Question

The signal flow graph of a system is shown in the figure. The transfer function $\frac{C\left(s\right)}{R\left(s\right)}$ of the system is

• A. $\frac{6}{s^2+29s+6}$
• B. $\frac{6s}{s^2+29s+6}$
• C. $\frac{s(s+2)}{s^2+29s+6}$
• D. $\frac{s(s+27)}{s^2+29s+6}$

Answer 1

The correct option is D $\frac{s(s+27)}{s^2+29s+6}$

$P_1=1$
${\Delta }_1=1+\frac{3}{s}+\frac{24}{s}=\frac{s+27}{s}$
$G\left(s\right)=\frac{P_1{\Delta }_1}{1-\left(\text{loop gain}\right)+\text{pair of non-touching loops}}$
$L_1=\frac{-3}{s},L_2=\frac{-24}{s},L_3=\frac{-2}{s}$
$L_1$ and $L_3$ are non-touching
$\therefore G\left(s\right)=\frac{\frac{(s+27)}{s}}{1-(\frac{-3}{s}-\frac{24}{s}-\frac{2}{s})+\frac{-2}{s}\times \frac{-3}{s}}$
$=\frac{\frac{(s+27)}{s}}{1+\frac{29}{s}+\frac{6}{s^2}}$
$\frac{C\left(s\right)}{R\left(s\right)}=\frac{s(s+27)}{s^2+29s+6}$