Complex plane analysis

Havriliak S, Negami S. A complex plane analysis of a-dispersions in some polymer systems. J Polym Sci C 1966 14 99-117. 

Polystyrene and polybutadiene homopolymers as well as random and block copolymers of these mers have been studied via dielectric relaxation spectroscopy and tensile stress-strain measurements. The results suggest that some block copolymer systems studied have styrene rich surfaces which appear to partially crosslink upon initial exposure to ozone even though surface oxygen concentrations are not significantly affected. After continued exposure these samples appear to then undergo chain scission. Complex plane analysis implies that after degradation... 

Havrihak, S. and Negami, S. 1966. A complex plane analysis of alpha-dispersion of some polymeric systems. J. Polym. Sci. C 14 99-117. 

The above paragraph gives, in brief, the analysis first published by the Sluyters [1964] for complex-plane analysis of the impedance spectrum of an electrode process at an interface exhibiting a double-layer capacitance hence its great importance as a basis for examining the impedance spectroscopy of supercapacitor systems, particularly those based (Conway [1999]) on double-layer capacitance (Grahame [1947]). 

A. Hooper [1977] A Study of the Electrical Properties of Single Crystal and Polycrystalline ]3-Alumina Using Complex Plane Analysis, J. Phys. D. Appl. Phys. 10, 1487-1497. 

N. Yoshuke, M. Ayusawa, and S. Kondo [1984] Electrochemical Properties of WO3 XH2O. in. Complex Plane Analysis of the Film on Sn02. J. Electrochem. 

It was shown in the section on complex plane analysis that the double layer capacitance Qi is obtainable from the maximum value of Z" in the semicircular region, where l/coC = / ct- There are situations, however, where Q is the prime quantity of interest, and in these cases the faradaic current is either zero or very small (/ ct Small-signal sinusoidal excitation is then a good way of measuring Ccn, and the analysis is a good deal simpler than the full complex plane analysis which is required when/ ct is small. 

Gobrecht, H., Hamann, K., and WUlers, G. (1971) Complex plane analysis of heat capacity of polymers in the glass transition region. /. Phys. E Sci. Instrum., 4, 21-23. 

FIGURE 230 Complex plane analysis of cell impedance for a charge-transfer process with kinetic control at a planar electrode. 

HOOPER, A., 1977. A study of the electrical properties of single-crystal and polycrystalline P-alumina using complex plane analysis. Journal of Physics D Applied Physics, 10(11), 1487-1496. 

In an analysis of an electrode process, it is useful to obtain the impedance spectrum —the dependence of the impedance on the frequency in the complex plane, or the dependence of Z" on Z, and to analyse it by using suitable equivalent circuits for the given electrode system and electrode process. Figure 5.21 depicts four basic types of impedance spectra and the corresponding equivalent circuits for the capacity of the electrical double layer alone (A), for the capacity of the electrical double layer when the electrolytic cell has an ohmic resistance RB (B), for an electrode with a double-layer capacity CD and simultaneous electrode reaction with polarization resistance Rp(C) and for the same case as C where the ohmic resistance of the cell RB is also included (D). It is obvious from the diagram that the impedance for case A is... 

The idea of a root locus plot is simple—if we have a computer. We pick one design parameter, say, the proportional gain Kc, and write a small program to calculate the roots of the characteristic polynomial for each chosen value of as in 0, 1, 2, 3,., 100,..., etc. The results (the values of the roots) can be tabulated or better yet, plotted on the complex plane. Even though the idea of plotting a root locus sounds so simple, it is one of the most powerful techniques in controller design and analysis when there is no time delay. 

On the other hand, the relative simplicity of expressions for the operational impedance (sometimes even as compared with the complex impedance) has led to the method of Laplace Plane Analysis of results of relaxation techniques [76—78, 81, 82]. The primary problem of the method is how to obtain Laplace transformed data of both the perturbation P(f) and the response R(t). To that end, several possiblities may be distinguished. 

There is one further important practical aspect that has to be considered when taking this approach to performing an experimental bifurcation analysis impedance measurements can only be carried out with stable stationary states it is not feasible to measure unstable stationary states, or states close to a bifurcation. However, as we discussed above, N-NDR and HN-NDR systems become unstable due to ohmic losses in the circuit, whereas they are always stable for vanishing R< >. Being aware that an ohmic series resistor causes only a horizontal shift of the impedance spectrum in the complex plane, it is apparent that it is possible to infer about the existence of bifurcations from impedance measurements at sufficiently low solution resistance (or when invoking an ZR-compensation, an option many potentiostats provide). This is illustrated with the schematic impedance spectrum shown in Fig. 12, which depicts a typical impedance spectrum of an N-NDR system. The spectrum possesses two... 

The principles of a.c. circuits necessary for the comprehension of some of the ideas and concepts presented here are given in Appendix 2. The impedance is the proportionality factor between potential and current if these have different phases then we can divide the impedance into a resistive part, R, where the voltage and current are in phase, and a reactive part, Xc = l (oCy where the phase difference between current and voltage is 90°. As shown in Appendix 2, it is often easier for posterior calculation and analysis to display the impedance vectorially in complex-plane diagrams. 

Fig. 8.8. Analysis of normalised IMPS response in the complex plane used to derive values...

EIS can also detect defects arising from lack of adhesion at adhesively bonded surfaces (111). The presence of such defects produces pronounced changes in the character of the data presented either in complex plane plots or in Bode plots. Figure 38 illustrates the measurement configuration and provides examples of EIS data for defective and defect-free samples. Studies have shown that the presence of defects is readily revealed and that the geometry of the defects and their spatial extent can be inferred from a detailed analysis of EIS spectra. 

The immittance analysis can be performed using different kinds of plots, including complex plane plots of X vs. R for impedance and B vs. G for admittance. These plots can also be denoted as Z" vs. Z and Y" vs. Y, or Im(Z) vs. Rc(Z), and Im( Y) vs. Re( Y). Another type of general analysis of immittance is based on network analysis utilizing logarithmic Bode plots of impedance or admittance modulus vs. frequency (e.g., log Y vs. logo)) and phase shift vs. frequency ( vs. log co). Other dependencies taking into account specific equivalent circuit behavior, for instance, due to diffusion of reactants in solution, film formation, or electrode porosity are considered in - electrochemical impedance spectroscopy. Refs. [i] Macdonald JR (1987) Impedance spectroscopy. Wiley, New York [ii] Jurczakowski R, Hitz C, Lasia A (2004) J Electroanal Chem 572 355... 

The generalized stress-strain relationships in linear viscoelasticity can be obtained directly from the generalized Hooke s law, described by Eqs. (4.85) and (4.118), by using the so-called correspondence principle. This principle establishes that if an elastic solution to a stress analysis is known, the corresponding viscoelastic (complex plane) solution can be obtained by substituting for the elastic quantities the -multiplied Laplace transforms (8 p. 509). The appUcation of this principle to Eq. (4.85) gives... 

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