The antenna current of an `AM` transmitter is `8A` when only the carrier is sent but increases to `8.96A` when the carrier is modulated sinusoidally . The percentage modulation is

Options

A. `60.1 %`
B. `70.1%`
C. `80.1 %`
D. `50.1 %`

Answer

To find the percentage modulation of the AM transmitter, we can follow these steps:

### Step 1: Identify the given values
- The antenna current when only the carrier is sent, \( I_C = 8 \, \text{A} \)
- The antenna current when the carrier is modulated, \( I_T = 8.96 \, \text{A} \)

### Step 2: Use the formula for the relationship between the currents
The relationship between the total current \( I_T \) and the carrier current \( I_C \) in amplitude modulation is given by:
\[ \frac{I_T}{I_C} = \sqrt{1 + \frac{M^2}{2}} \]
Where \( M \) is the modulation index.

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ \frac{8.96}{8} = \sqrt{1 + \frac{M^2}{2}} \]

### Step 4: Calculate the left side of the equation
Calculating the left side:
\[ \frac{8.96}{8} = 1.12 \]
So, we have:
\[ 1.12 = \sqrt{1 + \frac{M^2}{2}} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (1.12)^2 = 1 + \frac{M^2}{2} \]
Calculating \( (1.12)^2 \):
\[ 1.2544 = 1 + \frac{M^2}{2} \]

### Step 6: Rearrange the equation to solve for \( M^2 \)
Subtract 1 from both sides:
\[ 1.2544 - 1 = \frac{M^2}{2} \]
\[ 0.2544 = \frac{M^2}{2} \]
Now, multiply both sides by 2:
\[ M^2 = 0.5088 \]

### Step 7: Calculate \( M \)
Taking the square root of both sides:
\[ M = \sqrt{0.5088} \approx 0.713 \]

### Step 8: Calculate the percentage modulation
The percentage modulation is given by:
\[ \text{Percentage Modulation} = M \times 100 \]
Substituting the value of \( M \):
\[ \text{Percentage Modulation} = 0.713 \times 100 \approx 71.3\% \]

### Conclusion
The percentage modulation is approximately \( 71.3\% \).

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