Question

An amplitude modulator has output (in Volts): \[ s(t) = A \cos(400\pi t) + B \cos(360\pi t) + B \cos(440\pi t). \] The carrier power normalized to \(1 \, \Omega\) resistance is 50 Watts. The ratio of the total sideband power to the total power is \(1/9\). The value of \(B\) (in Volts, rounded off to two decimal places) is \(\_\_\_\_\).

Hint:

In amplitude modulation, calculate sideband power and total power using their respective relationships. The given power ratio helps in determining the unknown sideband amplitude.

Answer 1

Step 1: Calculate the carrier power.
The carrier power is given by: \[ P_c = \frac{A^2}{2 \times 1 \, \Omega} = 50 \, {Watts}. \] From this, we can determine: \[ A^2 = 100. \] Step 2: Compute the total sideband power.
In amplitude modulation, the total sideband power consists of the powers of the upper and lower sidebands: \[ P_{SB} = \frac{B^2}{2} + \frac{B^2}{2} = B^2. \] Step 3: Express the total power.
The total transmitted power is the sum of the carrier power and the sideband power: \[ P_{{total}} = P_c + P_{SB} = 50 + B^2. \] The problem states that the ratio of sideband power to total power is: \[ \frac{P_{SB}}{P_{{total}}} = \frac{1}{9}. \] Substituting \(P_{SB} = B^2\) and \(P_{{total}} = 50 + B^2\): \[ \frac{B^2}{50 + B^2} = \frac{1}{9}. \] Step 4: Solve for \(B^2\).
Simplify the equation: \[ 9B^2 = 50 + B^2 \implies 8B^2 = 50 \implies B^2 = \frac{50}{8} = 6.25. \] Taking the square root: \[ B = \sqrt{6.25} = 2.50 \, {Volts}. \] Final Answer: \[\boxed{{2.50}}\]