Transmission line plot

Figure 15 shows a set of complex plane impedance plots for polypyr-rolein NaC104(aq).170 These data sets are all relatively simple because the electronic resistance of the film and the charge-transfer resistance are both negligible relative to the uncompensated solution resistance (Rs) and the film s ionic resistance (Rj). They can be approximated quite well by the transmission line circuit shown in Fig. 16, which can represent a variety of physical/chemical/morphological cases from redox polymers171 to porous electrodes.172... 

In studies of these and other items, the impedance method is often invoked because of the diagnostic value of complex impedance or admittance plots, determined in an extremely wide frequency range (typically from 104 Hz down to 10 2 or 10 3 Hz). The data contained in these plots are analyzed by fitting them to equivalent circuits constructed of simple elements like resistances, capacitors, Warburg impedances or transmission line networks [101, 102]. Frequently, the complete equivalent circuit is a network made of sub-circuits, each with its own characteristic relaxation time or its own frequency spectrum. 

Figure 10. Nyquist plot of the impedance spectrum experimentally measured on the ACFCE at an applied potential of 0.1 V (vs. SCE) in a 30 wt % H2SO4 solution. Dotted and solid lines represent the impedance spectra theoretically calculated based upon the transmission line model (TLM) in consideration of pore size distribution (PSD) and pore length distribution (PLD), respectively. Reprinted with permission from G. -J. Lee, S. -I. Pyun, and C. -H. Kim, J. Solid State Electrochem., 8 (2004) 110. Copyright 2003, with kind permission of Springer Science and Business Media.

The first resistance Rs is the resistance of the electrolyte outside the pores the R, elements are the electrolyte resistances inside the pores of the electrode and are the double layer capacitances along the pores. This model is called the Transmission Line Model (TLM) by De Levie. A careful selection of a set of Rv C values allows to calculate back the experimental plot such as the one presented in Figure 1.23 [28]. It can be noted that constant phase element (CPE) can be used to replace the capacitance C for better fitting, the CPE impedance ZCPE being ZCPE = l//(Cco) . 

In contrast, Fig. 11.6 shows a typical Nyquist plot for the layer after switching between the oxidised and reduced forms in background electrolyte for several days (Fig. 11.4(c)). A pronounced semicircular region, Warburg 45° line and vertical capacitive region can clearly be seen. We have fitted these data to the transmission line circuit (Fig. 11.1). The value of Cs obtained is found to vary with dc potential (Fig. 11.7) and with the... 

When discussing the ionic conductivity of catalyst layers, one must mention the finite transmission-line equivalent circuit, which is widely used to model porous electrodes and was shown as Figure 4.33 in Chapter 4. For ease of discussion, the circuit is re-plotted here as Figure 6.23. 

FIGURE 2.4.9 Example of a transmission line Rj-ar vs. L) plot at a given Vq value. Extrapolation of the data to a channel length of zero yields the specific contact resistance R/ as the y-intercept. 

Let us now assume an open-circuit condition at the far end of the transmission line, i.e. no direct current can flow in the actual system. This is defined as diffusion in the case of the reflective boundary condition. At the far end complete blocking of diffusion occurs. This results in a vertical line at low frequencies in the Nyquist plot corresponding to a capacity only (Fig. II.5.6). Here, at very low frequencies, resistance and capacity C are in series. 

A plot of the scan impedance Za for an infinite x infinite array without a ground-plane and with the same interelement spacings as used in Chapter 6 as obtained from the PMM code is shown in Fig. D.3. The impedance is plotted in a Smith chart normalized to 100 ohms. Furthermore, the impedance in Fig. D.3 as well as in later figures includes the matching of a short transmission line with characteristic impedance 200 ohms and length 0.13 cm. This was typically true in the cases shown in Chapter 6, which should make comparisons between the various curves more meaningful. We remind the reader that the purpose of this transmission line (also referred to as a pigtail ) was to better center the impedances as well as compress them (for details see Chapter 6 and Appendix B). 

An example of response of the capacitive transmission line is given in the case study abstract. The plot of the current resulting from a tension step (Heaviside function) shows the current increasing extremely fast to a high value then decreasing more slowly as the inverse square root of the time. (This function is the Green function used in the convolution Equation G5.14.) Mathematically, the current should climb to infinite value, but instrumental and physical limitations prevent it. 

This model corresponds to the transmission line depicted in Eig. 9.19. The complex plane plots of the total impedance, Eq. (9.19), as well as the first and second terms of this equation, are displayed in Eig. 9.20. The total impedance starts at high frequencies at/ a p and initially displays a straight line at 45° followed by a semicircle (Eig. 9.20a). The first term in Eq. (9.19) shows a small inductive loop at high frequencies, followed by a semicircle, Eig. 9.20b, while the second term is similar to that of the porous electrode with solution resistance only, Eig. 9.20c. 

De Levie [1963,1964] was the first, it seems, to examine the frequency response of a wire-bmsh electrode of the above kind and provide the background mathematical treatment required to interpret the experimental behavior. The important aspect of the observed impedance spectroscopy over a wide range of frequencies was that transmission-line (Figure 4.5.22e) behavior resulted, as illustrated in Figure 4.5.30. The characteristic behavior is a linear complex-plane plot having a 45° phase-angle... 

FIGURE 1.55. Predicted dimensionless plots for analyzing chronoamperometric transients from transmission line or porous electrode model using Eqn. 286 (fidl curve). Note that the short-time limit given by Eqn. 283 dashed curve), and the long-time limit given by Eqn. 287 are also displayed. 

The shape of the chronoamperometric current response depends to a large extent on the value of the uncompensated solution resistance R . We note from Fig. 1.54 that is in series with the finite transmission line element. The presence of uncompensated solution resistance effects can be clearly identified by examining the current/time data when the latter is plotted in i(t) versus format. In many cases (see Fig. 1.56) such plots are nonlinear, so we observe deviation from the expected linear response at both short and long time periods. At short time... 

FIGURE 1.57. Effect of solution resistance on the shape of the normalized current/time plots according to the transmission line model. 

Figure 4.5 shows calculated impedance plots for the general model for different values of p. As resistances become more equal, the high-frequency limit on the x axis for Zl(Rx + Re) increases. For the extreme case when Rx = Re, a maximum value of 1/4 is found. There is then only a small difference between the real component at the high-frequency limit (1/4) and the real component at the low-frequency (1/3) limit. Under these conditions the Warburg region almost disappears, and the transmission line appears to be a capacitor. 

Typical impedance plots are shown in Fig. 4.6. In each plot we find a resistance on the real axis at a high frequency of / ,. This resistance is the sum of the electrolyte resistance Re and the resistance / n of the transmission line where, as discussed in Eqn. 25,... 

Values of y can be calculated from experimental results at high frequency for Rx and with Eqn. 32 for bic from results at low frequency for Figure 4.11 shows a typical plot of Eqn. 30. A reasonable straight line is obtained. This combination of data for the two different resistances shows the value of the transmission line with two resistances. 

In Figs. 4.19-4.25 we display typical results from the SigmaPlot analysis for a range of oxidation potentials of a polypyrrole film in aqueous solution. Some data are plotted in Fig. 4.13. In each case we obtain a very good fit of experimental data with results calculated from the derived parameters. Hence the Case A circuit involving the Randles circuit in series with the transmission line provides an excellent explanation of results at any particular potential. In Table 4.2 we collect results from the analysis. In addition to the Case A circuit, we include an... 

Fig. 20.29 (a) Equivalent circuit and (b) the electro-optical complex plane plot for a polypyrrole/polystyrenesulfonate composite film. Zd is a charge transport impedance within the polymer film (modeled as a transmission line circuit). ( ) Electrical data ( ) optical data. AE is the applied ac potential, and the total charge zlU is divided into doublelayer charging (A d) and Faradaic A Q ) components. (Reproduced with permission from Ref. 144.)... 

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