---
title: "AM modulation index; square law modulator"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:am-index-square-law-modulator"
source: https://www.bctengineer.com/q/semester_6/communication_systems/am-index-square-law-modulator
site: BCT Engineering
---

# AM modulation index; square law modulator

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2076 Bhadra (CS I), 1+5 marks
- Video: [Square Law Modulator (Basics, Working, Block Diagram & Circuit) Explained | AM Modulator](https://www.youtube.com/watch?v=l6i1XJlAwtw) — 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/am-index-square-law-modulator
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

What is modulation index of AM wave? Explain the process of generation of DSB-AM using square law modulator.

## Short answer

Modulation index μ = Am/Ac indicates the depth of amplitude variation. A square-law modulator sums message and carrier, passes through a nonlinear device (v2 = a1 v1 + a2 v1²), and a bandpass filter selects the DSB-AM term a1 Ac [1 + (2a2/a1)m(t)]cos ωct.

## Answer

#### Modulation index of AM

The **modulation index** $\mu$ (or $m_a$) of an AM wave is the ratio of the message amplitude to the carrier amplitude; it shows how deeply the carrier amplitude is varied.

$$
\mu = \frac{A_m}{A_c} = \frac{A_{max} - A_{min}}{A_{max} + A_{min}}
$$

For no envelope distortion, $0 < \mu \le 1$. Multiplied by 100 it is the percentage modulation.

#### Square law modulator for DSB-AM

A non-linear device (diode or FET in its square-law region) is fed with the sum of message and carrier; its square term produces the product $m(t)\cos\omega_ct$, and a band-pass filter selects the AM wave.

```text
 m(t) -->(+)--> v1 --> Non-linear --> v2 --> BPF --> s(t)
          ^            device              (fc,2W)
 Ac cos wc t
```

**Input:** $v_1(t) = m(t) + A_c\cos\omega_ct$

**Device:** $v_2(t) = a_1v_1(t) + a_2v_1^2(t)$

$$
\begin{aligned}
v_2(t) = {} & a_1m(t) + a_1A_c\cos\omega_ct + a_2m^2(t) \\
& + 2a_2A_cm(t)\cos\omega_ct + a_2A_c^2\cos^2\omega_ct
\end{aligned}
$$

The BPF centred at $f_c$ with bandwidth $2W$ keeps only the carrier and the product term:

$$
s(t) = a_1A_c\left[1 + \frac{2a_2}{a_1}m(t)\right]\cos\omega_ct
$$

This is DSB-AM with $k_a = 2a_2/a_1$.

Unwanted outputs removed: $a_1m(t)$ (0–$W$), $a_2m^2(t)$ (0–$2W$), and $a_2A_c^2\cos^2\omega_ct$ (DC and $2f_c$). For the filter to separate them, $f_c > 3W$.

The circuit is simple, but it is a low-level modulator and suffers distortion if the device is not purely square law.

## Check yourself

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

1. How is the AM modulation index defined?
   A. mu = Ac/Am
   B. mu = Am/Ac = (Amax - Amin)/(Amax + Amin)
   C. mu = Amax + Amin
   D. mu = Amax/Amin
   Answer: B. It is the ratio of message amplitude to carrier amplitude. (Key point: Modulation index)

2. What range of mu avoids envelope distortion?
   A. mu > 1
   B. mu < 0
   C. mu >= 2
   D. 0 < mu <= 1
   Answer: D. For mu > 1 the carrier phase reverses and envelope detection distorts. (Key point: Useful range)

3. Which term in the square-law output creates the AM sidebands?
   A. 2 a2 Ac m(t) cos wc t
   B. a1 m(t)
   C. a2 m^2(t)
   D. a2 Ac^2 cos^2 wc t
   Answer: A. The cross term of (m + Ac cos wc t)^2 gives the product. (Key point: Source of sidebands)

4. What does the BPF keep, and what is required of fc?
   A. Only baseband; fc < W
   B. Only 2fc; fc = W
   C. Carrier and product term; fc > 3W
   D. Everything; any fc
   Answer: C. The m^2(t) band reaches 2W and must not overlap fc - W. (Key point: Filter and condition)

## Related in Communication Systems

- [Square-law modulator for DSB-AM; characteristics of AM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asquare-law-dsbam-characteristics) — Communication Systems, chapter 2, asked 2×
- [DSB-FC generation using square law modulator](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbfc-square-law-modulator) — Communication Systems, chapter 2
- [Square law approximation for DSB-AM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asquare-law-approximation-dsbam) — Communication Systems, chapter 2
- [SSB vs AM and DSB-SC; balanced modulator](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Assb-vs-am-balanced-modulator) — Communication Systems, chapter 2
- [AM wave diagram, modulation index; why near 100%](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-diagram-index-100-percent) — Communication Systems, chapter 2
- [Switching modulator for DSB-FC AM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aswitching-modulator-dsbfc) — Communication Systems, chapter 2

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