Breakpoint frequency

The high-frequency asymptote intersects the L = 0 line at [Pg.429]

Thus the L curve can be easily sketched by drawing a line with slope of — 20 dB/decade from the breakpoint frequency on the L = 0 line. Notice also that the phase angle is -45° at the breakpoint frequency, which is the reciprocal of the time constant. The L curve has a value of — 3 dB at the breakpoint frequency, as shown in Fig, 12.15. 

Figure 12.19 shows the Bode plots for several values of damping coefficient (. The breakpoint frequency is the reciprocal of the time constant. The high-frequency... 

In most systems, the closedloop servo log modulus curves move out to higher frequencies as the gain of the feedback controller is increased. This is desirable since it means a faster closedloop system. Remember, the breakpoint frequency is the reciprocal of the closedloop time constant. 

If a transfer-function model is desired, approximate transfer functions can be fitted to the experimental curves. First the log modulus Bode plot is used. The low-frequency asymptote gives the steadystate gain. The time constants can be found from the breakpoint frequency and the slope of the high-frequency asymptote. The damping coefficient can be found from the resonant peak. 

Figure 43 Equivalent circuit model for a degraded polymer coating on a metal substrate and the corresponding EIS data presented in Bode plot format. The breakpoint frequency is indicated as/b in the plot. (From C. H. Tsai, F. Mansfeld. Corrosion 49, 726 (1993).)...

The breakpoint frequency analysis assumes that the pore resistance Rpo, the polarization resistance of the defect, Rp, and the double layer capacitance in the defects, Cdi, scale with the area of substrate-coating delamination, Ad by... 

The practical consequence of this analysis is that the breakpoint frequency is directly proportional to the delaminated area, and can be monitored with a fast... 

Despite inherent assumptions and objections to the breakpoint frequency method, experiments show that there is general agreement between/, and delaminated area (119). Figure 45 shows the relationship between/, and the estimated delamination area for opaque and transparent epoxy coatings on steel. The figure supports the positive relationship between /b and delaminated area indicated by Eq. (25), but the scatter in the relationship is relatively large, suggesting that sample-to-sample variation and frequency dispersion in the EIS data may affect the determination of fb for any specific amount of delamination. 

An example of the use of EIS measurements to make long-term corrosion predictions for polymer coated metals is shown in Fig. 46 (110). In this plot, the ASTM D610 and D714 visual rankings of corrosion damage after 550 days of exposure in simulated seawater are plotted against a protection index, < (/), determined after only 10 days of exposure. The damage protection index is determined from the breakpoint frequency, /, as... 

Figure 45 Relationship between the breakpoint frequency and the estimated active area for opaque and transparent epoxy coatings on steel. (From H. P. Hack, J. R- Scully. J. Electrochem. Soc. 138, 33 (1991).)...

There are two limits of the impedance (O = 0, Z = / and ro —> o°, Z = 0. The corresponding complex plane and Bode plots for the same values of R and C elements as those used in the series R-C model above, are shown in Fig. 3. The Nyquist plot shows a semicircle of radius RH with the center on the real axis and the frequency at the semicircle maximum equal to (0= RC. The circuit s characteristic breakpoint frequency (the inverse of the characteristic time constant), as observed in the impedance Bode graph, is the same as for the series and the parallel R-C circuits. The complex plane admittance plot represents a straight line parallel to the imaginary axis [Fig. 3(c)], which is similar to the impedance complex plane plot for the series R-C connection. 

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. 

These curves are shown in Fig. 10.16. The high-frequency asymptote has a slope of +20 dB/decade. The breakpoint frequency is I/t. The phase angle goes from zero... 

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