαD ª απDS the Ohmic-conduction-º is the numerical Kramers–Kronig transform, which DE∆' () ωτεαβ ZWH cos () 1 βE Φ+− ) can be calculated by numerical techniques described in free dielectric′c loss for rather broad« peaks [5].» For non- is an elegant way to remove Ohmic conduction from ∂wεH HN ¬ 2 ¼ −= 1+βE measuredliterature loss [3,4]. spectra. It transforms the real part of distributed∂w ϖY Debye relaxations, i.e. for single-relaxation− ln 2 An alternative to the numerical Kramers–Kronig time processes,ª the derivativeαD §παSD results· in 2peaksαD º that are the complex dielectric function into the imaginary « + () ωτZW cos21 ¨ ¸ + () ωτZW » part,transform which is is based now on solely the logarithmic based on relaxationderivative: phe- sharper [6]. ¬ © 2 ¹ ¼ nomena. It can be calculated by numerical techniques For the double-stretched Havriliak-Negami SwhwereHc 0 < D,E 1 are the shape parameters. There described in literaturecc ZH [3,4].)( wherefunction 0 < αwe,β have≤ 1 are for thethe shapeimaginary parameters. part İ": There are der w Z An alternative to the numericalln2 Kramers–Kronig three special cases. If α,β = 1 the relaxation peak is the

transform is based on the logarithmic derivative: sharpest possible and it is called a DebyeE peak caused ½ by a Debye-type relaxation. α and β make2 peaks broader °π ∂ε'′H ° ª D § SD · 2D º ′′ H ccωε )( −= ® ¾ EH in a symmetrical)' ZW cos21sin (β¨ = 1) or¸ in ZWan asymmetrically (α = 1) der ∂ ω D E « » ¯° 1 ln2 iZW ¿° way.¬ The first case© 2 is¹ called a¼ Cole-Cole, the latter a which approximately equals the Ohmic-conduction- Cole-Davidson peak. The frequency where the function freewhere dielectric loss for rather broad peaks [5]. For non- reachesfrequency its maximum where the equals function with the reaches characteristic its maximum fre- distributed Debye relaxations, i.e. for single-relaxation quencyequals for with Debye the andcharacteristic Cole-Cole frequency peak, but notfor forDebye Cole- and ª § SD · º time processes, the derivative« sinresults¨ ¸in peaks» that are DavidsonCole-Cole peak, peak, where but the notdifference for Cole-Davidson between them de- peak, « © 2 ¹ » E sharper [6]. ) arctanH. Haspel et al. / Processing and Application pendsof whereCeramics on the 4 [2]βdifference shape (2010) parameter. 87 between–93 them depends on the For the double-stretched« Havriliak-NegamiD § SD ·» function shapeFig. parameter. 1 shows the peak sharpening effect, which « ZW cos¨ ¸» we have for the imaginary¬ part ε”: © 2 ¹¼ makes Fig.possible 1 shows to resolve the peaknearby sharpening peaks better. effect, Because which themakes derivatives possible are sensitive to resolve to noise, E nearby a proper peaks numer- better. ǻİ = İs – İ is the relaxation strength, İs, and İ is the ½ Because the derivatives are sensitive2 to noise, a proper H.H.H. Haspel HaspelHaspel et etet al. al.al. / //Processing ProcessingProcessing and andand Application ApplicationApplication of ofof Ceramics Ceramics°Ceramics'∆ 4 4Hε 4[2] [2][2] (2010) (2010)(2010)° 87 8787––93–9393 ª ical techniqueD § shouldSD · be chosen2D º for the differentiation H. Haspel etH. H.al. Haspel Haspel/ Processingpermittivity et et al. al. / / andProcessing Processing Applicationε ascc′′ the−ℑ= and and frequency Application ofApplication Ceramics Ȧĺ of4of 0,[2]Ceramics Ceramics and (2010)= Ȧĺ 4 487 [2] [2]–93. (2010) (2010) 87 87––9393 H ® Eβ ¾ EH « )' of numerical theZW data.cos21sin Wübbenhorst¨ technique¸ ZW shouldand »van Turnhout be chosen suggested for the The logarithmic derivative ofDα this function is: 2 ¯° ()1+ ()iZWωτ ¿° ¬ to differentiation use either © one ofbased¹ the on data. a low¼ Wübbenhorst pass quadratic and least van DS Turnhout suggested to use either one based on a low where D ª º EEβE squares filter or a quadratic logarithmic-equidistant five DE' ZWH cos 1 E ) − E EEE ½½½ wH c « » 222 pass quadratic least squares filter or a quadratic °°° '''HHH °°° ½ HN ½½ ªªª DDαD §§§SD¬SDπαSD2··· 22D2DαDº¼ºº 2 point spline. These two techniques are special cases of 'H ''HH ª ªªD SDDD 2SDSDª D º 2 SD DDºº 2º22 HHHcccccc ®®® ° °°¾¾¾ ° ()EH EH βε EH °° )' )' +Φ∆= )' ()ZW ZWωτZW cos21sin cos21sin cos21sin ¨¨¨ § ¸¸¸+ · ()ZW ZWωτZW§§ 1E2 ·· § logarithmic-equidistant2·22 five point spline. These two H cc ®HHcccccc DDD E®E®E ¾ w lnY ¾¾ EH ««« )' EH EH ZW )' cos21sin )' ¨ZW ZW ¸cos21sin cos21sin ZW¨¨«»»» ¸¸sin ZW ¨ZW the¸ so-called» Savitzky-Golay method (SG) for differ- D E DD EEE ª « «« ©©©SD222 ¹¹¹ º 2 » techniques»» are special cases of the so-called Savitzky- ¯°¯°¯° 1 1 1° i iZWiZWZW °°¿°¿°¿° ° °° ¬¬¬ ¬ D ¬¬ § 2©· 2 ¹ 2D©© «¼22¼¼ ¹¹¼ © entiation2 ¹ ¼¼ » [7,8], which is a convolution of a uniformly- ¯ 1 iZW¯¯ 1 1 ¿ iiiZWZW ¿¿ « ZW cos21 ¨ ¸ ) ZW arctan» Golay method (SG) for differentiation [7,8], which is a where ¬ © 2 ¹ ¼ « D spaced§ SD ·(2» m + 1) point (with a positive integer m) data wherewherewherewhere wherewhere « ZW cos¨convolution¸» of a uniformly-spaced (2m + 1) point ¬ array© 2 with¹¼ a set of b coefficients derived from the least- D E (with a positive integerk m) data array with a set of bk where ª0ªª < , §§§SD SDSD1 are··· theººº shapeπα parameters. There are ««« ª sinsinsin¨¨¨ §¸¸SD¸ªªª · »»» º§§§SDSD··· ººº squares-fit formulas of n-degree polynomial: ǻİthree = İs –special İ is« the cases.©© ©relaxationsin222¨ ¹If¹¹« ««D¸,E =sinstrength, sinsin sin 1»¨ ¨the¨ relaxation¸¸¸ İs, and»»» İ peak is the is permittivitythe coefficients as the frequency derived from Ȧĺ 0,the and least-squares-fit Ȧĺ . formulas ««« 2 »»» 222 m ))) )arctanarctanarctanThe logarithmic«)) =Φ derivative© ««« ¹ »of©©© this¹¹¹ function»»» is: of n-degree polynomial: sharpestarctan««« possibleDDD arctanarctanarctan and§§§SDSDSD it ··is·»»» called a Debye peak caused k ZW ZWZW« coscosDcos¨¨¨««« §¸¸SD¸ −DαD·D» §§§SDπαSD···»»» der′′ = k ′ rb ωεωε )()( by a Debye-type««« ZW relaxation.cos () ZW ωτZW¨ »»» ¸+ coscoscosD ¨¨¨ and ¸¸¸E make peaks ∑ m ¬¬¬ « ©©©«2««22¹¹¹¼¼¼ » »»» D ªDS º −= mk ¬ ¬¬¬ © 2 ¹¼ ©©©E222 ¹¹¹¼¼¼DE' ZWH cos 1 E ) cc c kZHZH broader in a symmetrical c ( = 1) or in an « » der ¦ k rb )()( wH HN ¬ 2 For differentiating¼ the entire spectrum one has to slide ǻİǻİǻİ = == İ İsİs s – –– İ İİ is isis the thethe relaxation relaxationrelaxation strength, strength,strength, İ İsİs,s, , and andand İ İİ is isis the thethe permittivity permittivitypermittivityD as asas the thethe frequency frequencyfrequency Ȧ ȦȦĺĺĺ0,0,0, and andand Ȧ ȦȦĺĺĺ .. . mk ǻİ = İs – İǻİǻİ is = =the İ İss s – relaxation–– İ İ is isis the thethe relaxation strength,relaxationrelaxation İasymmetrically s strength, ,strength,strength, and İ is İ İs s,sthe, , and andand permittivity( İ İ = is is is1) the thethe way. permittivity permittivitypermittivity as The the frequencyfirst as asascase the thethe isfrequency Ȧfrequencyfrequency ĺcalled0, and a Ȧ Ȧĺĺ0,0, and.and ȦȦĺĺ 1.E. . Δε = εs – ε∞ is the relaxation strength,w lnY εs, and ε∞ is the this (2m + 1) point „window” through the measured data TheTheThe logarithmic logarithmiclogarithmic derivative derivativederivative of ofof this thisthis function functionfunctionCole-Cole, is: is:is: the latter a Cole-Davidson peak. The 2 The logarithmicTheThe logarithmiclogarithmic derivative derivativeofderivative thispermittivity function ofof thisthis is: functionasfunction the frequency is:is: ω→0, and ωª→ ∞. D § SDpoints· (except2D º for m points at each end of the data array « ZW cos21 ¨ ¸ ZW » DDD ªªDSªDSDS ººº ¬ © 2 ¹ ¼ DEDEDE''' ZWH ZWH ZWH coscoscosD ªDS 1 11DDDEEE) ) ) ªªDSDS º ºº ccc DE' ZWH DEDE««cos«'' ZWH ZWH cos1cosE»»») 1 1 EE ) ) wwwHHHHNHNHN ¬¬¬222« ««¼¼¼ » »» wH HNc wwHHHNcHNcc D E ¬ 2 ¬¬ 22 ¼ ¼¼ wh ere 0 < , 1 are the shape parameters.111EEE There wwwlnlnlnYYY 1E 111EEE w lnY ªªª wwlnlnlnYY SDSDSD ººº 222 ª DDD ª§ª§§ ·SD·· 22D2DDSDSDº 2 ºº 222 ««« ZW ZWZW cos21 cos21 cos21 D¨¨¨ §¸¸¸ ZW· ZWDZWDD §»§»»2D ·· 22D2DD « ZW «©«©cos21 ©222¨ ¹ ¹ZW ¹ZW¸ cos21 cos21 ZW¨¨ »¸¸ ZW ZW »» ¬¬¬ ¬ ¬¬ © 2 ¹ ©¼©¼¼22 ¼¹¹ ¼¼ DDDEEE whwhwhereereerewh 0 ere00 < << 0 ,<,, wh Dwh , 1E 1ere1 ere are areare 1 0 0 the are the

Figure 1. Comparison of dielectric loss ε”(ω) (lines with symbols) and ε” (ω) (dashed lines) for symmetric (left) and for Figure 1. Comparison of dielectric loss İ"(Ȧ) (lines with symbols) and İ" der(Ȧ) (dashed lines) for symmetric (left) and for asymmetric (right) HN functions for α,β = der0.2, 0.4, 0.6, 0.8, 1 asymmetric (right) HN functions for D,E = 0.2, 0.4, 0.6, 0.8, 1

88 xx H. Haspel et al. / Processing and Application of Ceramics 4 [2] (2010) 87–93 H. Haspel et al. / Processing and Application of Ceramics 4 [2] (2010) 87–93

Figure 2. Numerical approximation of the logarithmic derivative of the Debye peak at four different spectrum resolutions Figure 2. Numerical approximation of the logarithmic(r = derivative 2, 21/2, 21/4 ,of 21/8 the) Debye peak at four different spectrum resolutions (r = 2, 21/2, 21/4, 21/8) that must be treated separately, using special coefficients Fig. 2. One can easily compare the exact derived peak to calculateFor differentiating the derivative the [9]). entire The spectrum values of onethe coeffihas to- with the numerically calculated ones visually. The inset III. Results and discussion slidecients thisdepend (2 mon + the 1) degree point of „window”the fitted polynomial through theand graphs magnify the peaks at their maxima. As a general measuredthe chosen data width points of the (except „window” for m and points were at publishedeach end rule weThe can results state that of theas the numerical resolution differentiation of the analyzed with offor the a broaddata arrayrange thatof parameters must be treated in [7,9]. separately, using spectrumthe five techniques increases, allfor testedthe Debye approximations peak are presented provide in specialWe coefficients examined the to calculate effect of the the derivative polynomial [9]). degree The goodFig. results.2. One Thecan largereasily thecompare resolution, the exact the more derived precise peak valuesand the of widththe coefficients of the dataset, depend used on in the one degree calculation of the thewith technique the numerically is. At the same calculated time we ones have visually. to consid- The fittedstep, on polynomial the quality of and the theapproximation chosen width of the deriva- of the erinset that graphsin broadband magnify dielectric the peaks spectroscopy at their maxima. the number As a „window”tive at four and different were spectrum published resolutions for a broad for three range ba- of ofgeneral measurement rule we points can statecould that not asbe theincreased resolution arbitrari- of the parameterssic dielectric in [7,9].peaks. A normalized Debye, a Cole-Cole lyanalyzed because spectrumthe measurement increases, time all stronglytested approximations depends on andWe a Cole-Davidson examined the peak effect were of chosenthe polynomial for that with degree the it,provide especially good in the results. low frequency The larger regime. the resolution, Remember the andfollowing the width parameters: of the dataset,Δε = 1, τ used= 0.01, in (β one = 1, calculation α = 0.22), thatmore at 0.001 precise Hz the one technique total period is. takes At the ~17 same min, timehence we step,(α = 1, on β = the 0.2). quality The SG ofcoefficients the approximation of the used approx of the- ahave dielectric to measurement consider that in the in 10 -3 broadband–107 Hz range dielectric takes derivativeimations are at listed four in different Table 1. spectrum resolutions for hoursspectroscopy or even daysthe number at very of high measurement resolution. pointsThere couldare threeAt basic n = 2 dielectricand (2m + peaks.1) = 5 the A two normalized different Debye,set of co- a casesnot bewhere increased this can arbitraril be realizedy because (with stable the measurement sample, or Cole-Coleefficients andare thea Cole-Davidson quadratic five-point peak were SG chosenand the forqua- quenchedtime strongly measurement depends technique on it, especially[11]), but with in thesensi- low W thatdratic with spline, the following respectively parameters: [10]. r is ǻİ the = 1, logarithmic = 0.01, tivefrequency or time-dependent regime. Remember systems [12] that time at is 0.001 of primary Hz one (Espacing = 1, D between= 0.22), ( theD = measurement 1, E = 0.2). The points SG chosencoefficients as r importance.total period Therefore, takes ~17results min, at lower hence resolutions a dielectric are -3 7 of= the2, 2 used1/2, 2 1/4approximations, 21/8. Thus, spacing are listed is halved in Table in every 1. step ofmeasurement great practical in significance. the 10 –10 Hz range takes hours or (doubledAt n = the 2 andresolution) (2m + 1)on =a 5logarithmic the two different scale. set of evenThe daysmost atemphatic very high differences resolution. between There the areapplied cases coefficients are the quadratic five-point SG and the numericalwhere this techniques can be are realized at r = 2 (with logarithmic stable spacing. sample, It or quadraticIII. Results spline, and respectively discussion [10]. r is the logarithmic canquenched be seen, that measurement as the degree technique of the polynomial [11]), decreas but with- spacingThe between results ofthe the measurement numerical points differentiation chosen as with r = essensitive or the width or time-dependentof the dataset increases, systems the [12] numerically time is of 2, 21/2, 21/4, 21/8. Thus, spacing is halved in every step the five techniques for the Debye peak are presented in derivedprimary peak importance. is broadened Therefore,its maximum results is lowered. at lower (doubled the resolution) on a logarithmic scale. resolutions are of great practical significance. 89 xx H. Haspel et al. / Processing and Application of Ceramics 4 [2] (2010) 87–93

Table 1. bk coefficients used for the calculation of the logarithmic derivative

(2m + 1) n = 2 n = 3 (-2,-1,0,1,2) (1,-6,0,6,-1) (1,-8,0,8,-1) 5 10 ln r 8 ln r 12 ln r (-3,-2,-1,0,1,2,3) (22,-67,-58,0,58,67,-22) 7 28 ln r 252 ln r

The explanation of the first observation is that a The quantitative results for the Debye, Cole-Cole, higher degree polynomial can describe rapid changes and Cole-Davidson peaks are given in Fig. 3. Since the better. Since the Debye peak is the sharpest peak that quadratic Savitzky-Golay approximations are the worst theoretically exists in dielectric spectroscopy, such a ones, only the three other techniques (five-point qua- function fits better onto the data. However, higher de- dratic spline, five-point cubic SG, seven-point cubic gree polynomials are not considered in the following in SG) are the subjects of our further investigations. Fig. order to avoid overfitting which would describe random 3 is technically a matrix, in its rows results for the three error instead of the underlying process in a real, noisy peaks at one specific resolution, whilst in its columns measurement. calculations for one specific peak at four different reso- The latter statement is also easy to understand by re- lutions are presented. calling the calculation procedure applied in the convolu- As expected, the accuracy of the approximations tion method. A set of data points is multiplied with a set strongly differs with changing peak shapes. A broader of coefficients in order to get the smoothed derivative at peak could be fitted by any of the tested methods with the middle of the data set and then this “window” is slid an acceptable error. The difference in the absolute and forwards. Practically, this is a weighted moving average relative errors for the Debye and for the Cole-Cole peak which suppresses here not only the transients (in the case is about four-five order of magnitude at any resolution. of noisy data), but the whole peak. As the fitted dataset Furthermore, the accuracy depends on the measured broadens, the suppression is more and more notable. frequency as well, so results near and far from the max- The following order of decreasing accuracy of the ap- imum are discussed separately. plied methods can be deduced from Fig. 2 at r = 2: five- Hence the original accuracy order established on the point cubic SG (5pt n = 3), five-point quadratic spline basis of Fig. 2. must be reconsidered, or more exactly, (Spline), seven-point cubic SG (7pt n = 3), five-point a detailed one needs to be determined in the light of the quadratic SG (5pt n = 2), and seven-point quadratic SG error calculations. (7pt n = 2). It is also obvious that this order is just an im- At r = 2 spacing, near the maximum of the Debye pression from the figure and does not take the effect of peak the quadratic spline provides the best result, fol- the frequency into account. Furthermore, at higher reso- lowed by the five-point, and the seven-point SG with lutions the differences between the function and the nu- -0.9 %, -3.8 %, -11.7 % absolute, and -1.3 %, -5.0 % merical results decrease with each applied method, hence and -15.4 % relative errors. At lower and higher fre- with increasing resolution the use of lower degree poly- quencies the five-point SG claims the first place (-0.02 nomials or wider datasets provides also a good result. In % absolute, -15,1 % relative error), the spline is the sec- those measurements where huge amounts of data can be ond (-0.04 % absolute, -40.3 % relative error), and the collected and these data suffer from considerable noise, seven-point SG is the third best choice (-0.1 % absolute, the use of larger arrays should be considered. and -127.9 % relative error). Visual comparison allows important qualitative con- In the case of the Cole-Cole peak at the peak max- clusions to be drawn but it lacks the possibility of exact imum the five-point SG is the most accurate method numerical characterization of the applied methods. In our (-1.1×10-4 % absolute, -1.5×10-3 % relative error) fol- case the absolute and the relative error is defined as: lowed by the seven-point SG (-7.2×10-4 %, absolute, -8.9×10-3 % relative error) and the quadratic spline ∆ε ′′ ω = ε ′′ ω − ε ′′ ω)()()( der ,numder der (6.7×10-3 % absolute, and 8.4×10-2 % relative error). and In Fig. 3. points for spline divided by ten are plotted in order to scale all results to the same order of mag- ∆ ′′ ωε )( ε ′′ ω − ε ′′ ω)()( der = ,numder der nitude. Further from the maximum the same order of accuracy can be found with approximately half of ab- der′′ ωε )( der′′ ωε )( solute errors which means slightly better relative er- The lack of the absolute value function in the defi- rors (5pt n = 3: 5.4×10-5 % abs., 1.2×10-3 % rel.; 7pt n nition of the relative error allows this quantity to take = 3: 3.4×10-4 % abs., 7.2×10-3 % rel.; Spline: -1.4×10-3 negative values that show the direction of the deviation abs., -3.4×10-2 % relative error) in spite of the decreas- from the approximated function. ing function values.

90 H. Haspel et al. / Processing andand ApplicationApplication ofof CeramicsCeramics 44 [2][2] (2010)(2010) 87–9387–93

FigureFigure 3.3. Comparison of of the the quadratic quadratic spline spline (Spline), (Spline), the five-point the five-point cubic cubicSG (5pt SG n = (5pt 3) and n =the 3) seven-point and the seven-point cubic SG cubic SG (7pt n n = = 3) 3) numerical numerical derivatives derivatives for thefor investigatedthe investigated three differentthree different peaks at peaks four resolutions at four resolutions error).As the In Fig.Cole-Davidson 3. points for function spline describes divided by an tenasym are- tounchanged the peak value and thethe slopefive-point of the SG high-frequency is the most accurate wing metricallyplotted in order broadened to scale Debye all results peak to where the same the slopeorder of (0.2decreases % absolute, with decreasing0.7 % relative E, it error) could then be regardedthe quadratic as a themagnitude. low-frequency Further wing from ofthe the maximum peak is unchangedthe same order and splinemixture (0.3 of % a absolute,Debye and 1.1 a % Cole-Cole relative error) peak. and It thebehaves sev- theof accuracyslope of the can high-frequency be found with wing approximately decreases with half de of- en-pointin the low-frequency SG (0.9 % absolute, range likeand a3.9 Debye % relative peak error).and in creasingabsolute β errors, it could which be regarded means as slightly a mixture better of a relative Debye Sothe not high-frequency only the accuracy range order as but a Cole-Colethe direction peak of the as anderrors a Cole-Cole(5pt n = 3: peak.5.4×10 It-5 behaves % abs., in1.2×10 the low-frequen-3 % rel.; 7pt- differenceobservable was from changed. the error All calculations. tested methods Values behave at the at cyn =range 3: 3.4×10like a -4Debye % abs., peak 7.2×10 and in-3 the% high-frequency rel.; Spline: - lowcharacteristic frequencies frequency similar to are the best Debye approximated case: 1. five-point by the -3 -2 range1.4×10 as aabs., Cole-Cole -3.4×10 peak% relativeas observable error) fromin spite the of error the SGquadratic (-2.0×10 spline-3 % absolute, (-0.03 -15.3 % absolute, % relative -0.2 error), % relative2. qua- calculations.decreasing function Values values.at the characteristic frequency are draticerror), spline then the(-5.4×10 five-point-3 % absolute, SG (-0.3 -40.5% absolute, % relative -1.7 er %- best Asapproximated the Cole-Davidson by the quadratic function spline describes (-0.03 % ab an- ror),relative 3. seven-point error), and SG (-1.7×10 the seven-point-2 % absolute, SG and (-1.1 -130.9 % solute,asymmetrically -0.2 % relative broadened error), then Debye the five-point peak where SG (-0.3 the %absolute, relative anderror), -5.5 and % atrelative high frequencieserror). Close similar to the to peak the %slope absolute, of the -1.7 low-frequency % relative error), wing and of the the seven-point peak is Cole-Colevalue the case: five-point 1. five-point SG is the SG most (4.6×10 accurate-4 % absolute, (0.2 % SG (-1.1 % absolute, and -5.5 % relative error). Close 3.2×10-3 % relative error), 2. seven-point SG (1.1×10-2 % xx 91 H. Haspel et al. / Processing and Application of Ceramics 4 [2] (2010) 87–93 absolute, 7.2×10-2 % relative error), 3. quadratic spline ror). 3. quadratic spline (0.04 % absolute, 1.1 % rela- (-2.8×10-2 % absolute, -0.2 % relative error). tive error). Doubling the resolution of the spectrum (r = 21/2) re- At r = 21/8 spacing the previous order is valid for all arranges the accuracy order in the Debye and the Cole- types of peaks used to describe dielectric relaxation pro- Davidson cases. At the maximum value of the Debye cesses. Even if we use the least accurate one, the qua- peak the former two best choices are interchanged: 1. dratic five-point spline, the absolute error remains be- five-point SG (-0.4 % absolute, -0.5 % relative error), 2. low 0.1 %, the relative error under 0.3 % for a Debye quadratic spline (0.8 % absolute, 1.0 % relative error), peak, and below 0.01 % absolute, and 0.25 % relative 3. seven-point SG (-1.8 % absolute, and -2.4 % relative error for a Cole-Davidson peak error), whereas at much lower and higher frequencies the order remains the same: 1. five-point SG (-4.5×10-4 IV. Conclusions % absolute, -0.8 % relative error), 2. quadratic spline The accuracy of numerical techniques could depend (-2.9×10-3 % absolute, -5.3 % relative error), 3. seven- on many parameters, hence we have to apply them very point SG (-3.0×10-3 % absolute, -5.5 % relative error). carefully. Here we examined the effect of the frequency, A strong decrease in the errors could be observed, es- spectrum resolution and the shape of the peak on the ac- pecially far from the peak maximum for the quadratic curacy of the so-called numerical logarithmic derivative spline and the seven-point SG. In the case of the Cole- technique, a versatile tool in the evaluation of dielectric Cole function the order is the same as mentioned above, relaxation spectroscopy measurements. A well-known nu- but the absolute and the relative errors are so small even merical method, the Savitzky-Golay convolution method if we use quadratic spline (0.002 % absolute, and 0.02 for differentiation was chosen for the calculations. % relative error), that these are completely negligible in Synthetic dielectric data were generated in the 10-3– the evaluation of a real measurement. Since the errors 107Hz frequency range with four different resolutions decrease with increasing resolution, only the two other (r = 2, 21/2, 21/4, 21/8 logarithmic spacing). Three types of types of peaks are discussed below. peaks were used to investigate the shape effect: the log- The best method for the Cole-Davidson function at arithmic derivative of the normalized Debye, a normal- this resolution (r = 21/2) is the five-point SG with the ized Cole-Cole (α = 0.2), and a Cole-Davidson (β = 0.2) highest absolute error of 0.03 % (0.35 % relative error) functions. These are all empirical dielectric relaxation in the peak area, and -5.4×10-5 % (-0.8 % relative er- functions used frequently. ror) in the low-frequency range. The second best choice From visual comparison of the numerical results for is the quadratic spline with an absolute error of 0.12 % the Debye peak it can be concluded, that with increas- (0.54 % relative error) near the maximum and -3.5×10-4 ing polynomial degree or decreasing dataset width bet- % (-5.3 % relative error) at low frequencies. The abso- ter approximations could be reached even at lower res- lute error for the seven-point SG at the maximum is un- olution (i.e. at r = 2 spacing). As the resolution of the der 1 % (0.17 % absolute, 0.77 % relative error) and spectrum increases, all of the investigated methods give is close to the absolute error of the spline in the low- good results with an acceptable error. Although apply- frequency regime (-3.6×10-4 % abs., and -5.5 % rela- ing higher-than-third degree polynomials is not recom- tive error). At high frequencies both the absolute and mended in order to avoid overfitting, the use of wider the relative errors of all three methods are much lower, dataset could be beneficial due to its stronger noise sup- practically negligible, which is the effect of the similar- pression capability. ity to the Cole-Cole function. A quantitative analysis of the investigated methods At r = 21/4 logarithmic spacing for Debye peak the was also done for all the three types of peaks at the four accuracy order is the following in the whole frequen- different resolutions. In the case of the Cole-Cole func- cy range: 1. five-point SG with a highest absolute er- tion a definite order in the accuracy can be found. The ror of -0.04% and -0.05% relative error. 2. seven-point best choice for the differentiation of such a symmet- SG with maximum -0.2 % absolute and -0.3 % relative rically broadened peak is the five-point Savitzky-Go- errors. 3. quadratic spline with maximum of 0.3 % ab- lay method at all resolutions. In the case of Debye and solute and 1.1 % relative errors. From this resolution Cole-Davidson functions, at lower resolutions there is a on, all examined techniques are able to approximate the difference in the order of accuracy near the peak maxi- sharpest possible derived spectrum with an error (either mum and far from it. Generally the five-point quadratic absolute or relative) less than 1 % (even less than 0.5 spline approximates the peak values best, followed by %) which is an often used limit in the characterization the five-point SG, which is in turn the most accurate in of analytical methods. the low and high frequency regime followed by the qua- For the Cole-Davidson peak the same order holds dratic spline. As the resolution increases the five-point with very similar highest error values: 1. five-point SG SG overtake the first place in every range of frequency (-0.003 % absolute, and -0.05 % relative error). 2. sev- for any of the peak shape, and the accuracy of the seven-point SG (-0.02 % absolute, and 0.3 % relative er- en-point SG also increases. Hence if the spacing can be

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