Impedance plots in the complex plane

The second method is the most used, and frequently in terms of plots of Zf vs. Z , as will be explained in the next section. This type of analysis, developed by Sluyters et a/.1, is based on techniques used in electrical engineering. 

The experimental impedance is always obtained as if it were the result of a resistance and capacitance in series. We have already seen in (11.20) and (11.21) the relation between an RC series combination and the Rct + zw combination. It can be shown for the full Randles equivalent circuit for this simple charge transfer reaction, see Fig. 11.4, on separating the in-phase and out-of-phase components of the impedance, that 

These components are represented as a complex plane plot in Fig. 11.6 

It is interesting to consider two limiting forms of these equations  

This low-frequency limit is a straight line of unit slope, which extrapolated to the real axis gives an intercept of (Ra + Rct - 2o2Cd). The line corresponds to a reaction controlled solely by diffusion, and the impedance is the Warburg impedance, the phase angle being jt/4, see Fig. 11.6. 

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A simple circuit comprises a capacitor C shunted by a resistor R. Show that the frequency response of the circuit impedance, plotted in the complex plane, is represented by a semicircle of diameter R and with center at Z1 = -jR. 

Figure 3. Admittance data from a K +-conducting membrane and curve fits (solid curves) of eqs 2, 3, and 4 with Y /jf,) = 0 plotted in the complex plane [X(f) vs. R(f)] as impedance [Z(jf) = R(f) + jX(f) = Y 1(jf/)] loci (400 frequency points) over the 12.5 5000-Hz frequency range. These data were acquired rapidly as complex admittance data, as illustrated in Figure 1, at premeasurement intervals of 0.1 and 0.5 s after step voltage clamps to each of the indicated membrane potentials from a holding of —65 mV. The near superposition and similarity in shape of the two loci at 0.1 and 0.5 s, at each voltage, indicates that the admittance data reflect a steady state in this interval after step clamps. Axon 86-41 internally perfused with buffered KF and externally perfused in ASW + TTX at 12 °C. The membrane area is 0.045 cm2.

Figure 6. Admittance data from a Na+-conducting membrane and curve Jits (solid curves) of eqs 2, 3, and 5 with YK()f) = 0 plotted in the complex plane [X(f) vs. R(f)] as impedance loci (400 frequency points) over the frequency range 5 to 2000 Hz. Same axon and conditions as in Figure 5.

By taking into account the double-layer capacity, Q, and the electrolyte resistance, Re, one obtains the Randles equivalent circuit [150] (Fig. 10), where the faradaic impedance Zp is represented by the transfer resistance Rt in series with the Warburg impedance W. It can be shown that the high-frequency part of the impedance diagram plotted in the complex plane (Nyquist plane) is a semicircle representing Rt in parallel with Cd and the low-frequency part is a Warburg impedance. 

Fig. 5.20 Impedance as a vector quantity plotted in the complex plane.

An important finding by Yamamoto and Yamamoto (1976) was that the impedance of the removed SC layers did not produce a circular arc in the complex impedance plane. This is obvious from Figure 4.19, in which the admittance data from Figure 4.17 have been transformed to impedance values and plotted in the complex plane. 

If we merely plot the VSWR of the two impedances, they might exhibit some similarities, in particular for mediocre VSWR. This is by no means a proof of similarity. Impedance curves should preferably be plotted in the complex plane that always tells the complete story (and in particular what to do about matching see Chapter 6 and Appendix B). 

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. 

A critical problem in EIS (and in other electrochemical techniques) is the validation of the experimental data. This problem is more obvious in EIS than in time-domain techniques because of the manner in which the experimental data are displayed. For example, it is not uncommon to observe negative resistance (second quadrant) and inductive (fourth quad-rant) behavior when the experimental impedance data are plotted in the complex plane. Also, the impedance loci frequently take the form of depressed and/or distorted semicircles, and these may contain multiple loops. These features are not readily accounted for by using simple electric equivalent circuits. However, the inability to represent electrochemical impedance data by simple equivalent electric circuits is not in itself a problem, since there is no a priori reason why an interfacial impedance could be represented by such electrical analogs. Most companies selling impedance equipment are providing software... 

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. 

It is now possible to measure the real and imaginary parts of the impedance with the help of a vectorial analyzer in a large range of fiequencies. The experimental data are plotted in the complex plane (ReZ, -ImZ) and they are expected to draw out a semi-circle centered on the x-axis of diameter equal to R. At low frequency, the behavior is basically resistive at high frequency it is capacitive and the top of the semi-circle corresponds to the condition cot=. Experimentally, we often observe a deformation of this ideal curve, a flattening reflecting the fact that the capacitance (or equivalently the relative permittivity) is a function of the frequency (see section 11.2.1). 

One should note that the real and the imaginary parts of the Warburg impedance in Eq. (16.22) depend on frequency in the same way. Therefore the phase shift generated by the Warburg impedance is independent of frequency. Plotted in the complex-plane impedance format, this leads to a straight line with a slope of unity, as shown in Figure 16.6. 

Different kinds of plots based on impedance Z, admittance Z 1, modulus icoZ, or complex capacitance (z coZ) 1 can be used to display impedance data. In solid state ionics, particularly plots in the complex impedance plane (real versus imaginary part of Z) and impedance Bode-plots (log(Z) log(co)) are common. A RC element (resistor in parallel with a capacitor) has, for example, an impedance according to... 

Fig. 11.6. Plot of impedance in the complex plane of a simple electrochemical...

This is a vertical line in the complex plane impedance plot, since Z is constant but Z" varies with frequency, as shown in Fig. A2.2b. 

Figure 3.8. Impedance plots of real electrochemical systems in the complex plane...

In the complex-plane impedance diagram, the Nyquist plot of resistance and capacitance in parallel is an ideal semicircle, as depicted in Figure 4.3b. The diameter equals the value of the resistance, R. The imaginary part of the impedance reaches a maximum at frequency [Pg.146]

We shall refer to it as the complex-plane impedance plot, recognizing that the same data can also he represented in the complex-plane capacitance or the complex-plane admittance plots. The terms Cole-Cole plot, Nyqidst plot and Argand plot are also found in the literature. 

When 5 = 0, the faradaic impedance is real and equals One semicircle is observed in the complex plane plots (Fig. 4). 

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. 

The electrochemical impedance spectroscopic approach, which is largely based on similar methods used to analyze circuits in electrical engineering practice, was developed by Sluyters and coworkers (4) and later extended by others (8-12). It deals with the variation of total impedance in the complex plane [as represented in Nyquist plots (Section... 

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