Question

The equivalent transfer function of the system shown in block diagram is:
$G_1=\frac{1}{s+1};G_2=\frac{1}{s+4}and{G}_3=\frac{s+2}{s+5}$

• A. $\frac{s^3+4s^2+18s+20}{s^3+5s^2+10s+14}$
• B. $\frac{s^3+8s^2+24s+30}{s^3+10s^2+30s+25}$
• C. $\frac{s^3+4s^2+18s+20}{s^3+10s^2+30s+25}$
• D. $\frac{s^3+8s^2+24s+30}{s^3+5s+10s+14}$

Answer 1

The correct option is B $\frac{s^3+8s^2+24s+30}{s^3+10s^2+30s+25}$

For given block diagram.
In ineer loop feedback we can write.
$\frac{R\left(s\right)}{X\left(s\right)}=\frac{G_1\left(s\right)}{1+G_1\left(s\right)G_2\left(s\right)}$

$\therefore$ Overall transfer function,

$Y\left(s\right)=X\left(s\right)R\left(s\right)+G_3\left(s\right)X\left(s\right)$

$\frac{Y\left(s\right)}{X\left(s\right)}=R\left(s\right)+G_3\left(s\right)$

$\frac{\frac{1}{s+1}}{1+\frac{1}{(s+1)}\frac{1}{(s+4)}}=\frac{(s+4)}{(s+1)(s+4)+1}=\frac{s+4}{s^2+5s+4+1}$

$\frac{s+4}{s^2+5s+5}+\frac{s+2}{s+5}\Rightarrow \frac{(s+4)(s+5)+(s+2)(s^2+5s+5)}{(s^2+5s+5)(s+5)}$

$=\frac{s^2+9s+20+s^3+5s^2+5s+2s^2+10s+10}{s^3+5s^2+5s^2+25s+5s+25}$$=\frac{s^3+8s^2+24s+30}{s^3+10s^2+30s+25}$