Question

A signal \(\mathrm{e}_{\mathrm{m}}=20 \sin 2000 \pi \mathrm{t}\) amplitude modulates a carries wave \(e_{c}=80 \sin 200000 \pi t\). The percentage of modulation is (A) \(25 \%\) (B) \(0.25 \%\) (C) \(400 \%\) (D) \(40 \%\)

Short answer

The percentage of modulation is calculated using the modulation index, which is the ratio of the amplitude of the modulating signal \(E_m\) to the amplitude of the carrier signal \(E_c\). In this case, \(E_m = 20\) and \(E_c = 80\), so the modulation index is \(\frac{E_m}{E_c} = \frac{20}{80} = \frac{1}{4}\). To convert the modulation index to a percentage, multiply it by 100: \(Percentage\,of\,modulation = \frac{1}{4} \times 100\% = 25\%\). Therefore, the correct answer is (A) 25%.

Step-by-step solution

Define the modulating and carrier signal

The modulating signal is given by the equation: \(e_m = 20 \sin(2000 \pi t)\) and the carrier signal is given by the equation: \(e_c = 80 \sin(200000 \pi t)\).

Calculate the amplitude ratio

The modulation index \(m\) is defined as the ratio of the amplitude of the modulating signal \(E_m\) to the amplitude of the carrier signal \(E_c\): \[m = \frac{E_m}{E_c}\] Now we need to find the amplitude of both signals. From the equations, the amplitudes are as follows: \(E_m = 20\) and \(E_c = 80\) Now we can find the modulation index: \[m = \frac{20}{80}\]

Calculate the modulation index

Using the amplitude values from step 2, we have: \[m = \frac{20}{80} = \frac{1}{4}\]

Convert the modulation index to a percentage

To convert the modulation index to a percentage, multiply it by 100: \[Percentage\,of\,modulation = m \times 100\%\] Now, substituting the modulation index value from step 3: \[Percentage\,of\,modulation = \frac{1}{4} \times 100\% = 25\%\] The correct answer is (A) 25%.