The power of transmitter 19 kW. The power of the Carrier wave is, if the amplitude of modulated wave is 10 V and that of Carrier is 30 V,

Options

A. 18kW
B. 1kW
C. `4.75` kW
D. `9.5`kW

Answer

To find the power of the carrier wave (P_c) given the power of the transmitter (P_t), the amplitude of the modulated wave (V_m), and the amplitude of the carrier wave (V_c), we can follow these steps:

### Step 1: Identify the given values
- Power of the transmitter (P_t) = 19 kW = 19000 W
- Amplitude of the modulated wave (V_m) = 10 V
- Amplitude of the carrier wave (V_c) = 30 V

### Step 2: Calculate the modulation index (m)
The modulation index (m) is given by the formula:
\[ m = \frac{V_m}{V_c} \]
Substituting the values:
\[ m = \frac{10 \, \text{V}}{30 \, \text{V}} = \frac{1}{3} \]

### Step 3: Use the formula for the power of the carrier wave (P_c)
The power of the carrier wave can be calculated using the formula:
\[ P_c = P_t \left(1 + \frac{m^2}{2}\right) \]

### Step 4: Calculate \(m^2\)
Now, we calculate \(m^2\):
\[ m^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]

### Step 5: Substitute \(m^2\) into the power formula
Now substitute \(m^2\) into the power formula:
\[ P_c = 19000 \left(1 + \frac{1/9}{2}\right) \]
This simplifies to:
\[ P_c = 19000 \left(1 + \frac{1}{18}\right) \]

### Step 6: Simplify the expression inside the parentheses
Now simplify the expression:
\[ 1 + \frac{1}{18} = \frac{18}{18} + \frac{1}{18} = \frac{19}{18} \]

### Step 7: Calculate the power of the carrier wave
Now substitute back into the equation:
\[ P_c = 19000 \times \frac{19}{18} \]
Calculating this gives:
\[ P_c = 19000 \times 1.0556 \approx 20000 \, \text{W} \, \text{or} \, 20 \, \text{kW} \]

### Final Answer
The power of the carrier wave is approximately **20 kW**.

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