Bode angle plot

Fig. 13L Comparison of a) complex-plane impedance, (b) complex-plane admittance, (c) complex-plane capacitance and (d) Bode magnitude and Bode angle plots for the same equivalent circuit. C =20 uF R = 10 kFl R = I kO.. Values of li) (Rad/s) at which some of the points were calculated are shown.

It should be remembered that the curves shown in Fig. 13L are all simulated and therefore "ideal" in the sense that they follow exactly the equations derived for the given equivalent circuit. In practice, the points are always scattered as a result of experimental error. Also, the frequency range over which reliable data can be collected does not necessarily correspond to the time constant which one wishes to measure. For the case shown in Fig. 13L(a) the semicircle can be constructed from measurements in the range of 1 > o) > 20. In Fig. 13N(b) one would have to use data in the range of about 10 > to 200 to evaluate the numerical values of the circuit elements. From the Bode magnitude plots, can be evaluated from high-frequency measurements (to 100), while R can be obtained from low frequency data (to < 1). The capacitance can be obtained approximately as = l/co Z at the inflection point (which coincides with the maximum on the Bode angle plot), but this would be correct only if (p - 90 that is, if the... 

Fig. 5.14 Theoretical phase-angle plots simulating the experimental Bode plots during deactivation using Equations 5.3 and 5.8. Note that indicates Q for these cases is equal to the value noted divided by (a + 1). 

Figure 5 Nyquist, Bode magnitude and Bode phase angle plots for hypothetical corroding interfaces with Rp = 10, 100, or 1,000 ohms, Cd, = 100 tF, and Rs = 10 ohms using the electrical equivalent circuit model of Fig. 3a.

Figure 21 Common graphical representations of EIS data in corrosion studies, (a) Complex plane plot, (b) Bode magnitude and Bode phase angle plots. (From Gamry, EIS Manual, pp. 2-3, 2-5.)...

On a complex plane plot, a CPE exhibits a straight line whose angle is n/2 a (1 < a < -1) with respect to the real axis. In a Bode magnitude plot a straight line response like that of a capacitor is obtained. The slope deviates from an ideal value of -1 as a decreases below 1. 

Figure 30 Bode magnitude and phase angle plots of a 5 im thick sulfuric acid anodized film subject to sealing in hot water. Measurements were made on an interrupted basis in cold water. Immersion times are (1) 0 s, (2) 30 s, (3) 60 s, (4) 90 s, (5) 120 s, (6) 180 s, (7) 300 s, (From J. L. Dawson, G. E. Thompson, M. B. H. Ahmadun. p. 255. ASTM STP 1188. ASTM, Philadelphia, PA (1993).)...

In this expression, E and / are the magnitude of the potential and current noise at any given frequency, /. RRe and Rlm are the real and imaginary components of Rsn. Plots of spectral noise impedance versus frequency resemble Bode magnitude plots of EIS data as shown in Fig. 58. Meaningful phase angle information is not usually obtained, as this is not preserved by the MEM transform, and data are usually of insufficient quality for accurate phase information to be obtained from the EFT. 

The Bode plot contains a magnitude plot and a phase angle plot. For a Randles cell, the values of the electrolyte resistance and the sum of the electrolyte resistance and the polarization resistance can easily be identified from the horizontal line in the magnitude plot. At high or low frequencies, the phase angles are close to 0°. Otherwise, at intermediate frequencies, the phase angles fall between 0° and 90°. 

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. 

The corrected phase angle plots yield valuable information concerning the existence of CPE behavior that is obscured in the traditional Bode presentation. 

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