Impedance-experimental parameters limits

Cases in which Impedance Spectroscopy Becomes Limited. One might say that if one understands an interface well, the results of Z-to measurements can be readily understood. Of course, the interest is in the other direction, in using Z-to plots when one does not understand the interlace. Then the task is to find an interfacial structure and mechanism (and its resulting equivalent circuit) that provides a Z that is consistent in its dependence on to with the experimental results of the impedance measurement. This requires finding reasonable parameters to fit the value of the C s and R s as a function of to for the individual elements in the various equivalent circuits. If the shape of the calculated Z-to plot can only be made to match experiment by using C s and R s that are physically unreasonable, the proposed structure and the equivalent circuit to match it are not acceptable and another must be tried. 

In an age of computerized instrumentation, ac impedance and other measurement results are often presented to the user with four or more digits of precision, with little reminder of the intrinsic limitations of the measurement or computational techniques used. Even when considerable care is given to the electrical connections of the system under test and to analysis of subsequently produced data, the operation of the instrument is often transparent to the user. Since the choice of analyzer may determine acquisition precision, time, and other important parameters of data collection, it is of some value to the careful experimenter to understand as fully as possible the method of operation of the impedance analyzer. It is hoped that this chapter is useful in contributing to this understanding. 

Here functions R(v) and C(v) can be obtained by piecewise-linear interpolation of the dependence of R and C parameters obtained by fitting the experimental spectra at different voltages (such as in Figure 4.5.4) to the impedance function in Eq. (10). Any other suitable smooth interpolation can be used. The impedance function has to be expressed in terms of electric parameters, as described in Section 4.5.1.3. For use in a discretized equivalent circuit, the values obtained from the fit have to be divided or multiplied by the number of chains, depending on the series or parallel position of the electric element. So, for series resistors it has to be divided, and for parallel, multiplied. It should be considered that the low-frequency limit of Re Z), used as a fitting parameter in the equation, is not always a simple sum of the discrete elements that constitute a transmission line. In particular, in Eq. (10) the Ra is 1/3 of the specific resistance multiplied by the transmission line length, as can be seen from Eq. (8). Therefore resistance of single chain shown in Eig 4.55 will be Ra 3/N. 

The reaction penetration depths. Id or la, are highly insightful parameters to evaluate catalyst layer designs in view of transport limitations, uniformity of reaction rate distributions, and the corresponding effectiveness factor of Pt utilization, as discussed in the sections Catalyst Layer Designs in Chapter 1 and Nonuniform Reaction Rate Distributions Effectiveness Factor in Chapter 3. Albeit, these parameters are not measurable. The differential resistances, Rd or Ra, can be determined experimentally either as the slope of the polarization curve or from electrochemical impedance spectra (Nyquist plots) as the low-frequency intercept of the CCL semicircle with the real axis. The expressions in Equation 4.33 thus relate the reaction penetration depths to parameters that can be measured. 

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