Square law detector limitation; envelope detector

Limitation of the square-law detector

A square-law detector passes the AM wave through a non-linear device $v_2 = a_1v_1 + a_2v_1^2$ and then a low-pass filter. For $v_1 = A_c[1+k_am(t)]\cos\omega_ct$ the useful low-frequency output is

$$v_o(t) = \frac{a_2A_c^2}{2}\left[2k_am(t) + k_a^2m^2(t)\right]$$

The $k_a^2 m^2(t)$ term is distortion (second harmonic and intermodulation of the message). The ratio of distortion to wanted signal is about $k_a|m(t)|/2$, so it is small only when $\mu$ is small (lightly modulated, weak signals). It therefore needs low modulation depth, works only at low signal levels, and has low efficiency. For DSB-SC (no carrier) it gives only $m^2(t)$, so the message cannot be recovered at all.

Envelope detector

An envelope detector is a diode–RC circuit whose output follows the envelope $A_c[1+\mu m(t)]$ of a standard AM wave.

        D
 o----->|-----+--------+-----o
 AM           |        |
 input   C  ===       > RL    v_o
              |        >
 o------------+--------+-----o

Operation

1. In the positive half of each carrier cycle the diode is forward biased and the capacitor charges rapidly, through the small source resistance $R_s$, up to the peak value.
2. When the input drops below the capacitor voltage the diode turns off, and $C$ discharges slowly through $R_L$.
3. At the next peak the diode conducts again. The capacitor voltage thus traces the peaks, i.e. the envelope, with a small ripple at $f_c$.
4. An RC low-pass filter removes the ripple, and a blocking capacitor removes the DC level, leaving $m(t)$.

 Input AM          Output
 /\  /\/\  /\      _/\_   _
/\/\/\  \/\/\/  -> /    \_/
\/\/\/  /\/\/\
 \/  \/\/  \/

Conditions for correct working

1. Fast charging: $R_sC \ll 1/f_c$, so the capacitor reaches the peak in each cycle.
2. Slow discharge between carrier peaks: $R_LC \gg 1/f_c$, to keep ripple small.
3. Fast enough to follow the message: $R_LC \ll 1/W$. If $R_LC$ is too large, the output cannot follow a falling envelope (diagonal clipping). For a single tone the limit is $R_LC \le \dfrac{\sqrt{1-\mu^2}}{\mu\,\omega_m}$.
4. No over-modulation: $\mu \le 1$, otherwise the envelope crosses zero and no longer equals $m(t)$.
5. $f_c \gg W$, so the carrier ripple can be separated from the message.

Combined: $\dfrac{1}{f_c} \ll R_LC \ll \dfrac{1}{W}$.