Analysis of Single Impedance Arcs

The arc does not pass through the origin, either because there are other arcs appearing at higher frequencies and/or because 0. 

The center of an experimental arc is frequently displaced below the real axis because of the presence of distributed elements in the material-electrode system. Similar displacements may also be observed in any of the other complex planes plots (Y, M, or e). The relaxation time T is then not singlevalued but is distributed continuously or discretely around a mean, = 0)m. The angle 8 by which such a semicircular arc is depressed below the real axis is related to the width of the relaxation time distribution and as such is an important parameter. 

Arcs can be substantially distorted by other relaxations whose mean time constants are within two orders of magnitude or less of that for the arc under consideration. Many of the spectra shown in following chapters involve overlapping arcs. 

Here s = cotr is a normalized frequency variable, and h is the normalized, dimensionless form of Zzarc- Notice that it is exactly the same as the similarly normalized Cole-Cole dielectric response function of Eq. (1) when we set y/zc= 1 - a. We can also alternatively write the ZARC impedance as the combination of the resistance Rr in parallel with the CPE impedance Zcpe (see Section 2.2.22). The CPE admittance is (Macdonald [1984]) 

Let us start by considering two easy-to-use approximate methods of estimating the parameters, methods often adequate for initial approximate characterization of the response. The estimates obtained by these approaches may also be used as initial values for the more complicated and much more accurate CNLS method described and illustrated in Section 3.3.2. Note that the single RrCr situation, that where 0 = 0 and y/zc = 1, is included in the analysis described below. 

---