Question

Which condition is to be satisfied in an AM detector circuit so that the signal corresponds to an envelope of the carrier wave ? (A) \(\mathrm{T}_{\mathrm{C}}<<\mathrm{RC}\) (B) \(\mathrm{T}_{\mathrm{c}}>\mathrm{RC}\) (C) \(F_{C}<

Short answer

The correct condition for the AM detector circuit to work properly and the signal to correspond to an envelope of the carrier wave is (A) \(T_{C}<<RC\).

Step-by-step solution

1. Analyze the given options and identify the key parameters.

We are given four options for the condition to be satisfied in an AM detector circuit. We need to identify the key parameters related to the condition. In this case, the key parameters are the time constant of the circuit (RC), the frequency or period of the carrier wave (Fc or Tc), and the envelope signal.

2. Recall the properties of an AM detector circuit.

For an AM detector circuit to work properly, it must quickly discharge the capacitor when the voltage across it is higher than the incoming signal voltage, and slowly charge the capacitor when the voltage across it is lower than the incoming signal voltage. This means that the time constant (RC) should be large enough to keep the capacitor charged during the positive half cycle of the carrier wave and small enough to allow the capacitor to discharge quickly during the negative half cycle.

3. Analyze the options and compare with the properties in step 2.

Now let's analyze each of the given options and compare them with the properties of an AM detector circuit: (A) \(T_{C}<RC\) (C) \(F_{C}<<RC\) (D) \(\left(1 / f_{n}\right)<<RC\) Option (A) implies that the time constant, RC, is much larger than the time period of the carrier wave, Tc. This condition satisfies the property in Step 2, as the RC circuit can keep the capacitor charged during the positive half cycle and allow the discharge during the negative half cycle. Option (B) contradicts the property in Step 2, as it implies that the time constant is smaller than the time period of the carrier wave. This would cause the capacitor to discharge too quickly and fail to follow the envelope accurately. Option (C) relates the frequency of the carrier wave (Fc) to RC, but since the frequency is the inverse of the time period, the condition should be expressed using the time period (Tc) instead. Option (D) relates the envelope frequency (1/fn) to the time constant, which is not relevant to the problem, as we are looking for the relationship between the time constant and the carrier wave.

4. Determine the correct condition for the AM detector circuit.

From our analysis in Step 3, the correct condition to be satisfied for the AM detector circuit to work properly is: (A) \(T_{C}<<RC\) This condition ensures that the signal corresponds to an envelope of the carrier wave in the detector circuit.