When a broad cast AM transmitter is 50 percent modulated, its antenna current is 12 A. When the modulation depth is increased to `0.9` the current would be in amperes

Options

A. `4sqrt(11.24)`
B. `4sqrt(8.24)`
C. `2sqrt(11.24)`
D. `6sqrt(8.24)`

Answer

To solve the problem, we will use the relationship between the antenna current and the modulation index in an Amplitude Modulated (AM) transmitter. The formula we will use is:

\[ I_t = I_c \left( 1 + \frac{\mu^2}{2} \right)^{1/2} \]

where:
- \(I_t\) is the total antenna current,
- \(I_c\) is the carrier current,
- \(\mu\) is the modulation index.

### Step 1: Find the carrier current \(I_c\) when the modulation index \(\mu = 0.5\).

Given:
- \(I_t = 12 \, A\) (when \(\mu = 0.5\))
- \(\mu = 0.5\)

Using the formula:

\[ 12 = I_c \left( 1 + \frac{(0.5)^2}{2} \right)^{1/2} \]

Calculating the term inside the brackets:

\[ 1 + \frac{(0.5)^2}{2} = 1 + \frac{0.25}{2} = 1 + 0.125 = 1.125 \]

Thus, we have:

\[ 12 = I_c \sqrt{1.125} \]

Now, squaring both sides:

\[ 144 = I_c^2 \cdot 1.125 \]

Solving for \(I_c^2\):

\[ I_c^2 = \frac{144}{1.125} = 128 \]

Taking the square root:

\[ I_c = \sqrt{128} = 8\sqrt{2} \]

### Step 2: Find the total antenna current \(I_t\) when the modulation index \(\mu = 0.9\).

Now, we need to calculate \(I_t\) for \(\mu = 0.9\):

Using the same formula:

\[ I_t = I_c \left( 1 + \frac{(0.9)^2}{2} \right)^{1/2} \]

Calculating the term inside the brackets:

\[ 1 + \frac{(0.9)^2}{2} = 1 + \frac{0.81}{2} = 1 + 0.405 = 1.405 \]

Substituting \(I_c = 8\sqrt{2}\):

\[ I_t = 8\sqrt{2} \sqrt{1.405} \]

Calculating \(I_t\):

\[ I_t = 8\sqrt{2 \times 1.405} = 8\sqrt{2.81} = 8 \cdot \sqrt{2.81} = 8 \cdot \sqrt{11.24}/4 \]

Thus, we can express \(I_t\) as:

\[ I_t = 4\sqrt{11.24} \]

### Final Answer:

The total antenna current when the modulation depth is increased to 0.9 is:

\[ I_t = 4\sqrt{11.24} \, A \]