Impedance: Bode plot

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... 

The impedance ("Bode plot") and permittivity data are sometimes as revealing as the AC electrokinetic analysis in understanding the bulk-solution relaxations. The experimental data for monocyte blood cell solutions demonstrated the a- and P- bulk-solution relaxations in MHz and kHz regions and low-frequency interfacial processes (Figurell-7) [38]. 

Introducing the complex notation enables the impedance relationships to be presented as Argand diagrams in both Cartesian and polar co-ordinates (r,rp). The fomier leads to the Nyquist impedance spectrum, where the real impedance is plotted against the imaginary and the latter to the Bode spectrum, where both the modulus of impedance, r, and the phase angle are plotted as a fiinction of the frequency. In AC impedance tire cell is essentially replaced by a suitable model system in which the properties of the interface and the electrolyte are represented by appropriate electrical analogues and the impedance of the cell is then measured over a wide... 

The technique of AC Impedance Spectroscopy is one of the most commonly used techniques in electrochemistry, both aqueous and solid.49 A small amplitude AC voltage of frequency f is applied between the working and reference electrode, superimposed to the catalyst potential Uwr, and both the real (ZRe) and imaginary (Zim) part of the impedance Z (=dUwR/dI=ZRc+iZim)9 10 are obtained as a function of f (Bode plot, Fig. 5.29a). Upon crossplotting Z m vs ZRe, a Nyquist plot is obtained (Fig. 5.29b). One can also obtain Nyquist plots for various imposed Uwr values as shown in subsequent figures. 

Equivalent Circuit Analysis. IS measurements yield values of V and Z the real and imaginary components of the impedance, as a function of f, the AC frequency. The data are usually displayed as Nvauist plots (Z, vs. Z ) or Bode plots (impedance modulus,... 

The overall ability of a power supply to attenuate disturbances at its input is expressed as its PSRR (power supply rejection ratio). In graphs, PSRR is usually plotted as a function of frequency. We will invariably find that the rejection ratio is very low at higher frequencies. One reason for this is that the Bode plot cannot really help because the open-loop gain is very small at these frequencies. The other reason is, even a tiny stray parasitic capacitance (e.g., across the power switch and inductor) presents such a low impedance to noise frequencies (whatever their origin) that almost all the noise present at the input migrates to the output unimpeded. In other words, the power stage attenuation (which we had earlier declared to be Vo/Rin) is also nonexistent for noise (and maybe even ripple) frequencies. The only noise attenuation comes from the LC filter (hopefully). 

Electrochemical impedance spectroscopy (EIS) profiles, measured as a function of the interrogating frequency, can be represented on both Bode and Nyquist plots. The two-component Bode plot presents a comprehensive and... 

Fig. 1 Representative a Nyquist and b Bode plots from electrochemical impedance spectroscopy measurements (HubrechtJ (1998) Metals as Biomaterials, Helsen J, Breme H (eds) John WUey Sons Limited. Reproduced with permission)...

Equations 2.37-2.40 result in the commonly used presentation of the impedance, e.g. the Nyquist and the Bode plots. The first one shows the total impedance vector point for different values of co. The plane of this figure is a complex plane, as shown in the previous section. Electrochemical-related processes and effects result in resistive and capacitive behaviour, so it is common to present the impedance as ... 

In addition to the Nyquist representation, the Bode plot is as well applied for the description of impedance spectrometry data. In this case, the impedance data is represented in polar coordinates... 

Fig. 6. Impedance spectra in the Z plane and in the Bode plot (log(Z) vs. log(cu)) for (a) one RC element (b) two RC elements (c) three RC elements representing the situation of a polycrystal with non-ohmic electrodes and highly resistive grain boundaries and (d) three RC elements with two similar relaxation frequencies (cor2 — 3ror3) leading to overlapping semicircles.

Fig. 16.7. Impedance of AISI316 stainless steel in 3 per cent (wt) NaOH at 80°C (a) Complex plane plot (b) Bode plot (from Ref. 10 with permission).

Figure 19 Schematic Bode plots from EIS measurements and equivalent circuits that could be used to fit them for various possible corrosion product deposit structures (A) nonporous deposit (passive film) (B) deposit with minor narrow faults such as grain boundaries or minor fractures (C) deposit with discrete narrow pores (D) deposit with discrete pores wide enough to support a diffusive response (to the a.c. perturbation) within the deposit (E) deposit with partial pore blockage by a hydrated deposit (1) oxide capacitance (2) oxide resistance (3) bulk solution resistance (4) interfacial capacitance (5) polarization resistance (6) pore resistance (7) Warburg impedance (8) capacitance of a hydrated deposit.

The immittance analysis can be performed using different kinds of plots, including complex plane plots of X vs. R for impedance and B vs. G for admittance. These plots can also be denoted as Z" vs. Z and Y" vs. Y, or Im(Z) vs. Rc(Z), and Im( Y) vs. Re( Y). Another type of general analysis of immittance is based on network analysis utilizing logarithmic Bode plots of impedance or admittance modulus vs. frequency (e.g., log Y vs. logo)) and phase shift vs. frequency ( vs. log co). Other dependencies taking into account specific equivalent circuit behavior, for instance, due to diffusion of reactants in solution, film formation, or electrode porosity are considered in - electrochemical impedance spectroscopy. Refs. [i] Macdonald JR (1987) Impedance spectroscopy. Wiley, New York [ii] Jurczakowski R, Hitz C, Lasia A (2004) J Electroanal Chem 572 355... 

Generally, the impedance spectrum of an electrochemical system can be presented in Nyquist and Bode plots, which are representations of the impedance as a function of frequency. A Nyquist plot is displayed for the experimental data set Z(Zrei,Zim.,mi), (/ = 1,2,. ..,n) of n points measured at different frequencies, with each point representing the real and imaginary parts of the impedance (Zrei Zim4) at a particular frequency . 

A Bode plot is an alternative representation of the impedance. There are two types of Bode diagram, log z log (or z log ) and 9 log , describing the frequency dependencies of the modulus and phase, respectively. A Bode plot is normally depicted logarithmically over the measured frequency range because the same number of points is collected at each decade. Both plots usually start at a high frequency and end at a low frequency, which enables the initial resistor to be found more quickly. 

Figure 2.37 shows an example impedance spectrum of an electrochemical system with two time constants. Figure 2.37a, b, and c are the equivalent circuit, simulated Nyquist diagram, and Bode plot, respectively. 

The most common graphical representation of experimental impedance is a Nyquist plot (complex-plane diagram), which is more illustrative than a Bode plot. However, a Bode plot sometimes can provide additional information. 

In a situation where a charge transfer is also influenced by diffusion to and from the electrode, the Warburg impedance will be seen in the impedance plot. This circuit model presents a cell in which polarization is controlled by the combination of kinetic and diffusion processes. The equivalent circuit and the Nyquist and Bode plots for the system are all shown in Figure 2.40. It can be seen that the Warburg element is easily recognizable by a line at an angle of 45° in the lower frequency region. 

Experimental arcs in the spectrum are not always ideal semicircles, and this complicates parameter estimation. Nevertheless, there are still basic rules for estimating the initial values [8, 9], The key is to identify the region of the spectrum in which one element dominates and then estimate the value of the element in this region. For example, the resistor s impedance dominates the spectrum at a low frequency, while the impedance of a capacitor approaches zero at a high frequency and infinity at a low frequency also, individual resistors can be recognized based on the horizontal regions in a Bode plot. 

Electrochemical Impedence Spectroscopy (EIS) data are shown in Figure 33.15 (Bode plot) and Figure 33.16 (phase angle-frequency plot). All the samples exhibit... 

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 representation of impedance in terms of magnitude and phase angle as functions of frequency on a logarithmic scale are called Bode plots. ... 

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