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4.2.5  Bode cutoff

When the frequency components of the expected disturbances within the functional band f  < 1 have the same amplitudes, the loop gain within the functional band should be constant as shown in Fig. 4.14. The value Ao= 20 log > |T| must be maximized.

To find this response, Bode made use of the function (jf) defined by (3.15). The function

Ao + 2(1 - y)(jf) (4.3)

has the high-frequency asymptote with the slope 2(1 - y)n dB/oct. It replaces the constant-slope response in Fig. 4.2, as shown in Fig. 4.14(a). It is seen from the picture (and from the formulas) that this loop gain at f = 1 equals the value Ao that the constant-slope response has at f = 0.5. In other words, the functional bandwidth of AdB feedback in the Bode optimal cutoff becomes extended by one octave.

 
 
(a) (b)   (c)

Fig. 4.14   Bode optimal cutoff, (a) Bode diagram, (b) Nyquist diagram on L-plane,
(c) Nyquist diagram on T-plane

From the triangle shown in Fig. 4.14(a), the available loop gain Ao is the product of the slope 12(1 - y) dB/oct and the feedback bandwidth in octaves plus 1 (the extra octave is that from 0.5 to 1):

Ao 12(1 - y)(logfb + 1) . (4.4)

In the common case of 30° stability margin, i.e., y = 1/6,

Ao 10 (logfb + 1) .

(4.5)

This simple formula is quite useful for rapid estimation of the available feedback.

   

Example 1. The prescribed stability margins are 30° and 10 dB, the feedback is required to be constant over the bandwidth of [0,200] Hz, and the crossover frequency fb is limited by the system dynamics to 6.4 kHz. From (4.5) the available feedback is 60 dB.

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