Voigt model impedance

Selection of an appropriate initial value for the time constant is critical for regression of the Voigt model (see equation (20.5)) to impedance data. Inductive loops can be modeled by the Voigt model by allowing the resistance values to be... 

In some cases, the models used for impedance are strictly defined. Others, such as the Voigt model, allow use of an arbitrary number of parameters. The fit of a Voigt model can be improved by sequentially adding RC elements, and the best model is achieved when the x statistic reaches a minimum value. 

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. 

It is equivalent to a series of a number of RC parallel combinations in series with the solution resistance, which is also known as the Voigt model. If a sufficient number of terms is included, the model should be able to fit impedance data of any stationary electrochemical cell. Since the model is consistent with the Kramers-Kronig relations, any failure to fit experimental data using this model indicates a violation of the Kramers-Kronig relations. 

Orazem and coworkers [3,624,625] modified the method of statistical weighting in such a way that it could be used in cases of mildly nonstationary systems. Data should be acquired several times and fitted to the Voigt model, retaining a statistically significant number of circuit elements. From the differences between the experimental and model impedance values (each of which might be different for sequentially acquired impedances), the standard deviations might be determined at each frequency and then fitted to the model described earlier in (e). The calculated values are used in the calculation of the sum of squares, Eq. (14.18). The proposed procedure is detailed below ... 

Another way an incorrect or misleading large effective dielectric constant or capacitance can be obtained from even an N = 2 Voigt-model circuit is if experimental impedance data associated with such a circuit are interpreted in terms of a series capacitance and series resistance Z = + (io)C ) , where Z ... 

The above discussion demonstrates the possibility of using relatively simple models (the Voigt-type analog) in describing the impedance characteristics of Li electrodes. It should be noted, however, that other models have also been proposed, such as the space-charge approach proposed by Pejovnik et al. [98-100],... 

The complex-plane impedance diagram of the Voigt structure with three RC circuits is depicted in Figure 4.256. It is characterized by three time constants, r, t2, and t3. More examples of this model can be found in Appendix D (Model D22). 

Example 3.5 Evaluation of Chi-Squared Statistics Consider that, for a given measurement, regression of a model to real and imaginan/ parts of impedance data yielded = 130. Measurements were conducted at 70 frequencies. The regressed parameters needed to model the data included the solution resistance and 9 Voigt elements, resulting in use of 19 parameters. Under assumption that the variances used in the evaluation ofxf were obtained independently, evaluate the hypothesis that the x value cannot be reduced by refinement of the model. 

Example 19.1 Nonlinear Models Show that the equation for the impedance of a Voigt element is nonlinear with respect to parameters. 

The measurement model method for distinguishing between bias and stochastic errors is based on using a generalized model as a filter for nonreplicacy of impedance data. The measurement model is composed of a superposition of line-shapes that can be arbitrarily chosen subject to the constraint that the model satisfies the Kramers-Kronig relations. The model presented in Figure 21.8, composed of Voigt elements in series with a solution resistance, i.e.. 

The fit of the measurement model with 11 Voigt elements is presented in complex-impedance-plane format in Figure 20.7. The discrepancies evident in Figure 20.2 for the model presented in Section 20.2.1 are not apparent in Figure 20.7. 

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 extension of the previous models to a sphere coupled to the plate via a spring and a dashpot is straightforward. The coupling can be achieved either via a Voigt-type circuit (viscoelastic solid, Fig. 2e) or via a Maxwell-type circuit (viscoelastic liquid, Fig. 2f). Below, we assume that the object is so heavy that it does not take part in the motion. When the mass is infinite, the inertial term drops out of the load impedance. An infinite mass is graphically depicted as a wall. For Voigt-type couphng we find ... 

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