Impedance vector representation

Figure 3 Complex plane representation of impedance vector Z.

Fig. 9G Vector representation of the impedance of (a) series and (b) parallel combination of a capacitor and a resistor, showing how the phase angle changes with frequency (expressed in Radis on the vectors). The ends of the arrows show the absolute values of the vectors, except at the two lowest frequencies in the series combination.

Fig. IIL Vector representation of the impedance Z(co) in the complex plane. ReZ and ImZ are the real and the imaginary components of the impedance, respectively.

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.

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 5.21 Vector representation of impedance in the complex plane.

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. 

Impedance also is broken up into a vector in this complex representation — except that though it is frequency-dependent, it is (usually) not a function of time. 

There is no difference in principle in the presentation of impedance data. The vector must always be defined by two quantities and the choice of Z and Z" or Z" and (/> is a matter of convenience and preference. More specific situations described later will demonstrate the appropriate choice of data representation. 

The concept of electrical impedance was first introduced by Oliver Heaviside in the 1880s and was soon after developed in terms of vector diagrams and complex number representation by A. E. Keimelly and C. P. Steinmetz [1]. Since then the technique gained in exposure and popularity, propelled by a series of scientific advancements in the... 

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