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4.3.4  Non-minimum-phase shift

The n.p. lag in the feedback loop typically should be less than 1 rad at fc, or else to compensate for it, the Bode step would have to be very long. This limitation on the feedback feasibility can be critical for loops including a substantial delay. Transport delay, which causes a phase lag proportional to frequency, can be particularly large.

Let us consider two examples of audio systems with large transport delay.

   

Example 1. Since speaker systems are expensive and their frequency responses are difficult to equalize, since the sound wave reflections from the room walls change the frequency responses on the way to listener's ears, and since good quality inexpensive microphones are easily available, it would be commercially advantageous to make acoustical feedback from a microphone placed in the vicinity of one's ears. Is it possible to maintain good sound quality over the entire range of audio signals up to 15,000 Hz using a feedback system like that shown in Fig. 4.28? Probably not, since nobody does this. There must be a good reason. We might suspect that the reason is the excessive time of the signal propagation about the feedback loop. Let us check it out.


Fig. 4.28   This type of real time acoustical
feedback system is not feasible


Fig. 4.29  Sound suppression
feedback system

The speed of sound being 330 m/sec, and the distance l between the speakers and the microphone being 2 m, the transport delay is 6.6 msec. For the frequency fc = 15 × 4 = 60 kHz, the phase lag Bn(fc) = 2 × 60,000 × 0.0066  2500 rad which is 2500 times the allowable limit. Thus, real-time feedback in this system is not possible. (The response can be equalized by an adaptive system using plant identification.)

     
   

Example 2. A system for noise rejection is diagrammed in Fig. 4.29. The microphone is placed at the point where it is desired to keep the noise minimum. The sound from the speaker cancels the noise. The acoustic signal propagation between the speaker and the microphone introduces nonminimum phase lag into the feedback loop. To reduce the phase lag, the distance between the microphone and the compensating speaker should be short. The assembly of the microphone and the speaker is commonly mounted on a helmet.

When 30 dB of feedback is required up to 5 kHz, then frequency fd = 20 kHz. At this frequency, Bn should not exceed 1 rad. Therefore, the pure delay is (1/40,000)/2  0.000008 sec and the distance between the speaker and the microphone should be shorter than 2.6 mm.

Another source of non-minimum phase lag is the delay in the analog-to-digital conversion when the control is digital. The effects of this delay will be analyzed in the next chapter.

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