Question

The loop transfer function of a feedback control system is given by,
$G\left(s\right)H\left(s\right)=\frac{K(s+6)}{s(s+4)}$
The range of value of K for underdamped system will be

• A. $-4<K<1.07$
• B. $1.07<K<14.93$
• C. $K>14.93$
• D. $K<-4$

Answer 1

The correct option is B $1.07<K<14.93$

For this given system,
Break-away/Break-in points are

$K=\frac{-s^2-4s}{s+6}$

$\frac{dK}{ds}=0$

$\frac{(s+6)(-2s-4)+(s^2+4s)}{(s+6{)}^2}=0$

$(s+6)(-2s-4)+(s^2+4s)=0$
$s^2+12s+24=0$
$s=-2.5ands=-9.5$
Now value of 'K' at breakaway points indicates critical damping so,

${\left|K_{min}\frac{(s+6)}{s(s+4)}\right|}_{s=-2.5}=1$

${\left|K_{max}\frac{(s+6)}{s(s+4)}\right|}_{s=-9.5}=1$

$K_{min}=\frac{2.5\times 1.5}{3.5}=1.07$

$K_{max}=\frac{-s(s+4)}{s+6}{|}_{s=-9.5}=\frac{+9.5\times 5.5}{3.5}=14.93$

So range for underdamped system is
1.07 < K < 14.93