Measurement Models for Electrochemical Impedance Spectroscopy ...
Measurement Models for Electrochemical Impedance Spectroscopy ... Measurement Models for Electrochemical Impedance Spectroscopy ...
4156 J. Electrochem. Soc., Vol. 142, No. 12, December 1995 9 The Electrochemical Society, Inc.Table I. Parameter estimates for the error-structure model (Eq. 14) obtained by regression to replicate data for asingle-crystal n-GaAs/Ti Scholtky diode.ParameterUnfilteredNo movingAverageThree-point moving Five-point movingAverage AverageNo movingAverageFilteredThree-point movingAverageFive-point movingAveragecr 3.29-+ 0.13 • 10 -4 2.46_+ 0.48 • 10 -4 8.2-+ 5.6 • 10 3 8.12_+ 0.020 • 10 -4 9.45-+ 0.50 • 10 4(_+4.0%) (-+20%) (-+69%) (_+0.23%) (-+5.3%)1.20-+ 0.01 • 10 3 1.24_+ 0.03 x 10 -3 1.59-+ 0.07 • 10 -3 9.33-+ 0.011 • 10 4 1.49_+ 0.045 • 10 -3(-+0.88%) (+-2.8%) (-+4.1%) (-+1.1%) (+-3.0%)2.833 2.832 2.785 2.306 2.209-+0.0021 • 10 -4 _+0.0062_+ 10 ~ +-0.011 • 10 4 -+0.0017 _+ 10 -4 _+0.0090• 10 4(0.074%) (0.22%) (0.39%) (0.072%) (0.41%)X3/~ 4.74 1.77 1.64 15.2 2.189.43 -+ 0.64 • I0 -4(+_6.8%)1.12 _+ 0.052 • 10 -3(-+4.7%)2.268 -+ 0.010 • 10 .4(0.46%)1.83The square of the standard deviation of the standard deviationsfor the real and imaginary components of the impedancewas used, therefore, to weight the regression foridentification of the error structure. In addition, three- andfive-point moving averages were used to increase the samplesize for the standard deviation while retaining the generaltrends. The weighted X2 statistic (normalized by thedegrees of freedom for the regression)• 1 (Yexpt~i -- Ymodel,i) 2iwas improved for regressions using a moving average valuefor the variance. The use of a moving average did not appearto influence the fit of the model to cases where thestandard deviation for the impedance was large, but the fitwas qualitatively improved for cases where the standarddeviation of the impedance was small. The utility of themoving average should depend on the sampling rate. Thefive point moving average worked well for the ten points/decade sampling rate used here. The increased standarddeviation for the parameter estimates obtained using themoving average reflects the corresponding increased valuefor the average standard deviation used to weight the regression.By giving a better estimate for the standard deviationof the fitted quantity, the moving-ayerage approachyielded more reliable estimates for the confidence intervalsof the error-structure model parameters.Results of regression.--The parameter estimates obtainedare given in Table I. The results of the regression for GaAsat 320 K are shown in Fig. 14, and the results for 400 K arepresented in Fig. 15. The solid line represents the model forthe error structure given by Eq. 14. The dashed lines representthe contribution of the different terms in Eq. 14. Thejog in the line for the model corresponds to the frequency atwhich a change was made in the value of the current measuringresistor.The filtering algorithm described in the section Identificationof the noise model was applied to the above data setto give a better estimate for the stochastic errors. Equation14 was regressed to the filtered errors. The resultingparameters for the error structure model are shown inTable I. The filtered errors for GaAs at 320 and 400 K, withthe new model, are shown in Fig. 16 and 17, respectively.The standard deviations for the filtered data (Fig. 14) weresmaller than for the unfiltered case (Fig. 12), and the realand imaginary parts of the standard deviations are closer.The regression results in Fig. 16 and 17 show that the agreementof the model for the error structure (Eq. 14) to theexperimental standard deviations is good. Similar agreementwas found at all temperatures. The parameters fromTable I, for filtered errors, were used to predict the errorsfor corrosion of copper data shown in Fig. 3. The results arepresented in Fig. ii. The model shows a good agreementwith the experimentally obtained standard deviations. Thevalidity of the equation is supported since a three-parame-ter model provides a good agreement for solid-state systemsas well as for electrochemical systems, for data collectedunder various experimental conditions, and forerrors ranging in magnitude from 10 -3 to i0 ~ ~.Discussion and ConclusionThe measurement model provides much more than a preliminaryanalysis of impedance data in terms of the numberof resolvable time constants and asymptotic values, as suggested,for example, by Zoltowski. 23'24 As shown here, themeasurement model can be used as a filter for lack of replicacythat allows accurate assessment of the standard deviationof impedance measurements. As is discussed by manyauthors, 1'15,2~ this information is critical for selection ofweighting strategies for regression. This information alsoprovides a quantitative basis for assessment of the qualityof fits and can guide experimental design. In the subsequentpaper of this series, 9 the measurement model is usedto assess the bias component of the error structure.The results presented here show that impedance measurementsare heteroskedastic (in the sense that the standarddeviations are funetior~s of frequency). In spite of theapparent complexity, a definite structure for the errors isresolved. In the impedance plane, the standard deviationsfor the real and imaginary parts of the impedance areequal, even at frequencies sufficiently high or low that theimaginary part of the impedance asymptotically approacheszero. This result is consistent with the results presentedby Zoltowski. 17Aside from the obvious impact on the parsimony of themodel for the error structure (only three parameters areb"10 610 410 210 0-210' '""9 ........ I ....... '1 ' '"'"'1 ' ' .....--410 i , * ,iliil[ i I. lillJd , *liil,,l , ililild , %,,,i,.I0 ~ I01 I02 I0 ~ 104 105Frequency, HzFig. 14. Unfiltered standard deviation of the 320 K data presentedin Fig. I as a function of frequency. Circles represent the real and thetriangles represent the imaginarypart of the standard deviation. Thesolid line represents the error structure with unfiltered parametersshown in Table I. The dashed-dot line represents the contribution ofthe imaginary term to the error. The dashed-dot-dot line representsthe contribution of the real term, and the dashed line is the measuring-resistorterm."%
J. Electrochem. Soc., Vol. 142, No. 12, December 1995 9 The Electrochemical Society, Inc. 4157106104' ''"'"I ' '''""I ' ';'""I ' ''"'"I ' ''"I106104' ''"'"1 ' ''"'"t ' ''"'"l ' ''""~1102,102100100,i i %.--.410 -2-4ft\ -"i10 = ,,,,.,I , ,,,,,,,i , ,,,,,ul , ,,,,,,,I , i, ....s100 101 102 105 104 05Frequency, HzFig. 15. Unfiltered standard deviation of lhe 400 K data presentedin Fig. 1 as a function of frequency. Symbols and lines as defined inFig. 14.needed in Eq. 14 as compared to six in Eq. 13), the equalityof the real and imaginary standard deviations has implicationsfor the regression of models to impedance data. Thatthe information content of the imaginary part of theimpedance can be obscured by noise at the asymptotic tailsinfluences the manner in which the Kramers-Kronig relationscan be applied to assess the bias contribution to themeasurement. ~ In addition, the equality of the real andimaginary standard deviations becomes a criterion for selectionof appropriate weighting strategies. Among thecommonly applied weighting strategies, for example, proportionalweighting does not conform to this observation,but "no-weighting" and modulus weighting do conformand may be useful weightings for preliminary regressions.While the heteroskedastic nature of the measurementsshown here suggests that the no-weighting strategy is inappropriatefor experiments under potentiostatie modulation,this is not a general result since the standard deviationof some measurements [e.g., electrohydrodynamic imped-10104~106 ........, ........ I ........, ........ I .......J%',%102 "~.10 ~%,%,-2 "-10 4 ....... J ........ , ........ i ........, ....10 U 101 102 103 10 4Frequency, HzFig. 16. Filtered standard deviation of the 320 K data presented inFig. 1 as a function of frequency. Symbols and lines as defined inFig. 14 with exception that model parameters are those given asFiltered in Table I.10 -210 -41!f\ 14I i llillll j I ' illliii t I ill'ltJ t I lllilJ i I ilail)0 101 10 Z 105 104 05Frequency, HzFig. 17. Filtered standard deviation of the 400 K data presented inFig. 1 as a function of frequency. Symbols and lines as defined inFig. 16.ance spectroscopy (EHD)] is, to a first approximation, independentof frequency. 6'7Selection of inappropriate weighting strategies may havea severe impact on the amount of information that can beobtained by regression of models to impedance data sincethe weighting strategies discussed here can differ by manyorders of magnitude. To illustrate the influence of weighting,normalized weightings, defined byfor modulus weighting or by[ o'/IZl ~2to,z, = \(r l}~vJ [16a]~o = [16b]1.0E+01 1000 +001,0E-0mI0 o~I .OE-02 " ~".oE-o3 ..i ~~1.0E-04 1 "N 1.0E-06 ~.E1.0E_07 0"1 iZZ~1.0E-08.'_'_iiii'i~iiitn -No Weighting1.0E-09 ........................................... 0.01t 10 100 1000 10000 100000Frequency, HzFig. 18. Comparison of modulus weighting and no-weightingstrategies to the weighting by the error structure given in Eq. 14. Theterms plotted are defined by Eq. 16. The solid lines represent thecomparison of the modulus weighting strategy to weighting by thestandard deviation of the experiment determined by the methods ofthis paper. The dashed lines represent the corresponding comparisonfor the no-weighting weighting strategy. The upper set of lines wereobtained for the data collected at 320 K, and the lower set of lineswere obtained for the data collected at 400 K.
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J. Electrochem. Soc., Vol. 142, No. 12, December 1995 9 The <strong>Electrochemical</strong> Society, Inc. 4157106104' ''"'"I ' '''""I ' ';'""I ' ''"'"I ' ''"I106104' ''"'"1 ' ''"'"t ' ''"'"l ' ''""~1102,102100100,i i %.--.410 -2-4ft\ -"i10 = ,,,,.,I , ,,,,,,,i , ,,,,,ul , ,,,,,,,I , i, ....s100 101 102 105 104 05Frequency, HzFig. 15. Unfiltered standard deviation of lhe 400 K data presentedin Fig. 1 as a function of frequency. Symbols and lines as defined inFig. 14.needed in Eq. 14 as compared to six in Eq. 13), the equalityof the real and imaginary standard deviations has implications<strong>for</strong> the regression of models to impedance data. Thatthe in<strong>for</strong>mation content of the imaginary part of theimpedance can be obscured by noise at the asymptotic tailsinfluences the manner in which the Kramers-Kronig relationscan be applied to assess the bias contribution to themeasurement. ~ In addition, the equality of the real andimaginary standard deviations becomes a criterion <strong>for</strong> selectionof appropriate weighting strategies. Among thecommonly applied weighting strategies, <strong>for</strong> example, proportionalweighting does not con<strong>for</strong>m to this observation,but "no-weighting" and modulus weighting do con<strong>for</strong>mand may be useful weightings <strong>for</strong> preliminary regressions.While the heteroskedastic nature of the measurementsshown here suggests that the no-weighting strategy is inappropriate<strong>for</strong> experiments under potentiostatie modulation,this is not a general result since the standard deviationof some measurements [e.g., electrohydrodynamic imped-10104~106 ........, ........ I ........, ........ I .......J%',%102 "~.10 ~%,%,-2 "-10 4 ....... J ........ , ........ i ........, ....10 U 101 102 103 10 4Frequency, HzFig. 16. Filtered standard deviation of the 320 K data presented inFig. 1 as a function of frequency. Symbols and lines as defined inFig. 14 with exception that model parameters are those given asFiltered in Table I.10 -210 -41!f\ 14I i llillll j I ' illliii t I ill'ltJ t I lllilJ i I ilail)0 101 10 Z 105 104 05Frequency, HzFig. 17. Filtered standard deviation of the 400 K data presented inFig. 1 as a function of frequency. Symbols and lines as defined inFig. 16.ance spectroscopy (EHD)] is, to a first approximation, independentof frequency. 6'7Selection of inappropriate weighting strategies may havea severe impact on the amount of in<strong>for</strong>mation that can beobtained by regression of models to impedance data sincethe weighting strategies discussed here can differ by manyorders of magnitude. To illustrate the influence of weighting,normalized weightings, defined by<strong>for</strong> modulus weighting or by[ o'/IZl ~2to,z, = \(r l}~vJ [16a]~o = [16b]1.0E+01 1000 +001,0E-0mI0 o~I .OE-02 " ~".oE-o3 ..i ~~1.0E-04 1 "N 1.0E-06 ~.E1.0E_07 0"1 iZZ~1.0E-08.'_'_iiii'i~iiitn -No Weighting1.0E-09 ........................................... 0.01t 10 100 1000 10000 100000Frequency, HzFig. 18. Comparison of modulus weighting and no-weightingstrategies to the weighting by the error structure given in Eq. 14. Theterms plotted are defined by Eq. 16. The solid lines represent thecomparison of the modulus weighting strategy to weighting by thestandard deviation of the experiment determined by the methods ofthis paper. The dashed lines represent the corresponding comparison<strong>for</strong> the no-weighting weighting strategy. The upper set of lines wereobtained <strong>for</strong> the data collected at 320 K, and the lower set of lineswere obtained <strong>for</strong> the data collected at 400 K.
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