Nyquist plot modeling

Figures 5.29a and 5.29b show the Bode and Nyquist plot for a resistor, Ro, connected in series with a resistor, Rt, and capacitor, Ci, connected in parallel. This is the simplest model which can be used for a metal-solid electrolyte interface. Note in figure 5.29b how the first intersect of the semicircle with the real axis gives Ro and how the second intersect gives Ro+Rj. Also note how the capacitance, Ct, can be computed from the frequency value, fm, at the top of the semicircle (summit frequency), via C l JifmR .

Figure 5.29. Bode (a) and corresponding Nyquist plot (b) of the circuit shown in inset which is frequently used to model a metal/solid electrolyte interface. Effect (c) of capacitance C2 on the Nyquist plot at fixed R0, R( and R2.

Quite often we are face with the task of reducing the order of a transfer function without losing essential dynamic behavior of the system. Many methods have been proposed for model reduction, however quite often with unsatisfactory results. A reliable method has been suggested by Luus (1980) where the deviations between the reduced model and the original one in the Nyquist plot are minimized. 

After the parameters have been estimated, generate the Nyquist plots for the reduced models and the original one. Comment on the result at high frequencies. Is N=100 a wise choice ... 

Comment on this apparent discrepancy between the capacitive model and reported shapes of the real Nyquist plots. 

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.

Why such a difference between the real EDLC and the model proposed here In other words, what are the specific features the equivalent circuit does not take into account The oversimplified equivalent circuit presented in Figure 1.22 considers two planar electrodes face to face, with a constant thickness of dielectric material between them. The reality is much more complex since EDLCs use three-dimensional porous electrodes, and the porous electrodes are responsible for the particular shape of the Nyquist plot presented in Figure 1.23. 

Figure 18, taken from Ref. 77, describes several models proposed for the Li electrodes in solutions, their equivalent circuit analogs, and the expected impedance spectra (presented as Nyquist plots). Assuming parallel plate geometry for the solid electrolyte interface, as well as knowledge of the surface species involved from spectroscopy (and thus their dielectric constant, which is around 5 for many surface species formed on Li, including R0C02Li, Li2C03, LiF, ROLi, etc. [186]), it is possible to estimate the surface film s thickness from the electrode s capacitance (calculated from the model fitted to the spectra) ...

Figure 18 Various models proposed for the surface films that cover Li electrodes in nonaqueous solutions. The relevant equivalent circuit analog and the expected (theoretical) impedance spectrum (presented as a Nyquist plot) are also shown [77]. (a) A simple, single layer, solid electrolyte interphase (SEI) (b) solid polymer interphase (SPI). Different types of insoluble Li salt products of solution reduction processes are embedded in a polymeric matrix (c) polymeric electrolyte interphase (PEI). The polymer matrix is porous and also contains solution. Note that the PEI and the SPI may be described by a similar equivalent analog. However, the time constants related to SPI film are expected to be poorly separated (compared with a film that behaves like a PEI) [77]. (With copyrights from The Electrochemical Society Inc., 1998.)...

Figure 2.38. Typical Nyquist plots for electrochemical systems 2.6.3 Equivalent Circuit Models...

Figure 4.1. Nyquist plot showing the impedance spectra of an R/CPE electric circuit model [2], (Reproduced with permission from Research Solutions Resources LLC.)...

The simulated Nyquist plot of resistance and capacitance in series is a vertical line in the complex-plane impedance diagram, as shown in Figure 4.2(b). The effect of the parameter R on the position of the line is presented in Appendix D (Model Dl). 

Figure 4.4b shows the simulated Nyquist plot of resistance and a CPE in series connection, in a complex-plane impedance diagram. More examples of the effect of parameters on the spectra can be found in Appendix D (Model D3). 

Figure 4.6. a Resistor and inductor in series (Model D5) b Nyquist plot of resistor and inductor in series connection over the frequency range 1 MHz to 1 mHz (Model D5 R = 10 a, L = 0.0001 H)... 

Accordingly, the simulated Nyquist plot can be drawn as shown in Figure 4.86. On a Nyquist plot, the infinite Waiburg impedance appears as a diagonal line with a slope of 1. More examples can be found in Appendix D (Model D7). 

Therefore, the Nyquist plot can be simulated as shown in Figure 4.146. The deformation of the adsorption model caused by the mixing of the parameters is plotted in Appendix D (Model D13). 

Figure 4.16b shows the simulated Nyquist plot of the modified Randles cell shown in Figure 4.16a. Impedance diagrams with variations in the CPE parameters are included in Appendix D (Model D15).

Figure 4.20. a Equivalent circuit of modified Randles cell with bounded CPE in series with Rc, (Model D19) b Nyquist plot of modified Randles cell having a bounded CPE in series with Rc, over the frequency range 6 kHz to 1 mHz (Model D19 Rei = 10 2, Rcl = 20 Q, R0 =... 

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