Question

Electric current in a circuit is given by \(i = i_0 \frac{t}{T}\), then find the rms current for period t = 0 to t = T :

• A. \(\frac{i_0}{\sqrt{3}}\)
• B. \(\frac{i_0}{\sqrt{2}}\)
• C. \(\frac{i_0}{\sqrt{5}}\)
• D. \(\frac{i_0}{\sqrt{4}}\)
• E. None of these

Hint:

For any linear ramp current reaching \(I_{peak}\), the RMS value is always \(I_{peak} / \sqrt{3}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
RMS (Root Mean Square) current is calculated by taking the square of the current, finding its average over a period, and then taking the square root.
Step 2: Key Formula or Approach:
\[ I_{rms} = \sqrt{\frac{1}{T} \int_0^T i^2 dt} \]
Step 3: Detailed Explanation:
Substitute \(i = \frac{i_0}{T} t\) into the formula:
\[ I_{rms}^2 = \frac{1}{T} \int_0^T \left( \frac{i_0 t}{T} \right)^2 dt = \frac{i_0^2}{T^3} \int_0^T t^2 dt \]
\[ I_{rms}^2 = \frac{i_0^2}{T^3} \left[ \frac{t^3}{3} \right]_0^T = \frac{i_0^2}{T^3} \left( \frac{T^3}{3} \right) = \frac{i_0^2}{3} \]
Taking the square root:
\[ I_{rms} = \frac{i_0}{\sqrt{3}} \]
Step 4: Final Answer:
The RMS current for the period \(t = 0\) to \(t = T\) is \(\frac{i_0}{\sqrt{3}}\).