EFFECT OF GAIN

The effect of gain is well illustrated by the familiar public-address acoustic system, in which “whistling” or oscillation occurs when the volume control is set too high. As a model for this case, consider the transfer function Ke~Ts, a simple gain with time delay.

If the system input were a single short pulse (of duration < < T) as in Fig. 11.146, the signals in the e and у channels would be as shown, a sequence of alternating pulses at time interval T, all of the same width, but with magnitudes 1, if, A2 …. It is clear that if А < 1 the pulses form a dimin­ishing sequence that ultimately dies out, and that if A > 1, there is an in­creasing series which is a divergent, or unstable situation. This would correspond in the case of the P. A. system to an acoustic pulse travelling from the loudspeaker to the microphone and arriving there stronger than the one originally fed in.

EFFECT OF PHASE LAG

Suppose now that the input is a series of pulses, equally spaced but alternating in sign. If the time lag T is such that the feedback pulses fall in the “empty spaces” between the input pulses there is no interference of the pulses, each input can be considered individually, and the criterion for divergence is the same as above, i. e. A > 1. If, however, the time lag is such that each return pulse coincides exactly with the next input, as illus­trated by the dotted pulses in Fig. 11.146, then the error signal and the ouput form the sequences

e: 1 -(1+A) (1+ A + A2) -(1+ A + A2 + A3)— у: A – A(l + A) K(l + K + Kz) ••• A(1 + A +A2 + —)

The output is seen to contain the sum of a geometric progression of factor A, which is divergent if А > 1 and converges to the limit (1 — A)-1 if A < 1. Thus in the case of the alternating input we find again that the stability criterion is A < I. This is clearly the “worst” phase lag for a pulse train since each return pulse arrives at such a time that it provides the maximum reinforcement to the next input.