Modern Control Engineering

302 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method

of the G(s)H(s)plane, as shown in Figure 6–28. [Note that the complex plane employed

here is not the splane, but the G(s)H(s)plane.]

The root loci and constant-gain loci in the splane are conformal mappings of the loci

of and of constant in the G(s)H(s)plane.

Since the constant-phase and constant-gain loci in the G(s)H(s)plane are orthog-

onal, the root loci and constant-gain loci in the splane are orthogonal. Figure 6–29(a)

shows the root loci and constant-gain loci for the following system:

G(s)=

K(s+2)

s^2 +2s+ 3

, H(s)= 1

/G(s)H(s)=; 180 °(2k+1) ∑G(s)H(s)∑=

Re

Im

0

G(s)H(s) Plane

Re

Im

0

G(s)H(s) Plane

|G(s)H(s)|= constant

G(s)H(s)

= 180 ° (2k+ 1)

Figure 6–28

Plots of constant-

gain and constant-

phase loci in the

G(s)H(s)plane.

(a) (b)

s

jv

0

K= 6

K= 6

j 4

j 6

• j 4

K= 1

K= 2

K= 1

• 6 – 4246

K= 10

j (^2) K= 0.3

• j 2


• j 6

K= 0.3 K= 0.3

– (^2) s
jv
0
j 2
j 3

• j 2


• 3 – 212

j 1

• j 1


• j 3


• 1

B

C

A

Figure 6–29

Plots of root loci and constant-gain loci. (a) System with G(s)=K(s+2)/As^2 +2s+3B,

H(s)=1;(b) system with G(s)=K/Cs(s+1)(s+2)D,H(s)=1.

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