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 $\dfrac{1}{\sqrt{2}}\sin(t-\pi/4)$.

Answer 2

With \( x(t) = \sin(t)u(t) \) applied to the system whose block diagram gives \( H(s) = \dfrac{1}{s+1} \), the steady-state response to a sinusoidal input can be found by expressing the output as a magnitude-scaled, phase-shifted version of the input at the same frequency, then checking which option matches that scaled and shifted form.

1. Option \( \dfrac{1}{\sqrt{2}}\sin(t-\pi/4) \): Evaluating the frequency response at \( \omega = 1 \) (matching the input's frequency), \( H(j1) = \dfrac{1}{1+j} \); its magnitude is \( \dfrac{1}{\sqrt{1^2+1^2}} = \dfrac{1}{\sqrt{2}} \) and its phase is \( -\tan^{-1}(1) = -\dfrac{\pi}{4} \). Applying this magnitude and phase shift to the input sinusoid gives \( y(t) = \dfrac{1}{\sqrt{2}}\sin\!\left(t - \dfrac{\pi}{4}\right) \), which is exactly this option.
2. Option \( \dfrac{1}{\sqrt{2}}\sin(t+\pi/4) \): This has the correct magnitude but the wrong sign on the phase shift; the phase of \( H(j1) \) is negative (a lagging phase, since the imaginary part of the denominator is positive, making the overall angle negative), not a leading \( +\pi/4 \), so this option has the phase direction reversed.
3. Option \( \dfrac{1}{\sqrt{2}}e^{-t}\sin(t) \): This includes a decaying exponential factor \( e^{-t} \), which represents a transient (dying-out) term, not the steady-state part of the response; by definition steady-state response excludes such decaying transient components, so this cannot be the steady-state answer.
4. Option \( \sin(t) - \cos(t) \): This can be rewritten as \( \sqrt{2}\sin(t - \pi/4) \), which has the correct phase shift but the wrong magnitude, \( \sqrt{2} \) instead of \( 1/\sqrt{2} \); this does not match the magnitude computed from \( |H(j1)| \).

Only the option whose magnitude and phase both match the frequency response evaluated at the input's own frequency represents the true steady-state output.

So the correct answer is \( \dfrac{1}{\sqrt{2}}\sin(t-\pi/4) \).