Kramers-Kronig validation

The Kramers-Kronig validation [11] is a well-established method to assess consistency and quality of measured impedance spectra. The Kramers-Kronig relations are integral equations, which constrain the real and imaginary components of the impedance for systems that satisfy the conditions of causality, linearity, and... 

Figure 13.8 Ilustration of the validity of the Kramers-Kronig relations for the piezoelectric relaxation in Sm-modified lead titanate ceramics. The imaginary component is calculated from the real using numerical method and Kramers-Kronig relations and compared with experimentally determined data.

For a linear system, the phase and amplitude of the impedance relate to each other. Consequently, if we know the frequency dependence of the phase we can calculate the amplitude of the impedance as a function of frequency. Similarly, we can deduce the frequency dependence of the phase from that of the amplitude of the impedance. The calculation can be achieved by the Kramers-Kronig (K-K) transforms. This is a useful check on the validity of a measured impedance spectrum. For information on K-K transforms, see Appendix C. 

Data validation (for instance, using the Kramers-Kronig test)... 

As both the frequency domain and the time domain methods have disadvantages, Boukamp [87] recommended that both methods be combined using the CNLS-fit procedure, data validation (Kramers-Kronig transformation), and deconvolution. The Kramers-Kronig transformation can be found in Appendix C. 

Darowicki K, Kawula J (2004) Validity of impedance spectra obtained by dynamic electrochemical impedance spectroscopy verified by Kramers-Kronig transformation. Pol J Chem 78(9) 1255-60... 

The Einstein relation (159) or the expression (157) of the dissipative part ffiep(m) of the mobility constitute another formulation of the first FDT. Indeed they contain the same information as the Kubo formula (156) for the mobility, since p(co) can be deduced from 9ftep(oo) with the help of the usual Kramers-Kronig relations valid for real co [29,30]. Equation (156) on the one hand, and Eq. (157) or Eq. (159) on the other hand, are thus fully equivalent, and they all involve the thermodynamic bath temperature T. Note, however, that while p(oo) as given by Eq. (156) can be extended into an analytic function in the upper complex half-plane, the same property does not hold for D(co). 

The Kramers-Kronig relations are extremely general emd have been applied to a wide variety of research areas. In the field of optics, for which the validity of the Kramers-Kronig is not in question, the relationship between real and imaginary components has been exploited to complete optical spectra. In other areas where data cannot be assumed to satisfy the requirements of the Kramers-Kronig relations, the equations presented in Table 22.1 have been used to check whether real and imaginary components of complex variables are internally consistent. Failure of the Kramers-Kronig relations is assumed to correspond to a failure within the experiment to satisfy one or more of the constraints of linearity, stability, or causality. 

Qsp were applied for the validation of electrochemical impedance data. Agarwal et al. described an approach that eliminated problems associated with direct integration of the Kramers-Kronig integral equations and accoimted explicitly for stochastic errors in the impedance measurement. 

Let us finally notice that, investigating the dispersion theory for the effective third-order nonlinear susceptibility of nanocomposite media, Peiponen et al. established that Kramers-Kronig relations are not valid for whereas they are valid for other nonlinear processes such as frequency conversion [95]. 

It has been shown that instead of Kramers-Kronig transforms, another method involving a coherence function could be used to validate the data. The coherence function, y, is defined as... 

In principle, there is a complication concerning constant or slowly-varying background terms which have been omitted from (6.31) in the simple form used here, the Kramers-Kronig relations are valid only for... 

Scattering theory has very general validity. It is based on the existence of a Schrodinger-type equation and is consistent with causality requirements through the Kramers-Kronig dispersion relations. It is widely used outside atomic physics for the description of all resonance phenomena. There is therefore some advantage in using it in atomic physics it ensures unification and is more transparent to reseachers from other fields. 

The validity of El S data can be checked using the Kramers-Kronig transformations, which calculate the imaginary component of the impedance from the real component, and vice versa [17,18]. 

The Kramers-Kronig frequency domain transformations enable the calculation of one component of the impedance from another or the determination of the phase angle from the magnitude of the impedance or the polarization resistance Rp from the imaginary part of the impedance. Furthermore, the Kramers-Kronig (KK) transforms allow the validity of an impedance data set to be checked. Precondition for the application of KK transforms is, however, that the impedance must be finite-valued... 

Several other forms of these relations can be found in the literature [3]. Although the preceding equations were written for impedances, they are valid for any complex transfer function. Kramers-Kronig relations are very restrictive, and in EIS some of them might be slightly relaxed, and instead of the impedances, the admittances can be used. This will be discussed in what follows. 

Kramers-Kronig transforms require integration over frequencies from zero to infinity, which, in practice, is difficult to carry out. There exists a Hilbert logarithmic transform [602, 603] that can be used to validate impedance data in a limited frequency range. It is known under the name Z-HIT transform [585] and is used in the Zahner software [604]. The logarithm of a transfer function known for frequencies between tOs and cOo may be written as a function of the phase angle ... 

Experimental impedance data should be validated before further analysis. Raw data might be verified using Kramers-Kronig or Z-Hit transforms. It should be kept in mind that these transforms are not very sensitive to system nonlinearities, and an additional test with different amplitudes could be carried out. An alternative to the aforementioned transforms is the approximation to linear circuits (Figs. 13.2 and 13.4). 

Then the acquired data must be validated using Kramers-Kronig transforms. Such validated data can be used in subsequent analysis and modeling. 

Electrochemical impedance spectroscopy is a mature technique, and its fundamental mathematical problems are well understood. Impedances can be written for any electrochemical mechanism using standard procedures. Modem electrochemical equipment makes it possible to acquire data in a wide range of frequencies and with various impedance values. The validity of experimental data can be verified by standard procedures involving Kramers-Kronig transforms. Several programs either allow for the use of predefined simple and distributed elements in the construction of electrical equivalent circuits or directly fit data to equations (which should be defined by the user). 

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. 

An important requirement for a valid impedance function is that the system be linear. Theoretically, this implies that the real and imaginary components transform correctly according to the Kramers-Kronig relationships (discussed later in this section). Practically, linearity is indicated by the impedance being independent of the magnitude of the perturbation, a condition that is easily (although seldom) tested experimentally. 

Song, H. and Macdonald, D.D. (1991) Photoelectrochemical impedance spectroscopy I. Validation of the transfer function by Kramers-Kronig transformation. Journal of The Electrochemical... 

A critical problem in EIS, as well as any other scientific measurements, is the validation of the experimental data. The use of Kramers-Kronig (KK) transforms has been proposed to assess the quality of the measured impedance data [15]. These integral transforms were derived assuming four basic conditions ... 

In fact, these properties are not specific to the Debye-process, but have a deeper basis which extends their validity. According to the Kramers-Kronig relations, J and J" are mutually dependent and closer inspection of the equations reveals that it is impossible, in principle, to have a loss without a simultaneous change in J. Both effects are coupled, the reason being, as mentioned above, the validity of the causality principle. 

---