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Kruger, in "Passivity and Its Breakdown on Iron and Iron-Based Alloys," R. W. Staehle and H. Okada, Editors, p. 91, NACE, Houston, TX (1976). 42. M. A. Heine, D. S. Keir, and M. S. Pryor, This Journal, 112, 29 (1965). 43. M. J. Pryor, in "Localized Corrosion," R.W. Staehle, B.F. Brown, and J. Kruger, Editors, p. 2, NACE, Houston, TX (1974). 44. C. Y. Chao, L. F. Lin, and D. D. Macdonald, This Jour- nal, 128, 1187, 1194 (1981). 45. J. A. Richardson and G. C. Wood, Corros. Sci., 10, 313 (1970); This Journal, 120, 193 (1973). 46. K. Hashimoto and K. Asami, Corros. Sci., 19, 251 (1979). 47. R. Nishimura, M. Araki, and K. Kudo, Corrosion, 40, 465 (1984). 48. B. MacDougall, D. F. Mitchell, G. I. Sproule, and M. J. Graham, This Journal, 130, 543 (1983). 49. M. Urquidi-Macdonald and D. D. Macdonald, ibid., 134, 41 (1987). On the Application of the Kramers-Kronig Relations to Evaluate the Consistency of Electrochemical Impedance Data J. Matthew Esteban* and Mark E. Orazem* Department of Chemical Engineering, University of Florida, Gainesville, Florida 326I t ABSTRACT The use of the Kramers-Kronig (KK) relations to evaluate the consistency of impedance data has been limited by the fact that the experimental frequency domain is necessarily finite. Current algorithms do not distinguish between the re- sidual errors caused by a frequency domain that is too narrow and discrepancies caused by a system which does not sat- isfy the constraints of the KK equations. A new technique is presented which circumvents the limitation of applying the KK relations to impedance data which truncate in the capacitive region. The proposed algorithm calculates impedance values below the lowest experimental frequency which "force" the data set to satisfy the KK equations. Internally consist- ent data sets yield low-frequency impedance values which are continuous at the lowest measured experimental fre- quency. A discontinuity between the calculated low-frequency values and the experimental data indicates inconsistency which cannot be attributed to the finite experimental frequency domain. To facilitate the interpretation of impedance meas- urements, an investigator should know whether the exper- imental data is characteristic of a linear and stable system. It has been suggested that the Kramers-Kronig (KK) rela- tions can be employed to evaluate and analyze complex impedance data of electrochemical systems (1-3). These equations, developed for the field of optics, constrain the real and imaginary components of complex physical quan- tities for systems that satisfy the conditions of causality, linearity, stability, and finite impedance values at the fre- quency limits of zero and infinity (4-6). Bode (7) extended the concept to electrical impedance, and has tabulated var- ious forms of these equations. Macdonald and Urquidi- Macdonald (8) have demonstrated analytically that equiva- lent electrical circuits involving passive elements (R, C, L), the Warburg impedance, the pore diffusion model, R--C transmission lines, R--L transmission lines, and R--C--L transmission lines obey the KK relations. Consequently, experimental data that can be fitted to the analytical re- sponse of an equivalent circuit described above through nonlinear regression analysis satisfy the conditions of the KK relations. There is, however, controversy over the extent to which the KK relations can be used to validate electrochemical impedance data. The relationships are held to be valid, but Mansfeld and Shih (9-11) have argued that they are useless for data that do not include all the time constants for the system. Since the KK relations involved integrals over fre- quencies ranging from zero to infinity, valid data can ap- pear to be invalid if the measured frequency range is insuf- ficient. Macdonald et al. (2, 3, 12) have repeatedly emphasized the importance of collecting experimental * Electrochemical Society Active Member. data over a sufficiently wide frequency range in order to evaluate the KK relations with satisfactory accuracy, but this is not always possible. While the upper limit of mod- ern frequency analyzers (65 kHz to 1 MHz) is sufficient for most electrochemical systems, the lower measureable fre- quency limit for systems exhibiting large time constants is often governed by noise. It is this value that currently re- stricts the utility of the KK transforms. There is, therefore, a need to address the problem of ap- plying the KK relations to data sets with finite frequency domains which do not extend to a sufficiently low value. Once this problem is properly resolved then the sensitivity of the KK relations to evaluate experimental impedance data for violations of the causality, linearity, and stability conditions can be individually investigated . Previous au- thors have approached the problem differently: Mansfeld and Shih (9-11) stated that the KK transforms yield valid results only when the impedance data have reached a dc limit within the experimental frequency do- main. Application of a KK algorithm to valid data sets which truncate in the capacitive region result in discrepan- cies that may erroneously lead to the conclusion that the data sets are invalid. These discrepancies, however, are due to the neglected contributions to the integrals associ- ated with the inaccessible frequency domain (12). In one specific case, they were able to validate the experimental data using a KK algorithm only after having extrapolated the data below the lowest measured frequency using the fitting parameters of an equivalent circuit (10). This shows the importance of performing the integration over the widest frequency domain possible. It should be pointed out that, since the impedance response of electrical cir- cuits satisfy the KK relations, a "good fit" between the ex- perimental data and the analytic response of an equivalent