---
title: "Envelope detector and choice of time constant"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:envelope-detector-time-constant"
source: https://www.bctengineer.com/q/semester_6/communication_systems/envelope-detector-time-constant
site: BCT Engineering
---

# Envelope detector and choice of time constant

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2075 Baisakh (CS I), 4+2+2 marks; 2070 Bhadra (CS I), 6 marks (asked 2 times)
- Video: [Envelope Detector (Basics, Derivation, Working, Block Diagram & Circuit) Explained | AM detection](https://www.youtube.com/watch?v=W1_z05uN8eQ) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/envelope-detector-time-constant
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

Explain the envelope detection method for the demodulation of AM wave with necessary conditions for time constants and waveforms.

## Short answer

Envelope detector uses a diode and parallel RC. The time constant R_L C must satisfy 1/f_c << R_L C << 1/W and for a tone R_L C ≤ sqrt(1-μ²)/(μ ω_m) to avoid diagonal clipping. Charging time constant R_s C << 1/f_c.

## Answer

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

```text
        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

```text
 (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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What condition must the charging time constant satisfy?
   A. Rs C >> 1/fc
   B. Rs C << 1/fc so C charges to the carrier peak quickly
   C. Rs C = 1/W
   D. Rs C is irrelevant
   Answer: B. C must charge rapidly to the peak during each positive cycle. (Key point: Charging)

2. The correct condition on RL C is:
   A. 1/fc << RL C << 1/W
   B. 1/W << RL C << 1/fc
   C. RL C << 1/fc
   D. RL C >> 1/W
   Answer: A. Slow compared with carrier cycles to cut ripple, fast compared with message changes to follow the envelope. (Key point: Discharge conditions)

3. What causes diagonal clipping?
   A. RL C too small
   B. RL C too large, so the capacitor cannot follow a falling envelope
   C. Modulation index too low
   D. Carrier too high
   Answer: B. A large time constant makes the capacitor decay slower than the envelope falls, missing the troughs. (Key point: Diagonal clipping)

4. For mu = 0.5 and fm = 5 kHz, the maximum RL C without clipping is about:
   A. 5.5 us
   B. 55 us
   C. 550 us
   D. 5.5 ms
   Answer: B. RC <= sqrt(1-0.25)/(0.5 x 2pi x 5000) = 0.866/15708 = about 55 us. (Key point: Numerical limit)

## Related in Communication Systems

- [Envelope detector circuit and waveforms](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenvelope-detector-circuit) — Communication Systems, chapter 2
- [Phase error in coherent DSB-SC detection; AM demodulation by PLL](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aphase-error-dsbsc-am-pll) — Communication Systems, chapter 2
- [Envelope detector; can it demodulate DSB-SC?](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenvelope-detector-dsbsc) — Communication Systems, chapter 2, asked 2×
- [Envelope detector; why not for DSB-SC and SSB](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenvelope-detector-why-not-dsbsc-ssb) — Communication Systems, chapter 2
- [Square law detector limitation; envelope detector](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asquare-law-detector-limitation-envelope) — Communication Systems, chapter 2
- [ASK and FSK modulators/demodulators; PSK with envelope detector](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aask-fsk-mod-demod-psk-envelope) — Communication Systems, chapter 7

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