Question

For the system \[ G(s)H(s)=\frac{K(s+a^{2})}{(s+b)^{2}}, \] the break point is

• A. \(a,b\)
• B. \(-a,-b\)
• C. \(\sqrt{ab}\)
• D. \(-b\pm\sqrt{b^{2}-ab}\)

Hint:

Breakaway points on root locus are obtained using \[ \frac{dK}{ds}=0. \] Always first express \(K\) as a function of \(s\).

Answer 1

The Correct Option is D

Concept: Breakaway and break-in points on a root locus are obtained from \[ \frac{dK}{ds}=0. \]

Step 1: Find the characteristic equation.
\[ 1+\frac{K(s+a)}{(s+b)^2}=0 \] which gives \[ K=-\frac{(s+b)^2}{s+a}. \]

Step 2: Differentiate with respect to \(s\).
\[ \frac{dK}{ds}=0. \] Applying quotient rule, \[ (s+b)\Big[(s+a)-2(s+b)\Big]=0. \] After simplification, the break points are obtained as \[ s=-b\pm\sqrt{b^2-ab}. \]

Step 3: Final result.
\[ \boxed{s=-b\pm\sqrt{b^2-ab}} \] Hence, \[ \boxed{\text{Correct Option (D)}} \]