Bode magnitude plots

Fig. 9.26 Bode magnitude plot of weighted sensitivity function for Example 9.4.

Fig. 9.32 Plant singular value Bode magnitude plot.

Fig. 9.34 Singular value Bode magnitude plot of Tyi Ui (jo ) when 7 = 0.13.

Fig. 9.35 Controller single value Bode magnitude plot C( jo ). ...

EXERCISE 5-B For the inverting op-amp below, find the upper 3 dB frequency for mid-band gains of -1, -10, -100, and -1000. Show the Bode magnitude plots for all of the gains on the same graph. 

The maximum scan rate allowed to obtain accurate measurement of the polarization resistance has been addressed in the literature (35). The governing principles are best understood through the concepts of impedance and the Bode magnitude plot for the simplified circuit shown in Fig. 5. Here the maximum applied frequency allowed to obtain Rs + Rp from the low frequency plateau can be approximated by... 

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 24 Bode magnitude plots for CeCl3-, LaCl3-, and YCl3-passivated 6061-T6 exposed to 0.5 M NaCl solution, (a) CeCl3-passivated samples exposed for 7 d (curve 1), 15 d (curve 2), and 60 d (curve 3). (b) LaCl3-passivated sample (curve 1), YCl3-passivated sample (curve 2), and bare alloy (curve 3). (From H. Shih, F. Mansfeld. p. 180, ASTM STP 1134, ASTM, Philadelphia, PA (1992).)...

Figure 35 Bode magnitude plot for a leads unconnected measurement with a commercial electrochemical impedance system. Analysis of the data enable an estimate of the potentiostat input impedance (2 X 109 Q) and the stray capacitance (2 X 10 1° F).

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. 

Figure 2.39. Graphic presentations of the Randles cell a equivalent circuit, b Nyquist plot, c Bode magnitude plot, d Bode phase plot (Re/ = 20 2, Rct = 80 Q, CdI = 0.001 F)...

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... 

The discrepancy is not evident, however, in the Bode magnitude plot presented in Figure 20.3(a). In fact, the Bode magnitude representation is singularly incapable... 

Figure 5. Construction of the Bode magnitude plot for the circuit in Fig. 4(a) using Eq. (26). The solid line is a sum of all three contributions.

Armstrong and Firman ° analyzed a mechanism that included two successive electron-transfer reactions. A general approach to multistep mechanisms involving soluble species in semi-infinite diffusion was presented recently hy Harrington. It allows determination of the number of breakpoint frequencies on the Bode magnitude plot for an arbitrary mechanism and, in consequence, for the determination of the reaction mechanism and kinetics. 

Analysis of the data in Fig. 7 shows that, for the Fe/H2S04 system, Rp is about 98 f2 cm, and Cdl = 1/ (27t)(14 Hz) (98 f2 cm ) = 116 pF cm . TheRpvalue is similar to that determined by analysis of the Bode plot and by the computer fitting. However, the Cdl value is about a third of that determined from the Bode plot. The discrepancy is caused by the fact that the behavior of the interface is not exactly that of a perfect Randles circuit. The slope of the Bode magnitude plot is about —0.8 rather than —1, and the Nyquist plot is not a perfect semicircle, as the maximum imaginary impedance is less than twice the polarization resistance. So it is inappropriate to treat the data as if they were generated by a perfect RC circuit. 

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