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

To find the root mean square (RMS) current when the electric current in a circuit is given by the equation \(i(t) = i_0 \frac{t}{T}\), where \(t\) varies from 0 to \(T\), we can follow these steps:

1. The expression for the current is \(i(t) = i_0\frac{t}{T}\). The root mean square (RMS) value of an alternating current is given by: \[ i_{\text{rms}} = \sqrt{\frac{1}{T} \int_{0}^{T} i(t)^2 dt} \]
2. Substitute the expression for \(i(t)\) into the integral: \[ i_{\text{rms}} = \sqrt{\frac{1}{T} \int_{0}^{T} \left(i_0 \frac{t}{T}\right)^2 dt} \]
3. Simplify the expression inside the integral: \[ i_{\text{rms}} = i_0 \sqrt{\frac{1}{T^3} \int_{0}^{T} t^2 dt} \]
4. Perform the definite integration: \[ \int t^2 dt = \frac{t^3}{3} \] Evaluating from 0 to \(T\): \[ \int_{0}^{T} t^2 dt = \left[\frac{t^3}{3}\right]_{0}^{T} = \frac{T^3}{3} \]
5. Substitute the result of the integration back into the RMS formula: \[ i_{\text{rms}} = i_0 \sqrt{\frac{1}{T^3} \times \frac{T^3}{3}} \]
6. Simplify the expression: \[ i_{\text{rms}} = i_0 \sqrt{\frac{1}{3}} \] Which gives: \[ i_{\text{rms}} = \frac{i_0}{\sqrt{3}} \]
7. Thus, the RMS current for the given period is \( \frac{i_0}{\sqrt{3}} \).

Therefore, the correct answer is \(\frac{i_0}{\sqrt{3}}\), which matches the given option.