Question

A sinusoid \( (\sqrt{2} \sin t) \mu(t) \), where \( \mu(t) \) is the step input, is applied to a system with transfer-function \( G(s) = \frac{1}{s+1} \). The amplitude of the steady-state output is \(\underline{\hspace{2cm}}\).
 

Hint:

For a sinusoidal input, the amplitude of the steady-state output of a first-order system is the input amplitude multiplied by the magnitude of the transfer function at the input frequency.

Answer 1

Correct Answer: 0.95

The system is a first-order system with transfer-function \( G(s) = \frac{1}{s + 1} \). For a sinusoidal input of the form \( A \sin t \), the steady-state output amplitude \( Y_{\text{ss}} \) is given by: \[ Y_{\text{ss}} = |G(j\omega)| \cdot A \] where \( \omega = 1 \) (since the input is \( \sqrt{2} \sin t \)) and \( G(j\omega) = \frac{1}{j\omega + 1} \). We compute: \[ |G(j\omega)| = \frac{1}{\sqrt{1^2 + 1^2}} = \frac{1}{\sqrt{2}} \] Thus, the amplitude of the steady-state output is: \[ Y_{\text{ss}} = \frac{1}{\sqrt{2}} \cdot \sqrt{2} = 1 \] Thus, the amplitude of the steady-state output is \( 1 \).