Section 7–8 / Closed-Loop Frequency Response of Unity-Feedback Systems 481
3
1
1
2
2
X
Y
a= 20 °
a= 30 °
a= 40 °
a= 100 °
60 °
120 °
80 °
–120°
a=– 100 ° – 80 °
a=– 40 °
a=– 30 °
a=– 20 °
Figure 7–82
A family of constant
Ncircles.
of the G(jv)locus and the Mcircles and Ncircles give the values of MandNat fre-
quency points on the G(jv)locus.
The Ncircles are multivalued in the sense that the circle for a=a 1 and that for
a=a 1 ;180°n (n=1, 2,p)are the same. In using the Ncircles for the determination
of the phase angle of closed-loop systems, we must interpret the proper value of a.To
avoid any error, start at zero frequency, which corresponds to a=0°, and proceed to
higher frequencies. The phase-angle curve must be continuous.
Graphically, the intersections of the G(jv)locus and Mcircles give the values of M
at the frequencies denoted on the G(jv)locus. Thus, the constant Mcircle with the
smallest radius that is tangent to the G(jv)locus gives the value of the resonant peak
magnitudeMr. If it is desired to keep the resonant peak value less than a certain value,
then the system should not enclose the critical point (–1+j0point)and, at the same
time, there should be no intersections with the particular Mcircle and the G(jv)locus.
Figure 7–83(a) shows the G(jv)locus superimposed on a family of Mcircles. Figure
7–83(b) shows the G(jv)locus superimposed on a family of Ncircles. From these plots,
it is possible to obtain the closed-loop frequency response by inspection. It is seen that
theM=1.1circle intersects the G(jv)locus at frequency point v=v 1. This means
that at this frequency the magnitude of the closed-loop transfer function is 1.1. In Fig-
ure 7–83(a), the M=2circle is just tangent to the G(jv)locus. Thus, there is only one
point on the G(jv)locus for which @C(jv)/R(jv)@is equal to 2. Figure 7–83(c) shows the
closed-loop frequency-response curve for the system. The upper curve is the M-versus-
frequency vcurve, and the lower curve is the phase angle a-versus-frequency vcurve.
The resonant peak value is the value of Mcorresponding to the Mcircle of small-
est radius that is tangent to the G(jv)locus. Thus, in the Nyquist diagram, the resonant
