N Circles

Constant phase-angle loci (N-circles):

Fig: 2 (a) G(jω) locus superimposed on a family of M circles; (b) G(jω) locus superimposed on a family of N circles; (c) closed-loop frequency-response curves.

The use of the M and N circles enables us to find the entire closed-loop frequency response from the open-loop frequency response G(jω) without calculating the magnitude and phase of the closed-loop transfer function at each frequency. The intersections of the G(jω) locus and the M circles and N circles gives the values of M and N at frequency points on the G(jω) locus.

The N circles are multivalued in the sense that the circle for α = α₁ and that for α = α₁ ± 180⁰ n (n = 1,2, ...) are the same. In using the N circles 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 α = 0°, and proceed to higher frequencies. The phase-angle curve must be continuous. Graphically, the intersections of the G(jω) locus and M circles give the values of M at the frequencies denoted on the G(jω) locus.

Thus, the constant M circle with the smallest radius that is tangent to the G(jω) locus gives the value of the resonant peak magnitude M_r. 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 j0 point) and, at the same time, there should be no intersections with the particular M circle and the G(jω) locus.

Figure 2(a) shows the G(jω) locus superimposed on a family of M circles. Figure 2(b) shows the G(jω) locus superimposed on a family of N circles. From these plots, it is possible to obtain the closed-loop frequency response by inspection. It is seen that the M = 1.1 circle intersects the G(jω) locus at frequency point ω = ω₁. This means that at this frequency the magnitude of the closed-loop transfer function is 1.1.

In Figure 2(a), the M = 2 circle is just tangent to the G(jω) locus. Thus, there is only one point on the G(jω) locus for which |C(jω)/R(jω)| is equal to 2. Figure 2(c) shows the closed loop frequency-response curve for the system. The upper curve is the M versus frequency ω curve, and the lower curve is the phase angle α versus frequency ω curve.

The resonant peak value is the value of M corresponding to the M circle of smallest radius that is tangent to the G(jω) locus. Thus, in the Nyquist diagram, the resonant peak value M_r and the resonant frequency ω_r can be found from the M-circle tangency to the G(jω) locus. (In the present example, M_r = 2 and ω_r = ω₄.)