Complex-Impedance-Plane Representation

An impedance response can be interpreted graphically as a vector on the complex plane. The imaginary axis is the out-of-phase response (Z"), and the real axis is the in-phase response (Z ). The magnitude of the impedance response Z is the length of the vector, and the phase angle (]) describes its direction (Fig. 3). Each point on the plane defines an impedance response at a particular frequency. Such representations are commonly referred to as complex plane plots, Nyquist diagrams, or Cole-Cole plots. However, the Cole-Cole plot is actually the complex plane representation of the dielectric response of a material. 

Figure 3 Complex plane representation of impedance vector Z.

Fig. 12L Complex-plane representation of the impedance vector as a function of frequency for a simple circuit, consisting of a capacitor and resistor in parallel.

Fig. JOG Complex-plane representation of the impedance of an interphase. ReZ and ImZ are the real and imaginary components of the impedance, respectively.

Figure 1.4 Complex plane representation of the impedance of a typical polymer electrolyte.

The complex plane representation of the impedance of the above equivalent circuit is also shown in Figure 6. It is seen to give a semicircle which allows the determination of i , the ohmic resistance between the working... 

Figure 2.1.13. Complex plane representations of the impedance due to a finite-length diffusion process with (a) reflective, (b) transmissive boundary conditions at x = 1.

The complex plane representations of these two impedance behaviors are shown in Figure 2.1.13. 

In the complex-plane representation of the impedance behavior of a parallel RC circuit, it is convenient to identify the maximum (so-called top point ) in the semicircular plot which is at a critical frequency O) = l/RC, the reciprocal of the time constant for the response of the circnit. A similar sitnation arises for a series RC... 

This impedance is plotted in the complex plane representation in Fig. 6. Qualitatively, the impedance appears to be a pure capacitance at low frequencies, where the phase angle tends toward ir/2. At higher frequencies. 

Impedance plots - Fig. 19.4 shows experimental impedance plots (complex plane representation) obtained for both the as-deposited and the honeycomb diamond electrodes at 0.4 V. The plots for the pore types, 60 x 500 nm (Fig. 19.4c), 70 x 750 nm (not shown), and 400 nm x 3 mm (Fig. 19.4d), exhibit two distinct domains a high frequency domain, where the impedance behavior is that expected for a cylindrical pore electrode, with a characteristic linear portion at a 45° angle, and a low frequency domain, where the behavior is... 

A2.4 Representation in the complex plane A2.5 Resistance and capacitance in series A2.6 Resistance and capacitance in parallel A2.7 Impedances in series and in parallel A2.8 Admittance... 

Fig. A2.1. Representation in the complex plane of an impedance containing resistive and capacitive components.

According to the above calculations, a graphical representation of the AC impedance of a series RC circuit is presented in Figure 2.19. As shown in the complex plane of Figure 2.19, the AC impedance of a series RC circuit is a straight vertical line in the fourth quadrant with a constant Z value of R. 

The most common graphical representation of experimental impedance is a Nyquist plot (complex-plane diagram), which is more illustrative than a Bode plot. However, a Bode plot sometimes can provide additional information. 

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) ]. 

The complex-capacitance-plane plot is presented in Figure 16.11. The data are presented as a locus of points, where each data point corresponds to a different measurement frequency. As discussed for the impedance- and admittance-plane representations (Figures 16.1 and 16.6, respectively), the complex-capacitance-plane... 

As described in Sections 20.2.1 and 20.2.2, the quality of the regressions can be assessed to varying degrees of success by inspection of plots. The Nyquist or complex-impedance-plane representation given in Figure 20.13 reveals the difference between the finite-diffusion-length model and the models based on numerical solution of the convective-diffusion equation, but cannot be used to distinguish the models based on one-term, two-term, and three-term expansions. 

The above analysis shows that in the simple case of one adsorbed intermediate (according to Langmuirian adsorption), various complex plane plots may be obtained, depending on the relative values of the system parameters. These plots are described by various equivalent circuits, which are only the electrical representations of the interfacial phenomena. In fact, there are no real capacitances, inductances, or resistances in the circuit (faradaic process). These parameters originate from the behavior of the kinetic equations and are functions of the rate constants, transfer coefficients, potential, diffusion coefficients, concentrations, etc. In addition, all these parameters are highly nonlinear, that is, they depend on the electrode potential. It seems that the electrical representation of the faradaic impedance, however useful it may sound, is not necessary in the description of the system. The systen may be described in a simpler way directly by the equations describing impedances or admittances (see also Section IV). In... 

Figure 4.11 Representation of an impedance value Z(cu) in the complex plane.

Fig. 3.7 The complex-plane impedance plot representation (also called the Argand diagram or Nyquist diagram) of the ideal impedance spectra in the case of reflective boundary conditions. Effect of the ratio of the film thickness (L) and the diffusion coefiicient (D). L/D (7) 0.005 (2) 0.1 (2) 0.2 4) 0.5 and (5) 1 s /. i o = 2 Q, 7 ct = 5 Q, (7 = 50 cm fis / Cdi = 20 pFcm. The smaller numbers refer to frequency values in Hz...

In fact, ac electrogravimetry is the combination of electrochemical impedance spectroscopy with a fast quartz crystal microbalance. The fluxes of all mobile species are considered, and the usual conditions and treatments of EIS are applied. Beside the electrochemical impedance, an electrogravimetric transfer function, Aw/A ((o), can be derived which contains the dependences of the fluxes of anions, cations and solvent molecules, respectively, on the small potential perturbation. The complex plane plot representations of electrogravimetric transfer functions for PANI are shown in Figs. 3.17 and 3.18. 

S.3.3 Nyquist (or Argand) Complex-Plane Plots for Representation of Impedance Behavior... 

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