If modulation index is `1//2` and power of carrier wave is 2W. Then what will be the total power in modulated wave ?

Options

A. A) `0.5` W
B. B) 1 W
C. C) `0.25` W
D. D) `2.25` W

Answer

To solve the problem, we need to calculate the total power in a modulated wave given the modulation index and the power of the carrier wave. Here’s the step-by-step solution:

### Step 1: Identify the given values
- Modulation index (m) = 1/2
- Power of the carrier wave (Pc) = 2 W

### Step 2: Use the formula for total power in a modulated wave
The formula for the total power (Pt) in a modulated wave is given by:
\[ P_t = P_c \left(1 + \frac{m^2}{2}\right) \]

### Step 3: Substitute the values into the formula
Substituting the known values into the formula:
\[ P_t = 2 \left(1 + \frac{(1/2)^2}{2}\right) \]

### Step 4: Calculate the modulation index squared
Calculate \( (1/2)^2 \):
\[ (1/2)^2 = 1/4 \]

### Step 5: Substitute back into the equation
Now substitute this value back into the equation:
\[ P_t = 2 \left(1 + \frac{1/4}{2}\right) \]

### Step 6: Simplify the fraction
Calculate \( \frac{1/4}{2} \):
\[ \frac{1/4}{2} = \frac{1}{8} \]

### Step 7: Complete the equation
Now substitute this back:
\[ P_t = 2 \left(1 + \frac{1}{8}\right) \]

### Step 8: Add the terms inside the parentheses
Calculate \( 1 + \frac{1}{8} \):
\[ 1 + \frac{1}{8} = \frac{8}{8} + \frac{1}{8} = \frac{9}{8} \]

### Step 9: Multiply by the carrier power
Now multiply by 2:
\[ P_t = 2 \times \frac{9}{8} = \frac{18}{8} = \frac{9}{4} \]

### Step 10: Convert to decimal
Convert \( \frac{9}{4} \) to decimal:
\[ \frac{9}{4} = 2.25 \text{ W} \]

### Final Answer
The total power in the modulated wave is **2.25 W**.

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