Question

In the system shown below, $x(t)=\sin(t)u(t)$. In steady-state, the response $y(t)$ will be 

• A. $\dfrac{1}{\sqrt{2}}\sin(t-\pi/4)$
• B. $\dfrac{1}{\sqrt{2}}\sin(t+\pi/4)$
• C. $\dfrac{1}{\sqrt{2}}e^{-t}\sin(t)$
• D. $\sin(t)-\cos(t)$

Hint:

Steady-state sinusoidal response depends only on system frequency response at that frequency.

Answer 1

The Correct Option is A

Step 1: Identify the system transfer function.
From the block diagram, the transfer function is:
\[ H(s)=\frac{1}{s+1} \]
Step 2: Determine steady-state sinusoidal response.
For input $\sin(\omega t)$, steady-state output is obtained using frequency response:
\[ H(j\omega)=\frac{1}{1+j\omega} \]
Step 3: Evaluate magnitude and phase at $\omega=1$.
\[ |H(j1)|=\frac{1}{\sqrt{1^2+1^2}}=\frac{1}{\sqrt{2}}, \quad \angle H(j1)=-\tan^{-1}(1)=-\frac{\pi}{4} \]
Step 4: Write steady-state output.
\[ y(t)=\frac{1}{\sqrt{2}}\sin\left(t-\frac{\pi}{4}\right) \]
Step 5: Final conclusion.
Thus, the steady-state response of the system is $\dfrac{1}{\sqrt{2}}\sin(t-\pi/4)$.