Page 29 - Servo Motors and Industrial Control Theory -

2.3 Root Locus Method 21

Fig. 2.2 Root Locus for a –1 0
second order system x: =
–1 0
y: =
–1 –1
–1 1
–1 1.73
–1 –1.73
–1 3
–3
4

2

y
z –4 –2 0 2 4

–2

–4
x

Fig. 2.3 Block diagram of
a third order system with x – K 1 y
gain K – s(0.5s + 1)(s + 1)

With the characteristic equation of

0.5 s +· 3 1.5s ++ k : 0= (2.9)
2
s
The MathCAD Polyroots expression can be used to calculate the roots of character-
istic equation for various K. The root locus for this system is shown in Fig. 2.4. At
K = 0, there are three negative real roots. As K is increased the two real roots move
towards each other and the third real root moves towards infinity. The two real roots
break away from the real axis and become complex. As k is increased, the real part
of complex roots becomes positive showing that the system becomes unstable. The
root locus in this case enter the right-hand side of the s-plane.