Concept:An envelope detector requires the \( RC \) time constant to be large enough to filter out high carrier frequency fluctuations (\( f_c \)), but small enough to track the audio modulating frequency (\( f_m \)) without causing diagonal clipping distortion.
Formula:$$T_c \ll R C \ll T_m \implies \frac{1}{f_c} \ll R C \ll \frac{1}{f_m}$$
Solution:- If \( R C < 1/f_c \), the capacitor discharges too rapidly, failing to filter out the high-frequency carrier ripple.
- If \( R C > 1/f_m \), the capacitor discharges too slowly, causing the output voltage to fail to follow downward swings of the modulating envelope (diagonal clipping).
- Therefore, the optimal time constant must satisfy \( \frac{1}{f_c} \ll R C \ll \frac{1}{f_m} \).
Why other options are incorrect:- Option B: \( R C \gg 1/f_m \) causes diagonal clipping distortion because the capacitor cannot discharge fast enough to follow the audio envelope.
- Option C: \( R C \ll 1/f_c \) prevents proper rectification smoothing, allowing carrier ripple to pass to the output.
- Option D: \( R C = 1/(f_c f_m) \) is dimensionally incorrect (units of \( \text{s}^2 \)).
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