On the Error Structure of Impedance Measurements

Abstract

A new paradigm is introduced for the investigation of errors in frequency-domain measurements. The propagation of errors from time-domain measurements to the desired complex variables in the frequency domain was analyzed for the frequency response analysis ͑FRA͒ algorithm, one of two techniques commonly used for spectroscopy measurements. Errors in the frequency domain were found to be normally distributed, even when the errors in the time-domain were not normally distributed and when the measurement technique introduced bias errors. For additive errors in time-domain signals, the errors in the real and imaginary impedance were found to be uncorrelated, and the variances of the real and imaginary parts of the complex impedance were equal. The equality of variances was realized except for cases where the time-domain signals contained proportional errors. The statistical characteristics of the results were in good agreement with experimental observations. Weighted complex nonlinear regression techniques are typically used to extract information from electrochemical impedance data. The error structure of the measurement is used implicitly in regression analysis and has a significant influence on the quality and amount of information that can be extracted from impedance data. The variance of the stochastic error can be incorporated explicitly into the weighting strategy for the regression and can provide a means for determining whether observed features lie outside the noise level of the measurement. The stochastic errors can also influence the use of the Kramers-Kronig relations for determining the internal consistency of the data.For the purposes of the discussion presented here, the errors in an impedance measurement are expressed in terms of the difference between the observed value Z ob () and a model value Z mod () aswhere res represents the residual error, fit () is the systematic error that can be attributed to inadequacies of the model, bias () represents the systematic experimental bias error that cannot be attributed to model inadequacies, and stoch () is the stochastic error with expectation E͕ stoch ()͖ ϭ 0. The experimental bias errors, as referred to here, may be caused by a nonstationary behavior or by instrumental artifacts.Typically, the impedance is a strong function of frequency and can vary over several orders of magnitude through the experimentally accessible frequency range. The stochastic errors of the impedance measurement are strongly heteroskedastic, which means that the variance of the stochastic errors is also a strong function of frequency. 1-4 Selection of an appropriate weighting strategy is, therefore, critical for interpretation of data.A distinction is drawn, in the present work, between stochastic errors that are randomly distributed about a mean value of zero, errors caused by the lack of fit of a model, and experimental bias errors. The problem of interpretation of impedance data is therefore defined as consisting of two parts, one of identification of experimental errors, which ...

A systematic approach toward error structure identification for impedance spectroscopy

“…(3) is used under conditions where the error covariance terms cannot be neglected, the incorrect error structure will be reflected in the parameter estimates. Carson et al [27] used numerical simulations to show that, when a frequency-response analyzer algorithm is used to obtain the impedance response from time-domain signals containing normally distributed errors, the real and imaginary parts of the impedance were uncorrelated. In contrast, use of a phase-sensitivedetection algorithm yielded correlation between real and imaginary components of the impedance [28].…”

On the Error Structure of Impedance Measurements

“…Analytic expressions for all the coefficients involving derivatives in Eq. 30 These results were supported by instrument simulations conducted by Carson et al 23,24 The result that Z r 2 ϭ Z j 2 can be expressed in terms of the Taylor series expansions 14 and 15 as 1 and 2, respectively.…”

“…The approach was that described by Carson et al 23,24 for an electrochemical cell consisting of a leading 1 ⍀ resistor R e in series with a parallel combination of a resistor R and a capacitor C. The value of R/R e was allowed to vary between 1 and 10 4 , and the capacitance C was adjusted such that the RC time constant was always 10 Ϫ4 s. 16 was satisfied and for cases where it was not satisfied.…”

“…Both Milocco and Biagiola 20 and Carson et al [22][23][24] reported that the variance of the real and imaginary parts of the impedance were equal, but drew no conclusions concerning the behavior of the complex current and potential from which the impedance was obtained. Both Milocco and Biagiola 20 and Carson et al [22][23][24] reported that the variance of the real and imaginary parts of the impedance were equal, but drew no conclusions concerning the behavior of the complex current and potential from which the impedance was obtained.…”

Reliable Discrimination of Models Based on EIS Data

“…Following Ref. 12, where the covariance of real Z i Ј and imaginary Z i Љ parts of the collected data Z i at the frequency f i has been shown to be zero for FRA devices, we decompose the experimental datum and its associated range using the rectangular form…”