# Consider the system with $G(s)=\frac{K(s+2)}{(s^2+2s+3)}$ and H(s) = 1. The breakaway point(s) of the root loci is / are at

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Consider the system with $G(s)=\frac{K(s+2)}{(s^2+2s+3)}$  and H(s) = 1. The breakaway point(s) of the root loci is / are at
1. – 0.265 only
2. – 3.735 only
3. – 3.735 and -0.265
4. There is no breakaway point

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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.