Question

An alternating current at any instant is given by $i = \left[ 6 + \sqrt{56} \sin\left(100 \pi t + \frac{\pi}{3}\right) \right] \, \text{A}.$ The RMS value of the current is ______ A.

Answer 1

Understanding the RMS (Root Mean Square) Value:
The root mean square (rms) value $I_{\text{rms}}$ of a current $i = I_0 + I_1 \sin(\omega t + \phi)$ is given by:
$I_{\text{rms}} = \sqrt{(I_0)^2 + \frac{(I_1)^2}{2}}$ where $I_0$ is the DC component and $I_1$ is the amplitude of the AC component.

Identify $I_0$ and $I_1$:
In this case:
$I_0 = 6 \, \text{A} \quad \text{and} \quad I_1 = \sqrt{56} \, \text{A}$

Calculate the RMS Value:
Substitute $I_0 = 6$ and $I_1 = \sqrt{56}$ into the rms formula:
$I_{\text{rms}} = \sqrt{(6)^2 + \frac{(\sqrt{56})^2}{2}}$ $= \sqrt{36 + \frac{56}{2}}$ $= \sqrt{36 + 28}$ $= \sqrt{64} = 8 \, \text{A}$

Conclusion:
The rms value of the current is $8 \, \text{A}$.