Bode Representation

The magnitude of the impedance of the blocking circuit of Table 16.1a can be expressed as 

The magnitude tends toward Re as frequency tends toward oo and toward + R as frequency tends toward zero. The transition between low frequency and high frequency asymptotes has a slope of —1 on a log-log scale. 

Following equation (4.32), the phase angle for the blocking configuration can be expressed as 

The phase angle tends toward —90°s at low frequencies and toward zero at high frequencies. TTie phase angle at the characteristic angular frequency coc = ReC) is equal to —45° (see equation (16.11)). 

---

Fig. 6-14. Potentiostatic EHD impedance plots, in Bode representation (reduced amplitude A(pSc /3)/A(0) and phase shift, versus dimensionless frequency pSc / ) for the oxidation of hydroquinone on a 360 nm thick poly(TV-ethylcarbazole) film at E - 0.7 V (diffusion plateau).

Example 4.3 Bode Representation of Elemental Circuits Derive an expression for the magnitude and phase angle for the circuit elements shorm in Figure 4.1. 

The corresponding Bode representation of the impedance response is shown in Figure 4.8 as a function of frequency / in units of Hz and frequency a in units of s When plotted as a function of frequency o) in units of s the phase angle reaches an inflection point (—45 degrees) at a characteristic frequency u>c =... 

Figure 4.8 Bode representation of the impedance response for a 10 O resistor in parallel with a 0.1 F capacitor. The characteristic time constant for the element is 1 s.

Figure 15.2 Dimensionless function f 0) in Bode representation a) modulus versus the dimensionless frequency p and b) phase shift versus the dimensionless frequency p.

Figure 16.2 Bode representation of impedance data for Rg = 10 Qcm, R — 100 flcm, and C = 20 F/cm. The blocking system of Table 16.1(a) is represented by dashed lines, and the reactive system of Table 16.1(b) is represented by solid lines. Characteristic frequencies are noted as /rq = (27tRC) and fc = 2nRgC) a) magnitude and b) phase angle.

The popularity of the Bode representation stems from its utility in circuits analysis. The phase angle plots are sensitive to system parameters and, therefore, provide a good means of comparing model to experiment. The modulus is much less... 

For electrochemical systems, however, the Bode representation has drawbacks. The influence of electroljde resistance confoimds the use of phase angle plots such as shown in Figure 16.2(b) to estimate characteristic frequencies. In addition. Figure 16.2(b) shows that the current and potential are in phase at high frequencies wherecis, at high frequencies, the current and surface potential are exactly out of phase. This result is seen because, at high frequencies, the impedance of the surface tends toward zero, and the Ohmic resistance dominates the impedance response. The electrolyte resistance, then, obscures the behavior of the electrode surface in the phase angle plots. 

If an accurate estimate for electrolyte resistance Re at is available, a modified Bode representation is possible as... 

As discussed in Section 16.1, impedance data are often represented in complex-impedance-plane or Nyquist formats accompanied with Bode representations in... 

Figure 17.2 Traditional representation of impedance data for the Randles circuit presented as Figure 17.1(a) with a as a parameter, a) complex-impedance-plane or Nyquist representation (symbols are used to designate decades of frequency.) b) Bode representation of the magnitude of the impedance and c) Bode representation of the phase angle. (Taken from Orazem et al. ° and reproduced with permission of The Electrochemical Society.)...

The Bode representation is presented in Figures 17.8(b) and (b) for modulus and phase angle, respectively. The traditional values of modulus and phase angle are... 

The sensitivity of the impedance representation, presented in Figures 20.4(a) and (b) for real and imaginary parts respectively, is somewhat comparable to that seen for the Bode representation. The real part of the impedance is as insensitive to model quality as is the Bode modulus, and the imaginary impedance plots show a discrepancy between model and experiment at intermediate frequencies. 

Figure 11.18 Impedance data measured for polypyrrole as function of the potential. The polymer film was prepared by anodic oxidation in a perchlorate electrolyte. Bode representation (A) impedance versus frequency and (B) phase angle versus frequency.

The complex impedance data involves the interplay of three variables, the imaginary component of the impedance, the real component of the impedance, Zreai, and the phase angle, common types of representation for impedance data are, the Nyquist and the Bode representations. Nevertheless, these have become the most widely used graphical representations of impedance data. 

Figure 10.2 Electrochemical impedance diagrams (Bode representation) obtained for the FI and F2 samples after 24 h immersion in 0.1 M NaCl solution.

Fig. 20.26 Alternating current impedance data for a polypyrrole tosylate film in 1 M KCl in (a) complex plane format and (b) Bode representation. (Reproduced with permission from Ref. 96.)...

Figure 7.28 illustrates the complex-plane presentation of simulated data corresponding to the model circuit in Fig. 7.256 when R = 10 O, = 100 kfl, Cdi = 40 jiF, and the exponent n of the Warburg component = 0.4. Figure 7.29 shows the same data in a Bode representation. 

Bode representation of the same data illustrated in Fig. 7.26 in complex-plane format. 

The third circuit (Fig. 7.25c) has been proposed to describe EIS results containing two relaxation time constants. Such behavior is commonly encountered for corrosion under coatings or under scale, for corrosion-inhibited systems, or even for localized corrosion. The meaning of the circuit elements in Fig. 7.25c will vary with the physical systems represented, but their significance has been validated through additional measurements and calculations. Figure 7.30 illustrates the model circuit in Fig. 7.25c with simulated data obtained with Rs = 10 fi, = 40 kfl, and Qi = 40 piF with exponent ra = 1, i 2 = 20 kfi, and Q2 = 20 p-F with exponent n = 1. Figure 7.31 is a Bode representation of the same data illustrated in Fig. 7.30 in complex-plane format. 

For sensor applications, the data presentation is usually as a complex plane plot, with the real part (Z ) plotted on the x-axis and the imaginary part (Z") plotted on the y-axis, which is also referred to as a Nyquist plot . An alternative representation is of impedance magnitude and phase angle as a function of the logarithm of frequency—this Bode representation is most used in corrosion studies. 

Nyquist and Bode representation of complex impedance data for ideal electrical circuits... 

---