Question

For the block diagrm shown in figure, the transfor function $\frac{C\left(s\right)}{R\left(s\right)}$ is equal to

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

Answer 1

The correct option is B $\frac{s^2+s+1}{s^2}$

Method -1: Using block-diagram redius technique.

So, transfer function


$\frac{C\left(s\right)}{R\left(s\right)}=\frac{s^2+s+1}{s^2}$

Method -2: using signal flow graph



Thre forward paths,
$P_1=\frac{1}{s}.\frac{1}{s}=\frac{1}{s^2},P_2=\frac{1}{s}and{P}_3=1$

The number of individual loop = 0

So graph determinant $\Delta =1$

$and{\Delta }_1={\Delta }_2={\Delta }_3=1$

Applying Mason's gain formula

$G\left(s\right)=\frac{C\left(s\right)}{R\left(s\right)}$

$=\frac{P_1{\Delta }_1+P_2{\Delta }_2+P3{\Delta }_3}{\Delta }$

$\frac{S^2.1+\frac{1}{s}.1+1.1}{1}=\frac{s^2+s+1}{s^2}$