Question

The signal flow graph for a system is given below. The transfer function Y(s) / U(s) for this system given as

• A. $\frac{s+1}{5s^2+6s+2}$
• B. $\frac{s+1}{s^2+6s+2}$
• C. $\frac{s+1}{s^2+4s+2}$
• D. $\frac{1}{s^2+6s+2}$

Answer 1

The correct option is A $\frac{s+1}{5s^2+6s+2}$

Using Mason's gain formula
$\Delta =1-[-2s^{-1}-2s^{-2}-4-4s^{-1}]$
$\Delta =1+\frac{2}{s}+\frac{2}{s^2}+4+\frac{4}{s}$
$\Delta =\frac{5s^2+6s+2}{s^2}$
$P_1=s^{-2}=\frac{1}{s^2}$

$P_2=s^{-1}=\frac{1}{s}$
${\Delta }_1=1$
${\Delta }_2=1$
$\frac{Y\left(s\right)}{u\left(s\right)}=\frac{\Sigma P_k{\Delta }_k}{\Delta }=\frac{\frac{1}{s^2}\times 1+\frac{1}{s}\times 1}{\frac{5s^2+6s+2}{s^2}}$
$\frac{Y\left(s\right)}{u\left(s\right)}=\frac{s+1}{5s^2+6s+2}$