Analysis Argand diagram

AC susceptibility is a function of frequency, which means that, at a certain temperature, a series of in-phase and out-of-phase susceptibilities can be obtained by scanning the frequency. A plot of a series of versus x is an Argand diagram [102], which is a semicircle if only one relaxation process occurs. This type of plotting is referred to as Colo-Colo analysis. 

It has to be mentioned that such equivalent circuits as circuits (Cl) or (C2) above, which can represent the kinetic behavior of electrode reactions in terms of the electrical response to a modulation or discontinuity of potential or current, do not necessarily uniquely represent this behavior that is other equivalent circuits with different arrangements and different values of the components can also represent the frequency-response behavior, especially for the cases of more complex multistep reactions, for example, as represented above in circuit (C2). In such cases, it is preferable to make a mathematical or numerical analysis of the frequency response, based on a supposed mechanism of the reaction and its kinetic equations. This was the basis of the important paper of Armstrong and Henderson (108) and later developments by Bai and Conway (113), and by McDonald (114) and MacDonald (115). In these cases, the real (Z ) and imaginary (Z") components of the overall impedance vector (Z) can be evaluated as a function of frequency and are often plotted against one another in a so-called complex-plane or Argand diagram (110). The procedures follow closely those developed earlier for the representation of dielectric relaxation and dielectric loss in dielectric materials and solutions [e.g., the Cole and Cole plots (116) ]. 

Figure 11 shows a representation of e"(o)) = f(e ( o)), called an Argand diagram, for 0.48 M NaClO in a PC-DME mixture (20 weight % PC). Data analysis of the precedingly determined frequency-dependent permittivities of the solvent mixture without NaClO yielded two relaxation regions, one attributable to DME (relaxation time T = 4.7 ps) the other to PC (relaxation time t = 22 ps). The shifts of solvent relaxation times with reference to those of the two pure solvents, t(DME) = 3.6 ps and t(PC) = 39 ps, is correlated to the change in viscosity. Addition of the... 

Fig. 11. Argand diagram e" vs. e of a 0.48 M NaCI04 solution in propylene carbonate-dimethoxy-ethane (20 weight % of PC) at 25 °C (upper curve with measured points) Data analysis yields three relaxation times and appropriate Debye circles...

Frequency response analysis is another classical tool that has been used in the analysis and design of process control systems. The Laplace variable s is replaced by jfrequency response is then plotted using an Argand diagram approach. 

In addition to an examination of the frequency response of series and parallel components of the circuit impedance/admittance, another approach may be particularly valuable. This analysis method involves plotting the real versus imaginary parts of some such complex quantity as admittance or impedance as parametric functions of frequency. Such Argand or "circle diagrams" have been used for many years in electrical engineering when complex dielectric constant is the quantity considered, they are known as Cole-Cole plots. ... 

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