Envelope detector and choice of time constant

An envelope detector demodulates a standard AM wave $s(t)=A_c[1+\mu\cos\omega_mt]\cos\omega_ct$ by producing an output that follows its envelope $A_c[1+\mu\cos\omega_mt]$. It is a diode followed by a parallel RC load.

Circuit and operation

        D
 o---->|---+------+------o
    Rs     |      |
 AM in     C      RL    vo(t)
           |      |
 o---------+------+------o

1. Charging: on each positive peak of the carrier, the diode is forward-biased and $C$ charges rapidly to the peak value through the source and diode resistance $R_s$. Charging time constant $R_sC$ must be much smaller than the carrier period: $R_sC\ll1/f_c$.
2. Discharging: between peaks, the input drops below the capacitor voltage, the diode is cut off, and $C$ discharges slowly through $R_L$.
3. The capacitor voltage thus traces the envelope with a small sawtooth ripple at $f_c$. A coupling capacitor removes the DC part, giving the audio $m(t)$.

Necessary conditions on time constant

(a) Remove carrier ripple – discharge must be slow compared with the carrier period:

$$R_LC\gg\frac{1}{f_c}$$

(b) Follow the envelope – discharge must be fast compared with the message variations:

$$R_LC\ll\frac{1}{W}$$

Combined: $\frac{1}{f_c}\ll R_LC\ll\frac{1}{W}$, which is possible because $f_c\gg W$.

(c) Avoid diagonal clipping – for a tone, the envelope's fastest fall rate must not exceed the capacitor's discharge rate. This gives

$$R_LC\le\frac{\sqrt{1-\mu^2}}{\mu\,\omega_m}$$

For example, with $\mu=0.5$ and $f_m=5$ kHz, $R_LC\le\frac{0.866}{0.5\times2\pi\times5000}\approx55\ \mu\text{s}$.

Also, the modulation index must satisfy $\mu\le1$; otherwise the envelope crosses zero (over-modulation) and the output is distorted.

Waveforms

 (a) AM input      (b) RC too small   (c) RC correct
   |\ /\  /\ /|     follows carrier    smooth envelope
  /| V  \/  V |\          /\/\/\/\/\           .--.     .--.
 / |          | \                             /    \   /
                                                    '-'
 (d) RC too large: diagonal clipping
     '--.____         output cannot follow
             \___     the falling envelope

• (b) If $R_LC$ is too small, the capacitor discharges almost completely between cycles and the output contains large carrier ripple.
• (c) With the correct $R_LC$, the output follows the envelope with tiny ripple.
• (d) If $R_LC$ is too large, the capacitor voltage decays more slowly than the envelope falls, so the output misses the troughs (diagonal clipping).

Advantages: very simple, cheap, no local carrier needed; used in all AM broadcast receivers.