Question

The closed loop transfer function of a control system is given by
$\frac{C\left(s\right)}{R\left(s\right)}=\frac{1}{(1+s)}$
For the input $r\left(t\right)=sint,$ the steady state value of $c\left(t\right)$ is equal to

• A. $\frac{1}{\sqrt{2}}cost$
• B. 1
• C. $\frac{1}{\sqrt{2}}sint$
• D. $\frac{1}{\sqrt{2}}sin(t-\frac{\pi }{4})$

Answer 1

The correct option is D $\frac{1}{\sqrt{2}}sin(t-\frac{\pi }{4})$

$\because r\left(t\right)=sint$
$\Rightarrow \omega =1$
$Gain=\frac{1}{(1+s)}=\frac{1}{1+j\omega }=\frac{1}{1+j}$
$=\frac{1}{\sqrt{2}}\angle -{45}^{\circ }$
$\therefore output=sint\times \frac{1}{\sqrt{2}}\angle -{45}^{\circ }$
$=\frac{1}{\sqrt{2}}sin(t-{45}^{\circ })$