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

Correct Answer: 8

RMS Value Calculation:
The root mean square (rms) value \( I_{\text{rms}} \) for a current described by \( i = I_0 + I_1 \sin(\omega t + \phi) \) is determined using the formula:
\[ I_{\text{rms}} = \sqrt{(I_0)^2 + \frac{(I_1)^2}{2}} \] where \( I_0 \) represents the direct current (DC) component and \( I_1 \) is the amplitude of the alternating current (AC) component.

Component Identification:
 For the given current:
\[ I_0 = 6 \, \text{A} \quad \text{and} \quad I_1 = \sqrt{56} \, \text{A} \]

RMS Value Computation:
 Substituting \( I_0 = 6 \) and \( I_1 = \sqrt{56} \) into the rms formula yields:
\[ I_{\text{rms}} = \sqrt{(6)^2 + \frac{(\sqrt{56})^2}{2}} \] \[ = \sqrt{36 + \frac{56}{2}} \] \[ = \sqrt{36 + 28} \] \[ = \sqrt{64} = 8 \, \text{A} \]

Result:
The calculated rms value of the current is \( 8 \, \text{A} \).