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In some systems, for example in vibration suppression systems, the frequency band of functional feedback does not include dc. The band can be viewed as centered at some finite frequency f_{center}. Generally, the physically realizable band-pass transfer function can be found by substituting
s + (2f_{center})^{2} / s | (4.6) |
for s in a low-pass prototype transfer function [2]. The loop response obtained with (4.6) from the low-pass Bode optimal cutoff is shown in Fig. 4.15. Notice that in Fig. 4.15(b), two critical points for the Nyquist diagram to avoid are shown: -180° and 180°, each of the points being a mapping of the T-plane point -1 onto the L-plane.
(a) | (b) |
Fig. 4.15 Band-pass optimal Bode cutoff: Bode diagram (a) and Nyquist diagram (b)
The absolute bandwidth f of the available feedback is an invariant of the transform (4.6) as is illustrated in Fig. 4.16, and it equals the bandwidth of the low-frequency prototype f_{o}. (The bandwidths of the three responses in the picture do not look equal because the frequency scale is logarithmic.) It is seen that a higher f_{center} corresponds to a smaller relative bandwidth and steeper slopes of the band-pass cutoff.
Fig. 4.16 Preservation of operational |
Fig. 4.17 Bode diagrams for a wide-band |
When the relative functional bandwidth is fairly wide, more than 2 octaves, the steepness of the low-frequency slope has only a small effect on the available feedback since the absolute bandwidth of the entire low-frequency roll-off is rather small. This case is shown in Fig. 4.17.
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