Section 6–2 / Root-Locus Plots 271
The characteristic equation for this closed-loop system is obtained by setting the
denominator of the right-hand side of Equation (6–1) equal to zero. That is,
or
(6–2)
Here we assume that G(s)H(s)is a ratio of polynomials in s.[It is noted that we
can extend the analysis to the case when G(s)H(s)involves the transport lag e–Ts.]
SinceG(s)H(s)is a complex quantity, Equation (6–2) can be split into two equations
by equating the angles and magnitudes of both sides, respectively, to obtain the
following:
Angle condition:
(6–3)
Magnitude condition:
(6–4)
The values of sthat fulfill both the angle and magnitude conditions are the roots of
the characteristic equation, or the closed-loop poles. A locus of the points in the
complex plane satisfying the angle condition alone is the root locus. The roots of
the characteristic equation (the closed-loop poles) corresponding to a given value
of the gain can be determined from the magnitude condition. The details of applying
the angle and magnitude conditions to obtain the closed-loop poles are presented
later in this section.
In many cases,G(s)H(s)involves a gain parameter K,and the characteristic equa-
tion may be written as
Then the root loci for the system are the loci of the closed-loop poles as the gain Kis
varied from zero to infinity.
Note that to begin sketching the root loci of a system by the root-locus method we
must know the location of the poles and zeros of G(s)H(s).Remember that the angles
of the complex quantities originating from the open-loop poles and open-loop zeros to
the test point sare measured in the counterclockwise direction. For example, if G(s)H(s)
is given by
G(s)H(s)=
KAs+z 1 B
As+p 1 BAs+p 2 BAs+p 3 BAs+p 4 B
1 +
KAs+z 1 BAs+z 2 BpAs+zmB
As+p 1 BAs+p 2 BpAs+pnB
= 0
∑G(s)H(s)∑= 1
/G(s)H(s)=; 180 °(2k+1) (k=0, 1, 2,p)