Question

A first-order system with unity gain and time constant \( \tau \) is subjected to sinusoidal input having a frequency \( \omega \) of \( \frac{1}{\tau} \). The amplitude ratio for such system is:

• A. 0.25
• B. \( \frac{1}{\sqrt{2}} \)
• C. 1
• D. \( \infty \)

Hint:

For a first-order system, the amplitude ratio at the frequency \( \frac{1}{\tau} \) is \( \frac{1}{\sqrt{2}} \), which corresponds to the -3 dB point.

Answer 1

The Correct Option is B

Step 1: First-Order System Analysis. The amplitude ratio of a first-order system under sinusoidal input, specifically at a frequency of \( \frac{1}{\tau} \), is \( \frac{1}{\sqrt{2}} \). This frequency signifies a 3 dB reduction in the system's response.

Step 2: Option Evaluation. - Option (B) correctly states the amplitude ratio (\( \frac{1}{\sqrt{2}} \)) at the specified frequency (\( \omega = \frac{1}{\tau} \)). - Options (A) and (C) present incorrect amplitude ratios for this frequency. - Option (D), \( \infty \), is invalid as the amplitude ratio does not tend to infinity.

Final Answer: \[ \boxed{\text{B) } \frac{1}{\sqrt{2}}} \]