Representation of the Impedance Data

The impedance data will be represented in complex plane plots as Z vs Z (normally called Nyquist diagrams), 7 vs T diagrams and derived quantities as the modulus function M = jcoCcZ = M -f jM (Q is the capacity of the empty cell) 

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Generally, the impedance spectrum of an electrochemical system can be presented in Nyquist and Bode plots, which are representations of the impedance as a function of frequency. A Nyquist plot is displayed for the experimental data set Z(Zrei,Zim.,mi), (/ = 1,2,. ..,n) of n points measured at different frequencies, with each point representing the real and imaginary parts of the impedance (Zrei Zim4) at a particular frequency . 

Figure 17.1 Equivalent circuits used to demonstrate the graphical representation of reactive impedance data a) Randles circuit and b) blocking circuit.

This circuit is usually referred to as the Randles circuit and its analysis has been a major feature of AC impedance studies in the last fifty years. In principle, we can measure the impedance of our cell as a function of frequency and then obtain the best values of the parameters Rct,<7,C4i and Rso by a least squares algorithm. The advent of fast micro-computers makes this the normal method nowadays but it is often extremely helpful to represent the AC data graphically since the suitability of a simple model, such as the Randles model, can usually be immediately assessed. The most common graphical representation is the impedance plot in which the real part of the measured impedance (i.e. that in phase with the impressed cell voltage) is plotted against the 90° out-of-phase quadrature or imaginary part of the impedance. 

In addition to the Nyquist representation, the Bode plot is as well applied for the description of impedance spectrometry data. In this case, the impedance data is represented in polar coordinates... 

Figure 17.2 Traditional representation of impedance data for the Randles circuit presented as Figure 17.1(a) with a as a parameter, a) complex-impedance-plane or Nyquist representation (symbols are used to designate decades of frequency.) b) Bode representation of the magnitude of the impedance and c) Bode representation of the phase angle. (Taken from Orazem et al. ° and reproduced with permission of The Electrochemical Society.)...

The quantitative and qualitative analysis presented in Section 20.2.1 demonstrates that the finite-diffusion-layer model provides an inadequate representation for the impedance response associated with a rotating disk electrode. The presentation in Section 20.2.2 demonstrates that a generic measurement model, while not providing a physical interpretation of the disk system, can provide an adequate representation of the data. Thus, an improved mathematical model can be developed. 

A refined philosophical approach toward the use of impedance spectroscopy is outlined in Figure 23.1, where the triangle evokes the concept of an operational amplifier for which the potential of input channels must be equal. Sequential steps are taken until the model provides a statistically adequate representation of the data to within the independently obtained stochastic error structure. The different aspects that comprise the philosophy are presented in this section. 

The complex impedance data involves the interplay of three variables, the imaginary component of the impedance, the real component of the impedance, Zreai, and the phase angle, common types of representation for impedance data are, the Nyquist and the Bode representations. Nevertheless, these have become the most widely used graphical representations of impedance data. 

The information provided by EIS can be plotted in different graphical representations. Figure 8.2 shows one common representation of EIS data, called a Bode plot. In a Bode plot, the absolute magnitude of the impedance (Fig. 8.2a) and the phase shift (Fig. 8.2b), both of which are experimentally measured, are plotted against... 

The relationship of the impedance as a complex function is also what leads to the most common data representation in EIS, known as the Nyquist plot (Fig. 8.3). In the Nyquist plot, -Im(Z) is plotted versus Re(Z) over the entire frequency range of the EIS measurement. One of the main shortcomings of the Nyquist plot is that the frequency is not shown there is only an implicit understanding that high frequencies are at the lower Re(Z) values and decrease in the positive X-direction. In the example provided in Figure 8.3, which is the same as the system represented in the Bode plot in Figure 8.2, three separate resistances are apparent as intercepts on the X-axis. Sueh a response is often observed in fuel cells where reactants and products are supplied in excess, and the only resistances governing potential losses are the Ohmic resistance and the activation losses at the anode and the cathode. 

The resultant hybrid algorithm was implemented in home-made software (see screenshots in Figure 3). The software consists of 3 logical blocks in which the user can explore the raw data in different representations (imaginary and real parts of the impedance, absolute... 

It is hoped that the more advanced reader will also find this book valuable as a review and summary of the literature on the subject. Of necessity, compromises have been made between depth, breadth of coverage, and reasonable size. Many of the subjects such as mathematical fundamentals, statistical and error analysis, and a number of topics on electrochemical kinetics and the method theory have been exceptionally well covered in the previous manuscripts dedicated to the impedance spectroscopy. Similarly the book has not been able to accommodate discussions on many techniques that are useful but not widely practiced. While certainly not nearly covering the whole breadth of the impedance analysis universe, the manuscript attempts to provide both a convenient source of EK theory and applications, as well as illustrations of applications in areas possibly u amiliar to the reader. The approach is first to review the fundamentals of electrochemical and material transport processes as they are related to the material properties analysis by impedance / modulus / dielectric spectroscopy (Chapter 1), discuss the data representation (Chapter 2) and modeling (Chapter 3) with relevant examples (Chapter 4). Chapter 5 discusses separate components of the impedance circuit, and Chapters 6 and 7 present several typical examples of combining these components into practically encountered complex distributed systems. Chapter 8 is dedicated to the EIS equipment and experimental design. Chapters 9 through 12... 

It is convenient to start the discussion of the fundamentals of impedance data representation with an analysis of very simple systems. If a sinusoidal voltage is applied to a pure resistor of value R, then the measured complex impedance is entirely resistive at all frequencies asZ = R and the impedance magnitude IZI = R. If a sinusoidal voltage is applied across a pure capacitor, the measured impedance can be calculated according to the relationship Z = -jloaCy where C is the capacitance. The magnitude of the impedance for a pure capacitor is IZI = (coC) L This impedance depends on the frequency and is entirely capacitive (see Chapter 3). 

Figure 16.1 Impedance-plane or Nyquist representation of impedance data for R = 10 Qcm, R = 100 flcm, and C = 20 fiF/crn. The blocking system of Table 16.1(a) is represented by A and dashed lines, and the reactive system of Table 16.1(b) is represented by Q solid lines.

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