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.

Key concepts

Time Constant

In electrical circuits, the time constant, usually denoted as RC, plays a critical role in determining how a circuit responds to changes in voltage. It is a measure of the time required for a capacitor to charge or discharge by approximately 63% of the difference between its initial value and its final value. Imagine it as the time it takes for the capacitor to almost fully adjust to a new voltage level.
For an AM detector circuit, this concept is especially important. Here, the time constant ensures that the capacitor can hold its charge appropriately across the cycle of the carrier wave.

• When RC is large compared to the period of the carrier wave, the capacitor holds its charge longer, following the waveform's envelope.
• If RC is too small, the circuit fails to maintain the envelope as the capacitor discharges too quickly.

A good understanding of the time constant helps in designing circuits that can effectively demodulate a modulated signal by accurately following the envelope of the carrier wave.

Carrier Wave

The carrier wave is a high-frequency sine wave that serves as a basis for transmitting information over long distances. In amplitude modulation (AM), the amplitude of this carrier wave is varied in proportion to the information signal, which is often audio or another signal you aim to transmit.
The period of the carrier wave is crucial in determining how the AM detector circuit interacts with the modulated signal. A properly chosen carrier wave frequency ensures effective signal transmission without distortion.

• A high-frequency carrier allows the modulation of high-fidelity audio signals over considerable distances.
• Matching the carrier wave with the circuit's components, like the RC constant, is crucial for proper demodulation.

Thus, understanding the interplay between the carrier wave and the AM detector circuit is key to effective signal processing.

Envelope Signal

The envelope signal represents the varying amplitude of the modulated carrier wave, which contains the actual information you want to transmit. Imagine it as the visible shape or outline of the peaks of a wave when you look at it on an oscilloscope.
In an AM detector circuit, capturing the envelope accurately means reproducing the original information signal without distortion. To achieve this, the circuit needs to handle:

• Large variations quickly enough to follow the envelope without introducing delay.
• Ensuring that the time constant (RC) is adequately set so that the circuit neither discharges too slowly nor too fast.

Understanding how to correctly set up the detector circuit allows it to accurately track the envelope signal, effectively demodulating the transmitted information from the carrier wave.

Capacitor Discharge

Capacitor discharge in an AM detector circuit significantly impacts the signal's ability to track the envelope of the carrier wave accurately. The capacitor must discharge at a rate that allows the circuit to respond appropriately to changes in the modulated signal's amplitude.
The process works as follows:

• During the positive half-cycle of the carrier wave, the capacitor charges to the peak voltage.
• During the negative half-cycle, it discharges, but at a rate determined by the time constant, RC.

For the circuit to function correctly, the time constant must be optimized to ensure that the capacitor discharges quickly enough to follow the decreasing envelope, but not so fast that it introduces errors. Balancing this discharge process is crucial to preserving signal fidelity and ensuring that the demodulated output accurately represents the original message signal.