Measurement Models for Error Identification

While the line-shapes parameters may not be unequivocally associated with a set of deterministic or theoretical parameters for a given system, the measurement model approach has been shown to adequately represent the impedance spectra obtained for a large variety of electrochemical systems. The line-shape models represent the low-frequency stationary components of the impedance spectra (in a Fourier sense). Regardless of their interpretation, the measurement model representation can be used to filter and thus identify the nonstationary (drift) and high-frequency (noise) components contained in the same impedance spectrum. 

At first glance, it may not be obvious that such an approach should work. It is well known, for example, that the impedance spectrum associated with an electrochemical reaction limited by the rate of diffusion through a stagnant layer (either the Warburg or the finite-layer diffusion impedance) can be approximated by an infinite number of RC circuits in series (the Voigt model). In theory, then, a measurement model based on the Voigt circuit should require an infinite number of parameters to adequately describe the impedance response of any electrochemical system influenced by mass transfer. 

Thus the Voigt circuit can provide an adequate description of impedance data influenced by mass transfer or by distributed-time-constant phenomena such as is described in Chapter 13. In addition, inductive loops can be fitted by a Voigt circuit by using a negative resistance and capacitance in an element. Such an element will have a positive RC time constant. The Voigt circuit serves as a convenient generalized measurement model. 

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 

It should be noted that the error analysis methods using measurement models are sensitive to data outliers. Occasionally, outliers can be attributed to external influences. Most often, outliers appear near the line frequency and at the beginning of an impedance measurement. Data collected within 5 Hz of the line frequency and its first harmonic (e.g., 50 and 100 Hz in Europe or 60 and 120 Hz in the United States) should be deleted. Startup transients cause some systems to exhibit a detectable artifact at the first frequency measured. This point, too, should be deleted. 

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