478 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response MethodThe magnitude of the closed-loop transfer function at v=v 1 is the ratio of the magni-
tudes of to. The phase angle of the closed-loop transfer function at v=v 1 is
the angle formed by the vectors to —that is f-u, shown in Figure 7–80(b). By
measuring the magnitude and phase angle at different frequency points, the closed-loop
frequency-response curve can be obtained.
Let us define the magnitude of the closed-loop frequency response as Mand the
phase angle as a,or
In the following, we shall find the constant-magnitude loci and constant-phase-angle
loci. Such loci are convenient in determining the closed-loop frequency response from
the polar plot or Nyquist plot.
Constant-Magnitude Loci (Mcircles). To obtain the constant-magnitude loci, let
us first note that G(jv) is a complex quantity and can be written as follows:
whereXandYare real quantities. Then Mis given by
andM^2 is
Hence
(7–22)
IfM=1, then from Equation (7–22), we obtain This is the equation of a
straight line parallel to the Yaxis and passing through the point A-^12 ,0B.
X=-^12.
X^2 A 1 - M^2 B-2M^2 X-M^2 +A 1 - M^2 BY^2 = 0
M^2 =
X^2 +Y^2
(1+X)^2 +Y^2
M=
∑X+jY∑
∑1+X+jY∑
G(jv)=X+jY
C(jv)
R(jv)
=Meja
PA
!
OA
!
PA
!
OA
!
(a) (b)G(s)ImReO
P- 1 +j 0
AG(jv)u fv 1f –u
+
Figure 7–80 –
(a) Unity-feedback
system;
(b) determination of
closed-loop
frequency response
from open-loop
frequency response.Openmirrors.com