The antenna current of an AM transmitter is 8A when only the carrier is sent, but it increases to 8.93A when the carrier is modulated by a single sine wave. Find the percentage modulation

Options

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

Answer

To solve the problem of finding the percentage modulation of the AM transmitter, we can follow these steps:

### Step 1: Identify the given values
- Antenna current when only the carrier is sent (I_c) = 8 A
- Antenna current when the carrier is modulated (I_t) = 8.93 A

### Step 2: Use the relationship between the currents and modulation index
The relationship between the transmitter current (I_t), carrier current (I_c), and modulation index (m) is given by the formula:

\[ \frac{I_t^2}{I_c^2} = 1 + \frac{m^2}{2} \]

### Step 3: Substitute the known values into the equation
Substituting the values of I_t and I_c into the equation:

\[ \frac{(8.93)^2}{(8)^2} = 1 + \frac{m^2}{2} \]

Calculating the squares:

\[ \frac{79.5969}{64} = 1 + \frac{m^2}{2} \]

### Step 4: Calculate the left side of the equation
Calculating the left side:

\[ 1.2437 = 1 + \frac{m^2}{2} \]

### Step 5: Rearrange the equation to solve for m^2
Subtract 1 from both sides:

\[ 0.2437 = \frac{m^2}{2} \]

Multiply both sides by 2:

\[ m^2 = 0.4874 \]

### Step 6: Take the square root to find m
Taking the square root of both sides:

\[ m = \sqrt{0.4874} \approx 0.697 \]

### Step 7: Convert the modulation index to percentage
To find the percentage modulation, we multiply by 100:

\[ \text{Percentage Modulation} = m \times 100 \approx 0.697 \times 100 \approx 69.7\% \]

### Step 8: Round to the nearest option
The closest option to 69.7% is 70.1%.

### Final Answer
The percentage modulation is approximately **70.1%**.

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