In a circuit an a.c. current and a d, c. current are supplied together. The expression of the instantaneous current is given as `i=3+6sinomegat` Then the rms value of the current is –

Options

A. 3
B. 6
C. `3sqrt(2)`
D. `3sqrt(3)`

Answer

To find the RMS (Root Mean Square) value of the current given by the expression \( i = 3 + 6 \sin(\omega t) \), we can follow these steps:

### Step 1: Understand the Expression
The expression for the instantaneous current is given as:
\[ i(t) = 3 + 6 \sin(\omega t) \]
Here, \(3\) is the DC component and \(6 \sin(\omega t)\) is the AC component.

### Step 2: Identify the Components
The total current consists of:
- A DC component: \(I_{DC} = 3\)
- An AC component: \(I_{AC} = 6 \sin(\omega t)\)

### Step 3: Calculate the RMS Value
The RMS value of a current that has both AC and DC components can be calculated using the formula:
\[ I_{RMS} = \sqrt{I_{DC}^2 + I_{AC, RMS}^2} \]
where \(I_{AC, RMS}\) is the RMS value of the AC component.

### Step 4: Calculate \(I_{AC, RMS}\)
The RMS value of the AC component \(I_{AC} = 6 \sin(\omega t)\) is given by:
\[ I_{AC, RMS} = \frac{I_{peak}}{\sqrt{2}} = \frac{6}{\sqrt{2}} = 3\sqrt{2} \]

### Step 5: Substitute into the RMS Formula
Now substitute \(I_{DC}\) and \(I_{AC, RMS}\) into the RMS formula:
\[ I_{RMS} = \sqrt{3^2 + (3\sqrt{2})^2} \]

### Step 6: Calculate the Squares
Calculate each term:
\[ 3^2 = 9 \]
\[ (3\sqrt{2})^2 = 9 \cdot 2 = 18 \]

### Step 7: Add the Squares
Now add these values:
\[ I_{RMS} = \sqrt{9 + 18} = \sqrt{27} \]

### Step 8: Simplify the Result
Finally, simplify \(\sqrt{27}\):
\[ \sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3} \]

### Final Answer
Thus, the RMS value of the current is:
\[ \boxed{3\sqrt{3}} \text{ A} \]