EFFECT OF POLES AND ZEROS ON FREQUENCY RESPONSE
We have seen (7.2,9) that the transfer function of a linear/invariant system is a ratio of two polynomials in 5, the denominator being the characteristic polynomial. The roots of the characteristic equation are the poles of the transfer function, and the roots of the numerator polynomial are its zeros. Whenever a pair of complex poles or zeros lies close to the imaginary axis, a characteristic peak or valley occurs in the amplitude of the frequency-response curve together with a rapid change of phase angle at the corresponding value of u>. Several examples of this phenomenon are to be seen in the frequency response curves in Figs. 7.14 to 7.18. The reason for this behavior is readily appreciated by putting (7.2,9) in the following form:
G ~ Zi) • (s – z2) ••• (s – zj G – A,) • (s – A2) ••• (x – AJ
where the A, are the characteristic roots (poles) and the г, are the zeros of G(s). Let
(s – zk) = pke’ak (s – k) = гкефк
where p, r, a, /3 are the distances and angles shown in Fig. 7.8b for a point s = іш on the imaginary axis. Then
Figure 7.10 Frequency-response curves—first-order system.
When the singularity is close to the axis, with imaginary coordinate со’ as illustrated for point S on Fig. 1.8b, we see that as со increases through со’, a sharp minimum occurs in p or r, as the case may be, and the angle a or /3 increases rapidly through approximately 180°. Thus we have the following cases:
1. For a pole, in the left half-plane, there results a peak in |g| and a reduction in (p of about 180°.
2. For a zero in the left half-plane, there is a valley in g and an increase in <p of about 180°.
3. For a zero in the right half-plane, there is a valley in |g| and a decrease in <p of about 180°.