Circuit ambiguous

Here R12 and Ci,2 are defined such that RiCi R2C2. 

---

The subject of multiplicative fluctuations (in linear and especially nonlinear systems) is still deeply fraught with ambiguity. The authors of Chapter X set up an experiment that simulates the corresponding nonlinear stochastic equations by means of electric circuits. This allows them to shed light on several aspects of external multiplicative fluctuation. The results of Chapter X clearly illustrate the advantages resulting from the introduction of auxiliary variables, as recommended by the reduced model theory. It is shown that external multiplicative fluctuations keep the system in a stationary state distinct from canonical equilibrium, thereby opening new perspectives for the interpretation of phenomena that can be identified as due to the influence of multiplicative fluctuations. 

The lack of uniqueness of circuit models creates ambiguity when interpreting impedance response using regression analysis. A good fit does not, in itself, validate the model rised. As discussed in Chapter 23, impedance spectroscopy is not a standalone technique. Additional observations are needed to validate a model. 

Perhaps the major problem with the use of electrical circuit models to determine consistency is that interpretation of a poor fit is ambiguous. A poor fit could... 

While reaction parameters were not identified by regression to impedance data, the simulation presented by Roy et al. demonstrates that side reactions proposed in the literature can account for low-frequency inductive loops. Indeed, the results presented in Figures 23.4 and 23.5 show that both models can account for low-frequency inductive loops. Other models can also account for low-frequency inductive loops so long as they involve potential-dependent adsorbed intermediates. It is generally understood that equivalent circuit models are not unique and have therefore an ambiguous relationship to physical properties of the electrochemical cell. As shown by Roy et al., even models based on physical and chemical processes are ambiguous. In the present case, the ambiguity arises from uncertainty as to which reactions are responsible for the low-frequency inductive features. 

It should also be added that use of the equivalent circuits may introduce ambiguities. It is clear that in the Voigt circuit (a), permutation of the values of elements f 2 - C2 and f 3 C3 does not change the impedance values and the attribution of values to one or another set is arbitrary, i.e. when fitting one can converge on either set. This fact is important when the system is studied as a function of the electrode potential (or other experimental parameter) to not exchange of these parameters (e.g. set of parameters 2 and 3). However, such an ambiguity does not appear in the ladder circuit (b). 

The disadvantages of IS are primarily associated with possible ambiguities in interpretation. An important comphcation of analyses based on an equivalent circuit (e.g. Bauerle [1969]) is that ordinary ideal circuit elements represent ideal lumped-constant properties. Inevitably, aU electrolytic cells are distributed in space, and their microscopic properties may be also independently distributed. Under these conditions, ideal circuit elements may be inadequate to describe the electrical response. Thus, it is often found that Z/to) cannot be well approximated by the impedance of an equivalent circuit involving only a finite number of ordinary lumped-constant elements. It has been observed by many in the field that the use of distributed impedance elements [e.g. constant-phase elements (CPEs) (see Section 2.2.2.2)] in the equivalent circuit greatly aids the process of fitting observed impedance data for a cell with distributed properties. 

Another weakness of the CN LS analysis method is the ambiguity of the equivalent circuits, meaning that identical impedance spectra may be obtained from different circuits [1, 8]. Therefore, it is questionable to propose an a priori model without any knowledge about the real number and physical origin of the polarization processes contributing to the impedance response of the cell. 

Widely known is the approximation of the measured impedance spectra by a CN LS fit procedure, which results in a model function represented by an equivalent circuit [22]. However, this approach necessitates an a priori definition of the ECM. Consequently, the ECM is appHed without knowing the number and nature of ohmic and polarization processes contributing to the total cell impedance. This leads to a severe ambiguity of the adopted model [1]. 

Therefore, if an experiment can be interpreted in terms of circuit A, it can also be explained in terms of circuit B, and each set of circuit elements can be calculated from the other one. This ambiguity makes a physical interpretation doubtful. Whereas the transfer function [Eq. (7-41)] gives an unequivocal description of the system, the set of coefficients CQ-bi is determined and related to the parameters of a physical model. Therefore, it should be the aim of impedance analysis to derive a theoretical transfer function on the basis of a physical model of the system. 

The Nyquist plot is a very convenient representation of the R - R circuit process, as it shows an ideal semicircle as an indication of the activation-energy-controlled charge-transfer process. A depressed semicircle in the Nyquist plot is an indication of multiple processes with similar relaxation time constants, or distributed non-ideal kinetics. These ambiguities can be resolved in the "original" and modified Bode plots, as shown in Figures 2-3 and 2-4. The departure of the slope of log vs. log frequency dependency from the unity indicates a distributed process, with a characteristic frequency that may not even correspond to the highest peak value of... 

---