Impedance simulated complex-plane

The simulated complex-plane impedance diagram is shown in Figure 4.27b. As can be seen in the figure, this ladder structure is characterized by two semicircles with two time constants, r, = RclCd] and r2 = R3C2, accounting for the two-step reaction. The element C2 symbolizes the adsorption capacitance, and r2 represents the relaxation of the adsorbing process. 

Fig. 8.15 Simulated complex plane plots of impedance response for ideally polarized disk electrode (a) linear plot showing effect of dispersion at frequencies AT > 1 as deviation from vertical line (b) plot in logarithmic scale for imaginary impedances (From Ref [358], Reproduced with permission of Electrochemical Society)...

Figure 4.4.13. Simulated complex plane impedance diagrams for the electrodissolution of iron in sulfate media as a function of pH according to Keddam et al. [1981]. The potentials for which the diagrams are calculated are shown in Figure 4.4.12. The arrows indicate die direction of decreasing frequency. (From M. Keddam, O. R. Mattos, and H. J. Takenouti, Reaction Model for Iron Dissolution Studied by Electrode Impedance Determination of die Reaction Model, J. Electrochem. Soc., 128, 257—274, [1981]. Reprinted by permission of die publisher. The Electrochemical Society, Inc.)...

Figure 16,4 A simulated complex-plane representation of the impedance vector as a function of frequency for a simple circuit, consisting of a capacitor and resistor in parallel.

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). 

The impedance behavior of electrode reactions is often complex but can be conveniently simulated by computer calculations, especially in the case of the method based on kinetic equations (108, 113). The forms of the frequency response represented in terms of the Z versus Z" complex-plane plots and by relations of Z or phase angle to frequency ai or log (o (Bode plots) are often characteristic of the reaction mechanism and involvement of one or more adsorbed intermediates, and they thus provide diagnostic bases for mechanism determination complementary to those based on dc, steady-state rate versus potential responses. The variations of Z versus Z" plots with dc -level potential, in controlled-potential experiments, also give rise to useful diagnostic information related to the dc Tafel behavior. 

The assessment of the reported model for experimental passivation systems of say magnesium/magnesium perchlorate " and titanium/or titanium dioxide/sulphuric acid" has been made from the view-point of ffactality of the electrode surface and a good agreement between the model calculated complex plane impedance (Fig. 8 being a typical simulated plot) and the measured Nyquist-plots is obtained. 

Figure 9. FFT analysis of the sum of sine wave perturbation left side, no optimization right side, optimization of phases, (a) Perturbation voltage in the time domain, (b) Perturbation voltage in the frequency domain, (c) Complex plane plots of simulated impedance spectra with 5% noise added to the current response. Solid lines show response without noise.

Keiser etal. studied the impedance of arbitrarily shaped pores. They simulated the complex plane plots in the absence of a faradaic process (Fig. 40). Instead of a straight line at 45°, observed for cylindrical pores at high frequencies, different forms of plateaus or a semicircle were observed. 

In addition to comparing the sum of squares, the experimental and simulated data should be compared by using complex plane and Bode plots. The phase-angle Bode plot is particularly sensitive in detecting time constants. Boukamp proposed to study the residual sum of squares after subtracting the assumed model values from the total impedance data. If the model is valid, the residuals should behave randomly. If they display regular tendencies, it may mean that the model is not correct and further elements should be added. However, the variations of the residuals should be statistically important. 

Exercise 2.14 Simulate impedances for the circuit RCL in series for / = 1 2, C = 0.01 F, and L = 0.01 H. Make simulations using ZView and Excel. Make complex plane and Bode plots. 

Fig. 3.9 Complex plane plots of numerictilly simulated impedances, in Ci, for different perturbation waveforms with 1 % noise added (a) rectangular pulse, (b) exponentially decaying perturbation, (c) quasi-random noise, (d) sum of sine waves with constant amplitudes and zero phases (From Ref. [105] with permission of editorial board)...

Exercise 4.5 Write a program in Maple/Mathematica to simulate transmissive impedance and create the corresponding complex plane and Bode plots. The parameters are as in Exercise 4.3, with / = 0.01 cm. 

Fig. 8.21 Application of DIA to impedance data simulated using electrical equivalent model containing two time constants (a) electrical equivalent model and impedance complex plane plot (b) log of obtained parameters / 2, C, and time constant r = T as functions of log of inverse frequency log Tp = — log (c) and (d) cumulative spectral line s intensity plots as functions of log of values of parameters (From Ref. [392], copyright (2004), with permission from Elsevier)...

Fig. 9.29 Complex plane plot of the impedance simulated for the semi-infinite pore length using Eq. (9.44), continuous line, its CNLS approximations by de Levie s model, Eq. (9.7), dashed line, and fit to de Levie s model with CPE, dotted line-, for t]o = 0.5 V, jo = 10 A cm , pore parameters as in Fig. 9.26 (From Ref. [72] with kind permission from Springer Science and Business Media...

Figure 7-5 shows a simulation of the impedance of this circuit in Nyquist and Bode plot presentations. In the complex plane (Nyquist plot), an ideal capacitive semicircle with Rp as the diameter is displayed. Adopting this simplified model, analysis of corrosion systems is often reduced to the determination of the polarization resistance Rp available from the low frequency limit... 

Fig. 19.4. Complex-plane plots of the impedance for electrodes of (a) as-deposited diamond and pore types (b) 30 x 50 nm, (c) 60 x 500 nm, and (d) 400 nm x 3 un, at +0.4 V vs. Ag/AgCl.Experimental data points (O) and simulated curves (solid lines) calculated on the basis of equivalent circrdts involving modified transmission line models (see text), are shown. The parameters used in the calculated curves are given in Table 19. 2.

Figure 16.6 Complex-plane-impedance plot for an equivalent circuit with diffusion limitation at low frequencies. The values chosen for this simulation were Rs = Q cm Rf = SQ cm Cdi = 20 pF cm , Cb.ox = Cb.Red = 10 rnM ...

This section explains impedance and admittance formulas of nonuniform lines, such as finite-length horizontal and vertical conductors based on a plane wave assumption. The formulas are applied to analyze a transient on a nonuniform line by an existing circuit theory-based simulation tool such as the EMTP [9,11]. The impedance formula is derived based on Neumann s inductance formula by applying the idea of complex penetration depth explained earlier. The admittance is obtained from the impedance assuming the wave propagation velocity is the same as the light velocity in free space in the same manner as an existing admittance formula, which is almost always used in steady-state and transient analyses on an overhead line. 

---