Impedance graphical presentation

FIGURE 7.53 Push-pull amplifier (a) like transistors are configured in push-pull Class B. An input and an output transformer are needed to invert the phase of one transistor with respect to the other. The turns ratio of T1 is chosen so that the standard load impedance Rsl is transformed to the load impedance Ri presented to each collector. Then-graphical solution is shown in Fig. 7.52. (b) The phase of one transistor with respect to the other is inverted without transformers since one transistor is NPN whereas the other is PNP. 

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. 

A review of Chapter 1 may be useful. Summaries of relationships among complex impedance, real and imagmary parts of the impedance, and the phase angle and magnitude are foimd in Tables 1.1,1.2, and 1.3. A more complete discussion of the use of graphiced methods is presented in Chapters 16,17, and 18. 

Numerical solutions have been presented for the impedance response of semiconducting systems that accoimt for the coupled influence of transport and kinetic phenomena, see, e.g., Bonham and Orazem. Simplified electrical-circuit analogues have been developed to account for deep-level electronic states, and a graphical method has been used to facilitate interpretation of high-frequency measurements of capacitance. The simplified approaches are described in the following sections. 

The characteristic frequency evident as a peak for the imaginary part of the complex-capacitance in Figures 16.12(b) and 16.13(b) has a value corresponding exactly to fc = 27tReC) only for the blocking system. As found for data presentation in admittance format, the presence of a Faradaic process confounds use of graphical techniques to assess this characteristic frequency. Like the admittance format, the complex capacitance is not particularly well suited for analysis of electrochemical and other systems for which identification of Faradaic processes parallel to the capacitance represents the aim of the impedance experiments. It is particularly well suited for analysis of dielectric systems for which the electrolyte resistance can be neglected. 

Graphical methods provide a first step toward interpretation and evaluation of impedance data. An outline of graphical methods is presented in Chapter 16 for simple reactive and blocking circuits. The same concepts are applied here for systems that are more typical of practical applications. The graphical techniques presented in this chapter do not depend on any specific model. The approaches, therefore, can provide a qualitative interpretation. Surprisingly, even in the absence of specific models, values of such physically meaningful parameters as the double-layer capacitance can be obtained from high- or low-frequency asymptotes. 

A Remdles circuit is used here to demonstrate the graphical representation for reactive (nonblocking) systems. The impedance of the Randles circuit presented in Figure 17.1(a) is given by... 

The graphical representations presented here are intended to enhance analysis and to provide guidance for the development of appropriate physical models. While visual inspection of data alone cannot provide all the nuance and detail that can, in principle, be extracted from impedance data, the graphical methods described in this chapter can provide both qualitative and quantitative evaluation of impedance data. 

Under certain limiting conditions, graphical methods for analysis of impedance data can be beised on the physics of the system under study. Use of such methods does not provide the detailed information that may be available from use of regression techniques, presented in Chapter 19. The graphical methods may, however, complement the development of detailed process models by identifying frequency ranges in which the process model must be improved. 

A graphic illustration of these equations is presented in Fig. 11(b). Although, in simple cases, the process parameters may be obtained graphically, the best way to analyze the impedances is by the complex nonlinear least-squares approximation technique. The following parameters may be obtained from such fits 7 , R, and the Warburg coeffi-... 

One might be tempted to assume that the presentation above terminates our education in matching. Actually, it is just about to begin. The real challenge is to combine any of the three tools above in a design that transforms the arbitrary impedance curve Zl into a small cluster of impedance points located anywhere in the complex plane. There is in general no simple unique approach. In the anthor s opinion the best approach appears to be the graphical approach, as wiU now be illustrated with examples. (Oh yes, I admit for once, a little bit of experience is helpful as well.)... 

In the present state-of-the-art equipment it is possible to measure and plot the electrochemical impedance automatically. The generator can be programmed to sweep from a maximum to a minimum frequency in a number of required frequency steps. Thus the measurement frequency is changed automatically, and the total measurement time for one experiment can be predetermined. For example, for a measurement using five samples per every decade of frequency, it takes 31 seconds to take a sweep from 50 kHz to 0.1 Hz, 5 minutes 30 seconds for a sweep from 50 kHz to 0.01 Hz, and almost 54 minutes for a sweep from 50 kHz to 0.001 Hz. Automatic plotting of the experimental data can be performed by computer software in different graphical representa-... 

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