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.
Correct Answer: 8
Approach Solution - 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 \( 8 \, \text{A} \).
Approach Solution -2
The alternating current is given by:
\[ i = \left[ 6 + \sqrt{56} \sin\left(100\pi t + \frac{\pi}{3}\right) \right] \, \text{A} \] This represents an AC waveform with a DC component.
Step 2: Identify components.
DC component, \( I_{\text{DC}} = 6 \, \text{A} \)
AC component, \( I_{\text{AC}} = \sqrt{56} \, \text{A (amplitude)} \).
Step 3: Formula for RMS value of mixed AC and DC current.
The RMS value for a current having both DC and sinusoidal components is:
\[ I_{\text{RMS}} = \sqrt{I_{\text{DC}}^2 + \left(\frac{I_{\text{AC}}}{\sqrt{2}}\right)^2} \]
Step 4: Substitute the values.
\[ I_{\text{RMS}} = \sqrt{6^2 + \left(\frac{\sqrt{56}}{\sqrt{2}}\right)^2} \] Simplify: \[ I_{\text{RMS}} = \sqrt{36 + \left(\frac{56}{2}\right)} = \sqrt{36 + 28} = \sqrt{64} = 8 \]
Step 5: Final Answer.
\[ \boxed{I_{\text{RMS}} = 8 \, \text{A}} \]
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