Argand diagrams

Introducing the complex notation enables the impedance relationships to be presented as Argand diagrams in both Cartesian and polar co-ordinates (r,rp). The fomier leads to the Nyquist impedance spectrum, where the real impedance is plotted against the imaginary and the latter to the Bode spectrum, where both the modulus of impedance, r, and the phase angle are plotted as a fiinction of the frequency. In AC impedance tire cell is essentially replaced by a suitable model system in which the properties of the interface and the electrolyte are represented by appropriate electrical analogues and the impedance of the cell is then measured over a wide... 

Arelix Arenavirus h6-Arene A-Rest Arfonad Argand diagram Argatroban [74863-84-6] Argemomne [6901-16-2] Argentite... 

Fig. 7. (a) Simple battery circuit diagram where represents the capacitance of the electrical double layer at the electrode—solution interface, W depicts the Warburg impedance for diffusion processes, and R is internal resistance and (b) the corresponding Argand diagram of the behavior of impedance with frequency, for an idealized battery system, where the characteristic behavior of A, ohmic B, activation and C, diffusion or concentration (Warburg... 

Hence equation (6.14) can be plotted in the complex space (Argand Diagram) to produce a harmonic response diagram as shown in Figure 6.3. 

Since the real and imaginary parts of a complex number are independent of each other, a complex number is always specified in terms of two real numbers, like the coordinates of a point in a plane, or the two components of a two-dimensional vector. In an Argand diagram a complex number is represented as a point in the complex plane by a real and an imaginary axis. 

A mathematician would say that a plot of Z" (as y ) against Z (as jc ) forms an Argand diagram (or Argand plane ). As electroanalysts, we will call such a set of axes a Nyquist plot or simply an impedance plot (see Figure 8.9). 

Figure 6.2 Vector (Argand) diagram showing the reiationships between heavy-atom derivative (Fpn), native protein (Fp) and heavy atom (Fpi) ap is the phase angie for the native protein. The vectors are piotted in the compiex piane.

Figure 12.1. Argand diagram for the representation of complex numbers in the complex plane C.

Sluyters or Cole-Cole plot), in a form similar to the representation of complex numbers (Argand diagram). 

Fig. 3.1 AC losses in a dielectric (a) circuit diagram, (b) Argand diagram of complex current-voltage relationship.

Figure 4.100 shows the Argand diagram of water (curve 1) and the permittivity for 0.8 M KCl (curve 2) in water. The stractural part of the spectrum is represented by curve 3. The difference of curves 2 and 3 is the result of electrolytic conductance. 

An Argand diagram (also called a Cole-Cole plop is a diagram of the real e and imaginary e" components of the dielectric constant of the system. 

Fig. 4.100. Argand diagrams of a completely dissociated electrolyte and its pure solvent. Full circles experimental data from frequency domain measurements on aqueous potassium chloride solutions at 25 °C. Curve 1 Argand diagram of pure water. Curve 2 Argand diagram, ff = f(E ), of an 0.8 Waqueous KCI solution, Curve 3 Argand diagram, e"=f(e )r obtained from curve 2. (Reprinted from P. Turq, J. Barthel, and M. Chemla, in Transport, Relaxation and Kinetic Processes in Electrolyte Solutions, Springer-Verlag, Berlin, 1992, p. 78).

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. 

The dispersion and absorption curves of the pure solvents undergo drastic changes when an electrolyte is added, the most important being the superposition of conductivity shown in the absorption curve q"(i ) of fig. 5 a and in the Argand diagram tj" = f(e ) of fig. 5 b. Reduction of i7"( ) to t" v) is executed with the help of measured static conductivities cr. Two relaxation processes tire corroborated by two inflexion points of e v), two maxima... 

The Argand diagram of the 2.2 M solution of Et NCl shows three relaxation processes typical for aqueous electrolyte solutions (1) ion-pair relaxation (r l = tip), (2) low frequency relaxation (rj, as 8 ps) of water, (3) high frequency relaxation a 1 ps) of water, in contrast to that of the 2 M solution of Bu NBr where the relaxation process (2) splits up into two processes. Figure 7 shows the concentration dependence of the... 

Figure 6 Argand diagrams of aqueous solutions (25 °C) of (a) tetraethylammonium chloride (2.23 M Et NCl) and (b) tetrabutylamunonium bromide (2.00 M Bu NBr). For explanations see the text.

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 1.1 Argand diagram showing the position of a complex number and its complex conjugate on a complex plane.

Figure 1.2 Argand diagram showing relationships among complex impedance, magnitude, and phase angle.

Figure 1.3 Argand diagram showing 1 + (A) and the five roots of (1 ( ) as calculated...

Figure 1.4 Argand diagram showing the angle associated with l/jcor and the two roots 1/yJ](jOT as calculated in Example 1.7.

---