Complex plane impedance spectra

Fig. 5.25 Complex plane impedance spectra and their associated equivalent circuits (a) resistor, (b) capacitor, (c) resistor and capacitor connected in series, and (d) resistor and capacitor connected in parallel.

Fig. 5.27 Complex plane Impedance spectra (Nyquist plot) for corroding metal.

Show the complex-plane impedance spectra for the following electrochemical cells ... 

In the work conducted by Park and Macdonald [1983], impedance spectroscopy was used to measure the polarization resistance of corroding carbon steel in a variety of chloride (IM)-containing solutions at temperatures from 200 to 270°C. Typical complex plane impedance spectra for one such system as a function of time are shown in Figure 4.4.43. hi all cases (different exposure times), the impedance spectra are characteristic of a system that can be represented by an electrical transmission line. Also of interest is the observation (Park [1983]) that the inverse of the polarization resistance, which is proportional to the instantaneous corrosion rate, increases with increasing exposure time. Accordingly, the corrosion process is kineticaUy auto-catalytic, rather than being linear, as was previously reported. This autocatalytic... 

Figure 16.9 Example of complex-plane impedance results for the anode of a direct methanol fuel cell (DMFC). The numbers on the three parts of the spectrum show the frequency...

In an analysis of an electrode process, it is useful to obtain the impedance spectrum —the dependence of the impedance on the frequency in the complex plane, or the dependence of Z" on Z, and to analyse it by using suitable equivalent circuits for the given electrode system and electrode process. Figure 5.21 depicts four basic types of impedance spectra and the corresponding equivalent circuits for the capacity of the electrical double layer alone (A), for the capacity of the electrical double layer when the electrolytic cell has an ohmic resistance RB (B), for an electrode with a double-layer capacity CD and simultaneous electrode reaction with polarization resistance Rp(C) and for the same case as C where the ohmic resistance of the cell RB is also included (D). It is obvious from the diagram that the impedance for case A is... 

There is one further important practical aspect that has to be considered when taking this approach to performing an experimental bifurcation analysis impedance measurements can only be carried out with stable stationary states it is not feasible to measure unstable stationary states, or states close to a bifurcation. However, as we discussed above, N-NDR and HN-NDR systems become unstable due to ohmic losses in the circuit, whereas they are always stable for vanishing R< >. Being aware that an ohmic series resistor causes only a horizontal shift of the impedance spectrum in the complex plane, it is apparent that it is possible to infer about the existence of bifurcations from impedance measurements at sufficiently low solution resistance (or when invoking an ZR-compensation, an option many potentiostats provide). This is illustrated with the schematic impedance spectrum shown in Fig. 12, which depicts a typical impedance spectrum of an N-NDR system. The spectrum possesses two... 

Fig. 14. (a) Cyclic voltammogram of the reduction of 0.02 M H2O2 on a rotating Ag(pc) in 0.1 M HCIO4 scan rate 50 mV s-1. (b) Impedance spectrum in the complex impedance plane of Ag(pc) in the same electrolyte measured at Asmse = -0.20 V and a rotation speed lower than in (a) where the NDR branch is stable. Numbers indicate the frequency v. (Reproduced from C. Eickes, K. G. Weil and K. Doblhofer, PCCP 2 (2000), 5691-5697 by permission of The Royal Society of Chemistry on behalf of the PCCP Owner Societies.)... 

Fig. 17. Characteristic I-U curves of HN-NDR systems (a) under potential control for vanishing ohmic series resistance, Rq (b) under potential control for intermediate values of Rq and (c) under current control, (d) Typical impedance spectrum of an HN-NDR system in the complex impedance plane at the point indicated in (a).

Fig. 5. Complex-plane plot of impedance spectrum for a polycrystalline diamond film between two ohmic contacts. Frequency/kHz shown on the figure. Solid circles data obtained with ac bridge. Open circles data obtained with phase-sensitive analyzer. Top equivalent circuit [30].

Recall that the faradaic resistance can be determined as a low-frequency cut-off at the complex-plane plot of impedance spectrum (compare the equivalent circuit in Fig. 10b). Such plots measured in the Fc(CN)63 /4 solutions of different concentrations are given in Fig. 23a [104] (similar results were obtained in [111]). The plots are (somewhat depressed) semicircles, whose radii decreased with increasing redox couple concentration. Figure 23b shows the line plotted by using the data in Fig. 23a, in accord with Eq. (6). We notice that all three methods yielded similar results. 

Note that the quantities in bold are vectors represented in the complex plane. In Fig. 12.5 a transfer function spectrum obtained in 0.25 s is shown for 100 frequencies around 10 MHz. It can be used for real time evaluation of the quartz electro-acoustical impedance when the viscoelastic properties change. 

Usually an equivalent circuit is chosen and the fit to the experimental data is performed using the complex nonlinear least-squares technique. However, the model deduced from the reaction mechanism may have too many adjustable parameters, while the experimental impedance spectrum is simple. For example, a system with one adsorbed species (Section IV.2) may produce two semicircles in the complex plane plots, but experimentally, often only one semicircle is identified. In such a case, approximation to a full model introduces too many free parameters and a simpler model containing one time-constant should be used. Therefore, first the number and nature of parameters should be determined and then the process model should be constructed in consistency with the parameters found and the physicochemical properties of the process. 

FIGURE 15.6 Complex plane (or Nyquist) plot of the impedance spectrum for the equivalent circuit shown. An example impedance vector at some arbitrary frequency is illustrated by the dashed arrow. Frequency increases in the direction shown by the solid curved arrow. Circuit elements uncompensated solution resistance J s double layer capacitance Cji polarization resistance Rp and diffiasional (Warburg) impedance Z -... 

Figure 4.5 3. Complex-plane or Nyquist plot for the impedance spectrum of a simple series RC circuit.

The above paragraph gives, in brief, the analysis first published by the Sluyters [1964] for complex-plane analysis of the impedance spectrum of an electrode process at an interface exhibiting a double-layer capacitance hence its great importance as a basis for examining the impedance spectroscopy of supercapacitor systems, particularly those based (Conway [1999]) on double-layer capacitance (Grahame [1947]). 

The impedance Z co) of a system is a continuous complex function on the positive frequency domain, see Equation (2). This continuous function is sampled on a number of angular frequencies co. = 2nf by measuring the corresponding impedance points Z(co) in the complex plane. Hence, an impedance point consists of a real part Z (co) and an imaginary part and is thus 2-dimensional. The index i denotes the rth frequency of the discretized impedance spectrum. 

The major point to note from this analysis is that an impedance spectrum in its most basic representation is a set of points in the complex plane. A plot of -Z" versus Z is termed a Nyquist plot (see Fig. 1.67). 

The impedance Z[s) represents the state of a Li-ion cell in frequency domain and can be illustrated by an impedance spectrum, e.g. in a Nyquist plot in the complex plane, see Figure 3. The impedance is the system function or the frequency response locus which is in case of a Li-ion cell the complex voltage f/( ), i.e. the system response, divided by the complex current /(, i.e. the excitation signal. In frequency domain s = O jca is the complex frequency with a damping term cr and the angular frequency co=l7rf. 

One possible solution to the problem of recording a full impedance spectrum is, if there is full confidence in the impedance characteristics, to choose a smaller number of frequencies and make the spectrum recording take less time. If there is always a semi-circular (or depressed semi-circular) complex plane spectrum, then three or four carefully chosen frequencies should be sufficient to fit the data to the model. Other special cases are ... 

Fig. 4.5. Complex-plane presentation of (a) impedance spectrum for a HTHP single crystal in 2.5 M H2SO4 solution. (b) its high-frequency part [15]. The frequencies (kHz) are shown in the figure.

Parameters similar to those, for example, of the Cole equation, can also be computed without choosing a model. Jossinet and Schmitt (1998) suggested two new parameters that were used to characterize breast tissues. The first parameter was the distance of the 1 MHz impedance point to the low-frequency intercept in the complex impedance plane and the second parameter was the slope of the measured phase angle against frequency at the upper end of the spectrum (200 kHz—1 MHz). Significant differences between carcinoma and other breast tissues were found using these parameters. 

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