Distributed circuit elements

But an electrolytic cell or dielectric test sample is always finite in extent, and its electrical response often exhibits two generic types of distributed response, requiring the appearance of distributed elements in the equivalent circuit used to fit IS data. The first type, that discussed above, appears just because of the finite extent of the system, even when all system properties are homogeneous and space-invariant. Diffusion can lead to a distributed circuit element (the analog of a finite-length transmission line) of this type. When a circuit element is distributed, it is found that its impedance cannot be exactly expressed as the combination of a finite number of ideal circuit elements, except possibly in certain limiting cases. 

When plotted in the complex plane, Zw leads to an initial straight-line region with 8 = 45° it reaches a peak value of -Z = 0.417 Rdo at = 2.53 and then begins to decrease toward the real axis, finally approaching it vertically, as required by the limiting Rdo and Cdo in parallel 

When s 3, the tanh tenn approaches unity and Zw approaches Z, , given by 

Although we shall not discuss the general Z fu)) further here, there is one additional specific case which follows from it and deserves mention. Suppose that the finite-length transmission line analog is open-circuited (see Franceschetti and Macdonald [1979c]). Then no direct current can flow in the actual system, as it could with Zw (but not Zw, and the concentration of the diffusing particle increases at the 

the process leads to the limiting capacitance Cooc as m 0 further, the frequency response at this level is exactly the same as that for at the impedance level [see Eq. (10)] and thus involves an initial straight line at 0=45°. At the impedance level, Zdoc = (jo)Cc ) is given by 

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R. L. Hurt and J. R. Macdonald, "Distributed Circuit Elements in Impedance Spectroscopy A Unified Treatment of Conductive and Dielectric Systems," Solid State Ionics, 20 (1986) 111-124. 

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]... 

Impedance data are frequently fitted with an equivalent circuit made up of circuit elements, which are related to the physical processes in the system under investigation. In many cases, ideal circuit elements such as resistors and capacitors can be applied. Mostly, however, distributed circuit elements are required in addition to the ideal circuit elements to describe the impedance response of real systems adequately. Various distributed circuit elements and their applications are discussed in [3,15]. 

Historically, the Warburg impedance, which models semi-infinite diffusion of electroactive species, was the first distributed circuit element introduced to describe the behavior of an electrochemical cell. As described above (see Sect. 2.6.3.1), the Warburg impedance (Eq. 38) is also analogous to a uniform, semi-infinite transmission line. In order to take account of the finite character of a real electrochemical cell, which causes deviations from the Warburg impedance at low frequencies. 

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... 

Eor further information regarding distributed circuit elements, refer to [3, 15]. 

As indicated in Sect. 2.6.4, modeling of real electrochemical systems usually requires the aid of distributed circuit elements. In this section, the relationship between the morphological properties of rough or porous electrodes and their impedance behavior will be discussed. 

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. 

Figure 4. 1. The LEVM fitting circuit O. It may be used as shown for fitting immittance data in raw or specific form. The DE blocks may each be selected as any one of the many available distributed-circuit-element response models.

In most corrosion systems the capacitive semicircle exhibits significant deviation from an ideal semicircle. This has often been referred to as frequency dispersion attributed to surface inhomogeneities and distributed circuit elements. Detailed analysis of the experimental data shows that this deviation can be described by a rotation of the semicircle below the real axis by an angle Y, as shown in Fig. 7-7. A good approxima-... 

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