Question

The block diagram of a control system is shown in figure. The transfer function G(s) = Y(s)/U(s) of the system is

• A. $\frac{1}{18(1+\frac{s}{12})(1+\frac{5}{3})}$
• B. $\frac{1}{27(1+\frac{s}{6})(1+\frac{s}{6})}$
• C. $\frac{1}{27(1+\frac{s}{12})(1+\frac{5}{9})}$
• D. $\frac{1}{27(1+\frac{s}{9})(1+\frac{5}{3})}$

Answer 1

The correct option is B $\frac{1}{27(1+\frac{s}{6})(1+\frac{s}{6})}$

Integrator are representend as 1/s in s-domain


As per the block diagram, the corresponding signal flow graph is draw



One forward path $P_1=2/s^2$

The individual loops are,

$L_1=-\frac{3}{5},L_2=-\frac{12}{s}and{L}_3=-\frac{18}{s}$

$L_1$ and $L_2$ are non-touching loops

$L_1{L}_2=\frac{36}{s^2}$

The loops touches the forward path ${\Delta }_1=1$

The graph determinant is

$\Delta =1-(L_1+L_2+L_3)+L_1{L}_2$

$=1+\frac{3}{s}+\frac{12}{s}+\frac{18}{s^2}+\frac{36}{s^2}$

Applying mason's gain formula.

$G\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{P_1{\Delta }_1}{\Delta }$

$=\frac{2/s^2}{1+\frac{3}{s}+\frac{12}{s}+\frac{18}{s^2}\frac{36}{s^2}}$

$=\frac{2}{s^2+15s+54}=\frac{2}{(s+9)(s+6)}$

$\frac{1}{27(1+\frac{s}{6})(1+\frac{s}{6})}$