Root Locus Analysis 177

4

K = 1~1 I(SI + pJI

= 1(-1 + j1.333)( -1 + j1.333 + 2)(-1 + j1.333 + 1 + j.J3)( -1 + j1.333 + 1 - j.J3) I = 1.666 x 1.666 x 3.06 x 0.399

= 3.39

At this value of K, the other two closed loop poles can be found from the characteristic equation.

The characteristic equation is

s4 + 4s3 + 8s2 + 8s + 3.39 = 0

The two complex poles are s = - 1 ±j 1.333

.. The factor containing these poles i~

[(s + 1)2 + 1.777]

s2 + 2s + 2.777

Dividing the characteristic equation by this factor, we get the other factor due to the other two poles. The factor is

s2 + 2s + 1.223

The roots of this factor are

s=-l ±j 0.472

The closed loop poles with the required damping factor of 8 = 0.6, are obtained with K = 3.39. At this value of K, the closed loop poles are,

s = - 1 ±j 1.333, - 1 ±j 0.472

Note: The Examples 5.2 and 5.3 have the same real poles at s = 0 and s = - 2. The complex poles are different. If the real part of complex poles is midway between the real poles, the root locus will have one breakaway point on real axis and two complex breakaway points. If real part is not midway between the real roots there is only one breakaway point. In addition, if the real part of the complex roots is equal to the imaginary part, the root locus will be as shown in Fig. 5.18.

K Fig. 5.18 Root locus of G(s) H(s) = 2

s(s + 2)(s + 2s + 2)