Checking Kramers -Kronig Compliance by Approximations

It is possible to replace the Kramers-Kronig integration by approximation. If the system can be well approximated by a linear circuit, then it must be Kramers-Kronig transformable. Orazem and coworkers [572, 573] proposed using the Voigt circuit displayed in Fig. 13.2. 

Each element of this circuit consists of the connection of R and C in parallel, and it is, of course, transformable therefore, their sum is also transformable. The impedance of the Voigt circuit is described by the following equation  

Boukamp and Macdonald [574] and Boukamp [575] proposed fitting impedances to a Voigt circuit with a fixed distribution of time constants, taking six to seven time constants per decade or simply the time constant equal to the inverse of the experimental frequency , tj = 1/ in the latter case, a perfect approximation is obtained even in the presence of the experimental noise, which should, in principle, be avoided. By fixing the values of tj, the only unknown parameters in Eq. (13.10) are R. The CNLS approximation becomes linear and no initial guess of parameters R is necessary (Chap. 14). The approximating function becomes 

In the case of blocking electrodes, the impedance increases to infinity as the frequency approaches zero. In such cases, approximation with the Voigt circuit is not appropriate. When a high-frequency impedance is finite, the easiest way to verify the Kramers-Kronig compliancy is to fit the impedances to the admittance representation of the circuit containing a ladder of (RC) element series (Fig. 13.4) [575]. In addition, capacitance, Cq, or inductance can be added in parallel. 

Analysis of the plot in Fig. 13.6b using linear approximation, Eq. (13.11), is displayed in Fig. 13.7. Note that important deviations appear at 10 rad s . In further analysis, these data points were eliminated without any important loss of information in the high-frequency zone in Fig. 13.6a [575]. 

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