The output current of an 80% mondulating amplitude modulated generator is 1.8A. To what value will the current rise if the generator is additionally modulated by another audiowave of modulation index 0.6?

Options

A. 1.71A
B. 1.81A
C. 1.91A
D. 2.01A

Explanation

To solve the problem step by step, we will use the given information and the relevant formulas for amplitude modulation.

### Step 1: Identify the given values
- The output current \( I_T \) of the generator is given as \( 1.8 \, A \).
- The modulation index for the first modulation \( \mu_1 \) is \( 80\% \) which can be converted to a decimal as \( \mu_1 = 0.8 \).
- The modulation index for the second modulation \( \mu_2 \) is \( 0.6 \).

### Step 2: Use the formula to find the carrier current \( I_C \)
The formula relating the total current \( I_T \) to the carrier current \( I_C \) and the modulation index \( \mu_1 \) is given by:

\[ I_T = I_C \sqrt{1 + \frac{\mu_1^2}{2}} \]

Rearranging this formula to find \( I_C \):

\[ I_C = \frac{I_T}{\sqrt{1 + \frac{\mu_1^2}{2}}} \]

### Step 3: Substitute the values to calculate \( I_C \)
Substituting the known values into the equation:

\[ I_C = \frac{1.8}{\sqrt{1 + \frac{(0.8)^2}{2}}} \]

Calculating \( \frac{(0.8)^2}{2} \):

\[ \frac{(0.8)^2}{2} = \frac{0.64}{2} = 0.32 \]

Now substituting this back into the equation for \( I_C \):

\[ I_C = \frac{1.8}{\sqrt{1 + 0.32}} = \frac{1.8}{\sqrt{1.32}} \]

### Step 4: Calculate \( \sqrt{1.32} \)
Calculating \( \sqrt{1.32} \):

\[ \sqrt{1.32} \approx 1.1487 \]

Now substituting this value back to find \( I_C \):

\[ I_C \approx \frac{1.8}{1.1487} \approx 1.568 \]

### Step 5: Calculate the new total current \( I_{net} \)
The new total current \( I_{net} \) when modulated by the second audio wave can be calculated using the formula:

\[ I_{net} = I_C \sqrt{1 + \frac{\mu_1^2}{2} + \frac{\mu_2^2}{2}} \]

Substituting the values we have:

\[ I_{net} = I_C \sqrt{1 + \frac{(0.8)^2}{2} + \frac{(0.6)^2}{2}} \]

Calculating \( \frac{(0.6)^2}{2} \):

\[ \frac{(0.6)^2}{2} = \frac{0.36}{2} = 0.18 \]

Now substituting this into the equation:

\[ I_{net} = I_C \sqrt{1 + 0.32 + 0.18} = I_C \sqrt{1.5} \]

### Step 6: Calculate \( \sqrt{1.5} \)
Calculating \( \sqrt{1.5} \):

\[ \sqrt{1.5} \approx 1.2247 \]

Now substituting \( I_C \) back into the equation for \( I_{net} \):

\[ I_{net} \approx 1.568 \times 1.2247 \approx 1.92 \]

### Final Answer
The new total current \( I_{net} \) will rise to approximately \( 1.92 \, A \).

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