Impedance constant phase angle

Logarithmic relationship between measured impedance at phase-angle shift of zero and the distance between the electrodes for constant length of the electrodes (60 mm) and diameter of the electrodes (0.2mm) and different electrolyte concentrations (1) 1 x10" (2) 1x1(T2, (3) 1 x 10 3 and (4) 1x10"4molL. ... 

In the 1920s, impedance was applied to biological systems, including the resistance and capacitance of cells of vegetables and the dielectric response of blood suspensions. ° Impedance was also applied to muscle fibers, skin tissues, and other biological membranes. " The capacitance of the cell membranes was found to be a function of frequency, and Fricke observed a relationship between the frequency exponent of the impedance and the observed constant phase angle. In 1941, brothers Cole and Cole showed that the frequency-dependent complex... 

Most of the literature on the dynamic properties of solid interfaces is based on the analogy between membrme and interfacial kinetics tnd the electric response of rough electrodes, for which an anomalous impedance scaling with frequency w, referred to as Constant Phase Angle behavior, Z[oj) = / 4- 0 < w <, i / has been... 

The speeifie frequeney dependence in Eq. (36) is known as the constant phase angle (CPA) dependenee (78-80). This impedance behavior oeeurs for a wide elass of elee-trodes (75-77) and suggests the introduction of a new equivalent circuit element with impedance eharaeteristies similar to those of Eq. (36). We call this element a reeap element (derived from resistance and capacitance). The electric and fractal properties for this recap complex impedance Cy( ) are given by ... 

The important feature of these lines is that they produce a constant phase angle, like a Warburg impedance, but with the phase angle not restricted to ti/4. This is exactly the behavior often found at the electrode-electrolyte interface and has been termed a constant-phase element (CPE). It appears to be true that roughness is an important contributing factor to the observed frequency dispersion. Scheider s model, however, remains qualitative, and the microscopic link between the topology and the circuit is absent. 

The capacitance observed is not ideal capacitance as observed by the inclined line at low frequency in impedance and usually represented by constant phase angle element (CPE) shown in Fig. 20 [5, 7, 57, 63, 76, 93]. 

This is the phenomenon of the so-called constant phase element (CPE) as it follows from eqn. (115) that Yc is composed of an imaginary and a real component, with a frequency-independent phase angle cor/2. Though the phenomenon is most clearly discovered in impedance or admittance analyses, its effect in time-domain methods should not be ignored. 

In recent years it has been demonstrated by many researchers16,172 173 that the frequency dispersion or capacitance dispersion is intimately related to PSD or pore length distribution (PLD). In this case, the frequency dispersion is not called CPE behavior since the phase angle of the impedance spectra did not show a constant value over the whole frequency range. The phase angle of the impedance spectra measured on the porous electrode with broad PSD or PLD is larger than 45° in value at high frequencies and smaller than 90° in value at low frequencies. 

It must be emphasized that the mathematical simplicity of equations (13.1) and (13.2) is the consequence of a specific time-constant distribution. As shown in this chapter, time-constant distributions can result from nonuniform mass transfer, geometry-induced nonuniform current and potential distributions, electrode porosity, and distributed properties of oxides. At first glance, the associated impedance responses may appear to have a CPE behavior, but the frequency dependence of the phase angle shows that the time-constant distribution differs from that presented in equation (13.7). 

In addition to comparing the sum of squares, the experimental and simulated data should be compared by using complex plane and Bode plots. The phase-angle Bode plot is particularly sensitive in detecting time constants. Boukamp proposed to study the residual sum of squares after subtracting the assumed model values from the total impedance data. If the model is valid, the residuals should behave randomly. If they display regular tendencies, it may mean that the model is not correct and further elements should be added. However, the variations of the residuals should be statistically important. 

ASTM D150 Standard Test Methods for AC Loss Characteristics and Permittivity (Dielectric Constant) of Solid Electrical Insulation includes the determination of relative permittivity, dissipation factor, loss index, power factor, phase angle, and loss angle through specimens of solid electrical insulating materials when the standards used are lumped impedances. The frequency range that can be covered extends from less than 1 Hz to several hundred megahertz. 

He introduced a constant phase element (CPE), defined in the paper by the phase angle ( )3 = arccotan (m), and m = accordingly using m completely differently from Fricke ideal resistor has m = oo and < )3 = 0°, and found the impedance locus for such a system was a circular arc with the center below the real axis in the Wessel diagram. A plot of complex immittance or immittivity in the Wessel diagram with the purpose of searching for circular arcs, may according to this book, be called a Cole-plot. 

It is by now well known that the variation with angle of incidence of the scan impedance of phased arrays as well as the bandwidth of hybrid radomes can be reduced by using dielectric slabs placed between free space and the device in question. To be sure, the dielectric constant should in general be less than 2 (for a single slab) and the thickness should be somewhat thicker thau A./4 in the dielectric. An example of applying this technique is shown in Fig. C.15. Compared to the uncompensated case in Fig. C.13, we observe some improvement... 

The resistance of membranes can be measured by AC impedance methods [85,86], using the four-point-probe technique. The test membrane is placed in a cell consisting of two Pt-foil electrodes, spaced 3 cm apart, to feed the current to a sample of 3 x 1 cm and two platinum needles placed 1 cm apart, to measure the potential drop (see Fig. 4.3.26). The cell is placed in a vessel maintained at constant temperature by circulating water. The impedance measurements are then carried out at 1-10 kHz using a frequency-response analyzer (e.g., Solatron Model 1255HF frequency analyzer). After ensuring that there are no parasitic processes (from the phase angle measurements, which should be zero), one can measure the resistance directly. The membrane resistance can also be obtained directly from the real part of the impedance (see typical data in Fig. 4.3.27). 

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