Stability of Electrochemical Systems

In this section we will look in more detail at system stability [587]. In control system theory, a stable system is one that produces a bounded response to a bounded input. In general, system stability depends on the proprieties of the transfer function, in this case of the impedance or the admittance [588]. Impedance and 

Therefore, there are real and complex conjugated zeros and roots. Examples of such polynomials representing the impedance of a system were presented, for example, in Eq. (2.91) for an R-C connection, Eq. (2.94) for an R-L coimection, and Eq. (2.97) for an R-L-C connection in series. This expression can be simplified into partial fractions  

When the poles Aq are negative, the current relaxes to zero and the system is stable. In the case of complex poles, Eq. (13.19), p must be negative or zero, that is, the real part of the root cannot be negative to assure stability. Solutions for different values of Aq and p are presented in Fig. 13.14. Stable systems are obtained only when these values are negative or zero. When Aq = 0, a time-independent constant 

Let us consider a simple circuit Ro(RiCi), shown in Eig. 2.26. Its impedance is described by Eq. (2.128), which can be rearranged into the following form, Eq. (13.22)  

If all circuit parameters are positive, then the pole and zero are negative and the system is stable. For the impedances in Fig. 13.15 the poles and zeros are displayed on the s-plane in Fig. 13.16. When all the circuit parameters are positive, the poles and zeros are always negative. The complex plane plot in this case, which displays well known behavior, is shown in Fig. 13.15a. 

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Another contemporary and noteworthy review article by Koper follows yet another concept. Koper first stresses the importance of the electric circuit by evaluating, in a rigorous way, the stability of electrochemical systems by frequency response methods. He then thoroughly discusses the dynamics of selected examples, including some semiconductor systems, which are not included in this chapter, with special emphasis on how they relate to the frequency response theory. 

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