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: Understanding the first-order system.
The amplitude ratio for a first-order system subjected to sinusoidal input at the frequency equal to \( \frac{1}{\tau} \) is \( \frac{1}{\sqrt{2}} \), which corresponds to the frequency at which the system's response is reduced by 3 dB.

Step 2: Explanation of options.
- (B) \( \frac{1}{\sqrt{2}} \) is the correct amplitude ratio at \( \omega = \frac{1}{\tau} \). - (A) and (C) These values are incorrect for the specified frequency. - (D) \( \infty \) is incorrect because the amplitude ratio does not go to infinity.

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