The Use of Kramers-Kronig Transforms

The use of a frequency domain transformation first described by Kramers [1929] and Kronig [1926] offers a relatively simple method of obtaining complex impedance spectra using one or two ac multimeters. More important, retrospective use of Kramers-Kronig (KK) transforms allows a check to be made on the validity of an impedance data set obtained for linear system over a wide range of frequencies. Macdonald and Urquidi-Macdonald [1985] have applied this technique to electrochemical and corrosion impedance systems. 

The KK transforms of interest in analyzing corrosion and electrochemical systems are 

These equations show that the real component of the impedance can be calculated from the imaginary component and vice versa, the phase angle p(o)) can be com- 

Practical limitations are imposed at low frequencies, however, where the rectification-smoothing function necessary to transduce the ac voltage magnirnde to a dc level becomes inaccurate. Ac voltmeters typically become seriously in error at frequencies below 20Hz. To obtain an accurate KK transform, it is necessary to extend the measurement frequency range significantly beyond the limits of frequency needed to elucidate the equivalent circuit under test. Thus, the method described here is not appropriate for aqueous electrochenfical systems for which the diffusional impedance is prominent. This method can be useful for systems in which the lowest frequency of interest is greater than 50 Hz or so, as is usually the case for solid ionic conductors, oxide films, and semiconductor surfaces. 

A more subtle limitation is imposed by the use of the method described here in that all of the four assumptions implied in the use of KK transforms are subsumed when the magnitude-to-phase transformation is made. That is, the unknown impedance is given the properties of linearity, invariance, and causality whether or not they apply, and there is no independent check of this assumption. In current practice, this limitation is not very severe since experimenters frequently report and draw conclusions from impedance data sets, normally derived, that have not been subjected to the scrutiny of the KK rules or other simple tests of experimental validity. 

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The use of measurement models to identify consistency with the Kramers-Kronig relations is equivalent to the use of Kramers-Kronig transformable circuit analogues. An important advantage of the measurement model approach is that it identifies a small set of model structures that are capable of representing a large... 

If the transfer function H is in accordance with the causality rule, the components R and X are no longer independent of each other. Causality in the meaning of system theory forces couplings between the real and imaginary part, which are known as Kramers-Kronig relations (KKT) or Hilbert relations (HT), for details see Section 3.1.2.9 (The use of Kramers-Kronig Transforms). 

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