Causality, Stability, Finiteness

The impedance technique is often applied to electrochanical systems that have not been studied before. The complex plane and Bode plots obtained often displayed shapes that had never been encountered previously. Before starting the analysis and modeling of the experimental results, one should be certain that the impedances are valid. There is a general mathematical procedure that allows verification of the impedance data. It was introduced by Kramers and Kronig, further developed by Bode, and later applied to During the impedance measurements, a small ac 

Linearity. A system is linear when its response to a sum of individual input signals is equal to the sum of the individual responses. This also implies that the system is described by a system of linear differential equations [see e.g., Eqs. (2) and (7)]. Electrochemical systems are usually highly nonlinear and the impedance is obtained by the linearization of equations [see e.g., Eqs. (42) and (130)] for small amplitudes. For linear systems, the response is independent of the amplitude. It is easy to verify the linearity of the system if the impedance obtained is the same when the amplitude of the applied ac signal is halved, then the system is 

Causality The response of the system must be entirely determined by the applied perturbation that is, the output depends only on the present and past input values. A causal system cannot predict what its future input will be. Causal systems are also called physically realizable systems. If the system is at rest and a perturbation is applied at f = 0, the response must be 0 for f 0. In the complex plane, the above criterion requires that for f 0, oo = 0. Moreover, the integral on and inside a closed path C of an analytic function 

If the function Z(s) has singularities, then the sum of the residuals of the poles a, must equal 0  

Equations (227) and (228) are mathematical forms of causality. The system is causal if it does not have any singularities Eq. 

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The concept of impedance is only applicable if the system under study complies with the conditions of stability, causality, finity, and linearity. It is good practice to verify whether these conditions hold before giving further interpretation to the data. A stable Systran has a Zfym) independent of time. Causal systems do not show any response if the input signal is zero (in the dark for IMPS). All practical systems are finite. Linear systems show a response linearly related to the input the current density is hnearly related to the irradiation intensity. A convenient technique... 

In this section, we discuss in detail how the selection of various experimental parameters affects each of these conditions. One of the first studies on EIS measurements in MXC applications by Strik et al. [29] covers some of these conditions very well, but we provide an expanded explanation here. While all these conditions are especially difficult to fulfill in a typical electrochemical cell, the conditions used in MXCs further exacerbate the problem. For example, it is known that polarization curves for microbial anodes exhibit nonlinear, Nernstian responses [30]. Thus, there are regions in the polarization curve where the system may not behave linearly even when small amplitudes are applied. The irreversibility of the enzymatic responses also leads to regions where finiteness is not met (Fig. 8.6). These cases would run also into difficulties in terms of the signal-to-noise ratio when small amplitudes are applied at potentials on the saturation region of the polarization curve. Similarly, as MXCs are biological reactors and can have changes in microbial responses due to small perturbations outside the conttol of researchers, conditions of both stability and causality are difficult to fulfill. 

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