Correct Answer - Option 2 : asynchronous detector
Synchronous detector:
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A synchronous detector is a device that recovers information from a modulated signal by mixing the signal with a replica of the un-modulated carrier.
- This can be locally generated at the receiver using a phase-locked loop or other techniques.
Asynchronous detector:
- It does not require the presence of any carrier signal at the receiver in sync with that of the transmitter carrier signal.
An envelope detector is an asynchronous detector.
Envelope Detector:
For the output of the envelope detector to closely follow the modulating signal, the slope of vo is always greater than that of the envelope of AM signal input.
The output vo is the voltage across the capacitor which is exponentially decaying
\(\begin{array}{l} {v_0}\left( t \right) = {v_i}{e^{ - \frac{t}{{RC}}}}\\ = {V_i}\left[ {1 - \frac{t}{{RC}} + \frac{1}{{2!{{\left( {\frac{t}{{RC}}} \right)}^2}}} + \ldots } \right] \end{array}\)
Neglecting higher power terms as RC>>t
\(= {V_i}\left[ {1 - \frac{t}{{RC}}} \right]\)
For successful detection, the rate of discharge of capacitor i.e. magnitude of the slope of v0(t) must be more than that of the envelope
The envelope of AM signal is \({V_i}\left( t \right) = {A_c}\left[ {1 + {A_m}\cos {\omega _m}t} \right]\)
The slope of the message signal
\( \frac{{d{V_i}\left( t \right)}}{{dt}} = - {A_c}{A_m}\sin {\omega _m}t\)
The slope of output voltage
\(\frac{{d{v_0}}}{{dt}} = \frac{{{-V_i}}}{{Rc}}\)
For DETECTION slope of capacitor voltage should be always more than the envelope of AM signal
\(\begin{array}{l} \frac{{{V_i}}}{{RC}} \ge - {A_C}{A_m}\sin {\omega _m}t\\ \frac{{{A_c}}}{{RC}}\left[ {1 + {A_m}\cos {\omega _m}t} \right] \le {A_c}{A_m}\sin {\omega _m}t\\ RC \le \frac{{1 + {A_m}\cos {\omega _m}t}}{{{A_m}{\omega _m}\sin {\omega _m}t}} \end{array}\)
RC must be less than the minimum of R.H.S.
\(RC \le \frac{1}{{{\omega _m}}}\left[ {\frac{{\sqrt {1 - {\mu^2}} }}{\mu}\;} \right]\)
