Correct Answer - Option 2 : – 3.735 only
Concept:
A breakaway point is a point on a real axis segment of the root locus between two real poles, where the two real closed-loop poles meet and diverge to become complex conjugates.
The breakaway/ break-in/ saddle point is calculated from the solution of:
\(\frac{{dK}}{{dS}} = 0\) .
Calculation:
Given:
\(G(s)=\frac{K(s+2)}{(s^2+2s+3)}\)
1 + G(s)H(s) = 0
\(1 + \frac{k(s+2)}{s^2+2s+3}=0\)
\(K=-\frac{(s^2+2s+3)}{(s+2)}\)
\(\frac{dK}{ds}=-\frac{d}{ds}\frac{(s^2+2s+3)}{(s+2)}=0\)
On solving we'll get:
s = -2 ± √3
s = -3.735, -0.265
Only -3.735 lies on the root locus.
Note:
Breakaway point is formed when an underdamped system is converted into an overdamped system, i.e. it is the point at which the system becomes critically stable.
