Bode diagram phase shift

Figure 4.5.49. Bode diagram (phase-shift vs. log f) of the impedance spectra measured in the potential range from OCP to —170mV during ORR on a Ag GDE in 25wt% KOH at IS C,...

This heuristic argument forms the basis of the Bode stability criterion(22,24) which states that a control system is unstable if its open-loop frequency response exhibits an AR greater than unity at the frequency for which the phase shift is —180°. This frequency is termed the cross-over frequency (coco) for reasons which become evident when using the Bode diagram (see Example 7.7). Thus if the open-loop AR is unity when i/r = —180°, then the closed-loop control system will oscillate with constant amplitude, i.e. it will be on the verge of instability. The greater the difference between the open-loop AR (< I) at coc and AR = 1, the more stable the closed-loop... 

The polar plot is an alternative to the Bode diagram for representing frequency response data and is the locus of all points occupied by the tip of a vector in the complex plane whose magnitude and direction are determined by the amplitude ratio and phase shift, respectively, as the frequency of the forcing function applied to the system is varied from zero to infinity. 

The Bode diagrams (in honour of H. W. Bode) constitute a convenient way to represent the frequency response characteristics of a system. As we can see from Eqs. (17.14a) and (17.14b), the amplitude ratio and the phase shift of the ultimate response of a system are functions of the frequency to. The Bode diagrams consist of a pair of plots showing ... 

A line of 45° versus the coordinate axis represents the Warburg impedance in the complex plain presentation (Nyquist plot. Figure 5.7a). The representation in the Bode diagram is shown in Figure 5.7b. The phase shift has a constant value of 45°, whereby the modulus of the impedance, IZI is linearly decreasing with increasing frequency. 

Figure 9.3 shows an example of how a model series circuit will look like in a Bode diagram. Most of the phase shift is within a frequency range of two decades, centered on the characteristic frequency 1.592 Hz where cp = 45°. Z, however, has hardly started to increase at the eharaeteristic frequency when passing toward lower frequencies. 

The frequency response (although called transfer function) is a conmum function in signal analysis and control engineering when the dynamic behavior of a system must be analyzed. Therefore, the input and output parameters of the system will be compared as a function of frequency. For example, when the system is stimulated with a harmonic input signal of a certain frequency, the system will answer with the same frequency, but with attenuated amplitude and a shifted phase. Since the amplitude attenuation and the phase shift are both functiOTis of the stimulation frequency, it is common to plot them in Bode diagrams, where the amplirnde response and the phase response are displayed separately over the frequency. 

The impedance of an electrode can also be represented in Bode diagram, which displays the modulus IZI and the phase shift as a function of the logarithm of the radial frequency co. Figure 5.27 shows the Bode diagram for the circuit drawn in Figure 5.26 using the same values of R, Ra and C. Equations (5.115) and (5.117) with (5.144) and (5.145) give the modulus and the phase shift. 

The graphs in which the amplitude ratio and phase shift are plotted as a function of the frequency o), are called Bode diagrams. In the Bode plot, log(AR) and (j) are shown as a function of first-order process, this means ... 

As was derived in chapter 9, the amplitude ratio for a dead-time process is 1.0 and the phase shift -0)6. The amplitude ratio for the process becomes then AR (second-order process) x AR(dead-time process). The phase shift of the process becomes then (second-order process) + dead-time process). Figure 32.3 shows the Bode diagram in which the logarithm of the amplitude ratio and the phase shift are plotted against the frequency O). For the amplitude ratio two asymptotes emerge, one for low frequencies a>- ) (static behaviour) en one for high frequencies 0)- °° (high-frequency behaviour). The values can easily be calculated from ... 

Figure 32.4 shows the Bode diagram of a PI eontroller with integral action = 10. Rather than plotting the amplitude ratio, the ratio between amphtude ratio and controller gain is plotted on the vertieal axis. As can be seen from Eqn. (32.25), plotting log(AR) versus log(comer frequency, where the asymptotes interseet is at 1/t . The phase shift ranges from -90° to 0°. 

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