Page 520 - Mechanical Engineers' Handbook (Volume 2)

3 Hall Chart 511

Figure 10 Family of constant-M circles.

tan tan
Y
Y
1
1
X 1 X (3)
Taking the tangent on both sides of Eq. (3) and rearranging yields
Y
X 1 2 2
2
1
1
1
2 2N 4 2N (4)
where N tan .
Equation (4) represents a family of circles with center at ( 1/2, 1/2N) and with radius
2
1/4 1/(2N). The constant-phase loci are shown in Fig. 11.
The use of constant-magnitude and constant-phase circles enables one to find the entire
closed-loop frequency response from the open-loop frequency response G( j ) without cal-
culating the magnitude and phase of the closed-loop transfer function at each frequency. The
intersections of the G( j ) locus and the M circles and N circles give the values of M and
N at frequency points on the G( j ) locus.
3.3 Closed-Loop Frequency Response for Nonunity Feedback Systems
The constant-M and constant-N circles are limited to closed-loop systems with unity negative
feedback, whose transfer function is given by Eq. (1). When a system has nonunity feedback,
the closed-loop transfer function is
C(s) G(s)
(5)
R(s) 1 G(s)H(s)