Distributed circuit elements constant phase

In many cases, the use of ideal equivalent circuits is convenient but not always appropriate. Nonideal behavior might arise from interactions of species, resulting in frequency-dependent capacitances [C((D)]. Under these conditions, the physical process is more accurately described by a range of relaxation time constants instead of a unique value. Such distributed relaxation events are usually manifested as semicircles depressed below the real axis in the complex plane, and the angle of depression is related to the degree of nonideality. Various distribution functions and constant phase elements have been employed to describe such events. These nonidealities are especially evident in biological systems. 

Greszczuk et al. [252] employed the a.c. impedance measurements to study the ionic transport during PAn oxidation. Equivalent circuits of the conducting polymer-electrolyte interfaces are made of resistance R, capacitance C, and various distributed circuit elements. The latter consist of a constant phase element Q, a finite transmission line T, and a Warburg element W. The general expression for the admittance response of the CPE, Tcpr, is [253]... 

In all real systems, some deviation from ideal behavior can he observed. If a potential is applied to a macroscopic system, the total current is the sum of a large number of microscopic current filaments, which originate and end at the electrodes. If the electrode surfaces are rough or one or more of the dielectric materials in the system are inhomogeneous, then all these microscopic current filaments would be different. In a response to a small-amplitude excitation signal, this would lead to frequency-dependent effects that can often be modeled with simple distributed circuit elements. One of these elements, which have found widespread use in the modeling of impedance spectra, is the so-called constant phase element (CPE). A CPE is defined as... 

The two circuits in Fig. 2.37 in series and nested are described by / o(Ci/ i) RiCi) and R(, C R C2Rt))). Other distributed circuit elements can also be used Q represents a constant phase element, CPE, W a semi-infinite Warburg element, Ws a finite length transmissive element, Wo a finite length reflecting element, and so forth. In the case of distributed elements, it is preferable to define them specifically. 

There were significant difficulties associated with fitting models to impedance data. The electrochemical systems frequently did not conform to the assumptions made in the models, especially those associated with electrode uniformity. Constant-phase elements (CPEs), described in Chapter 13, were introduced as a convenient general circuit element that was said to account for distributions of time constants. The meaning of the CPE for specific systems was often disputed. 

The impedance response of electrodes rarely show the ideal response expected for single electrochemical reactions. The impedance response typically reflects a distribution of reactivity that is commonly represented in equivalent electrical circuits as a constant phase element (CPE). ° For a blocking electrode, the impedance can be expressed in terms of a CPE as... 

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. 

Determination of Parameters from Randles Circuit. Electrochemical three-electrode impedance spectra taken on electrochromic materials can very often be fitted to the Randles equivalent circuit (Randles [1947]) displayed in Figure 4.3.17. In this circuit R /denotes the high frequency resistance of the electrolyte, Ra is the charge-transfer resistance associated with the ion injection from the electrolyte into the electrochromic film and Zt, is a Warburg diffusion impedance of either semi-infinite, or finite-length type (Ho et al. [1980]). The CPEdi is a constant phase element describing the distributed capacitance of the electrochemical double layer between the electrolyte and the film having an impedance that can be expressed as... 

ABSTRACT State determination of Li-ion cells is often accomplished with Electrochemical Impedance Spectroscopy (EIS). The measurement results are in frequency domain and used to describe the state of a Li-ion cell by parameterizing impedance-based models. Since EIS is a costly measurement method, an alternative method for the parameterization of impedance-based models with time-domain data easier to record is presented in this work. For this purpose the model equations from the impedance-based models are transformed from frequency domain into time domain. As an excitation signal a current step is applied. The resulting voltage step responses are the model equations in time domain. They are presented for lumped and derived for distributed electrical circuit elements, i.e. Warburg impedance, Constant Phase Element and RCPE. A resulting technique is the determination of the inner resistance from an impedance spectrum which is performed on measurement data. 

Due to the high measurement and computational complexity as well as cost factors, frequency domain EIS measurements are not likely to be implemented on board in vehicles in the near future. An alternative approach is shown to parameterize impedance-based models with time domain data available on board, i.e. currents, battery or cell voltages and temperatures. Therefore, in this work, a method is proposed for the transformation of electrical circuit model equations from frequency domain into time domain model equations. Particularly for electrical circuit models containing distributed elements, e.g. Warburg impedances (WB), Constant Phase Elements (CPE), RCPE or ZARC elements, these transformations require fractional calculus methods, as will be presented in detail. 

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