Question

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

• 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. $\frac{1}{5s^2+6s+2}$

Answer 1

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

Using Mason's gain formulae

$\Delta =1-[-2s^{-1}-2s^{-2}-4-4s^{-1}]$
$\Delta =1+\frac{2}{s}+\frac{2}{s^2}+4+\frac{4}{s}\frac{5s^2+6s+2}{s^2}$

$P_1=S^{-2}=\frac{1}{s^2}$
$P_2=S^{-1}=\frac{1}{s}$
${\Delta }_1=1,{\Delta }_2=2$

$\frac{Y\left(s\right)}{U\left(s\right)}=\frac{\sum P_1{\Delta }_1}{\Delta }=\frac{\frac{1}{s^2}\times 1+\frac{1}{s}\times 1}{\frac{5s^2+6s+2}{s^2}}$

$\frac{Y\left(s\right)}{U\left(s\right)}=\frac{s+1}{5s^2+6s+2}$