The Question is from neet-physics alternating-current , and it was created by the author NAGA SAI

Question

A complex AC waveform has an instantaneous current given by $i(t) = 3 + 4\sin(\omega t) + 2\sin(3\omega t)$. What is the RMS value of this current?

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Root Mean Square (RMS) Value of a Complex Waveform
	**Related Topics**: AC circuits, Fourier series, Power calculations
	**Prerequisites**: Understanding of RMS values, trigonometric identities, and integration.
## SOLUTION PATHWAY
**Given Information**:
	$i(t) = 3 + 4\sin(\omega t) + 2\sin(3\omega t)$
**Method & Steps**:
	1. **Identify the DC and AC components**: The current has a DC component (3) and two AC components ($4\sin(\omega t)$ and $2\sin(3\omega t)$).
		Reasoning: The RMS value of a complex waveform is calculated by considering the RMS values of its individual components.
	2. **Calculate the RMS value of each component**:
		DC component: $I_{DC_{RMS}} = 3$
		AC component 1: $I_{AC1_{RMS}} = \frac{4}{\sqrt{2}}$
		AC component 2: $I_{AC2_{RMS}} = \frac{2}{\sqrt{2}}$
		Working: The RMS value of a sinusoidal wave is its amplitude divided by $\sqrt{2}$.
	3. **Calculate the overall RMS value**:
		$I_{RMS} = \sqrt{I_{DC_{RMS}}^2 + I_{AC1_{RMS}}^2 + I_{AC2_{RMS}}^2}$ 
		$I_{RMS} = \sqrt{3^2 + (\frac{4}{\sqrt{2}})^2 + (\frac{2}{\sqrt{2}})^2}$ 
		$I_{RMS} = \sqrt{9 + 8 + 2}$ 
		$I_{RMS} = \sqrt{19}$ A
		Result: $\sqrt{19}$ A
		Verification: The RMS value represents the equivalent DC current that would produce the same average power. 
## OPTION ANALYSIS
[A] :x: 5 A: This is incorrect. It seems to be a simple addition of the amplitudes, which is not the correct method for RMS calculation.
[B] ✓: $\sqrt{19}$ A: This is the correct answer, obtained by applying the RMS formula to each component and then combining them.
[C] :x: $\sqrt{17}$ A:  This is incorrect, likely a miscalculation during the squaring and summing of the RMS values of individual components.
[D] :x: $\sqrt{29}$ A: This is incorrect, likely a miscalculation during the squaring and summing of the RMS values of individual components.
## LEARNING AIDS
	**Common Mistakes**:
		1. Simply adding the amplitudes:  The RMS value is not a simple sum of amplitudes.  Remember to divide the amplitude of sinusoidal components by $\sqrt{2}$ before squaring.
		2. Forgetting the DC component:  The DC component directly contributes to the RMS value.  Don't forget to square it and add it to the squared RMS values of the AC components.
	**Quick Tips**:
		1. Remember the formula: $I_{RMS} = \sqrt{\frac{1}{T} \int_0^T i(t)^2 dt}$.  This is the general formula, but for waveforms with distinct DC and sinusoidal AC components, the simplified method described above is usually faster.
		2. Visualize: The RMS value is related to the average power.  Think of it as the equivalent DC current that would produce the same heating effect.

NAGA SAI · Answer

5 A

NAGA SAI · Answer

The correct Option is "sqrt(19) A"

NAGA SAI · Answer

sqrt(17) A

NAGA SAI · Answer

sqrt(29) A

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