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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 A_{o}= 20 log > |T| must be maximized.
To find this response, Bode made use of the function (jf) defined by (3.15). The function
A_{o} + 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 A_{o} that the constant-slope response has at f = 0.5. In other words, the functional bandwidth of A_{o }dB 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 A_{o} 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):
A_{o} 12(1 - y)(log_{2 }f_{b} + 1) . | (4.4) |
In the common case of 30° stability margin, i.e., y = 1/6,
A_{o} 10 (log_{2 }f_{b} + 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 f_{b} is limited by the system dynamics to 6.4 kHz. From (4.5) the available feedback is 60 dB. |
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