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4.2.6  Band-pass systems

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 fcenter. Generally, the physically realizable band-pass transfer function can be found by substituting

s + (2fcenter)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 fo. (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 fcenter corresponds to a smaller relative bandwidth and steeper slopes of the band-pass cutoff.

Fig. 4.16  Preservation of operational
bandwidth of the band-pass transform

Fig. 4.17  Bode diagrams for a wide-band
band-pass system

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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