--- title: "Square-law modulator for DSB-AM; characteristics of AM" subject: "Communication Systems" chapter: 2 key: "semester_6/communication_systems:square-law-dsbam-characteristics" source: https://www.bctengineer.com/q/semester_6/communication_systems/square-law-dsbam-characteristics site: BCT Engineering --- # Square-law modulator for DSB-AM; characteristics of AM Communication Systems · Chapter 2: Amplitude Modulation - Asked in: 2072 Asoj (CS I), 5+3 marks; 2065 Kartik (CS I), 6+2 marks (asked 2 times) - 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/square-law-dsbam-characteristics - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2 ## Question Show that a square-law modulator can be used to generate DSB-AM signal with explanation of its diagram and also sketch spectrum at the input of the BPF. What are the basic characteristics of AM modulation? ## Short answer A square-law modulator uses a nonlinear device and BPF to generate DSB-AM. The output is a_1 A_c [1 + (2a_2/a_1) m(t)] cos ω_c t. AM characteristics: envelope follows message, carrier plus sidebands, bandwidth 2W, modulation index ≤1, efficiency at most 33.3%. ## Answer A **square-law modulator** generates standard AM (DSB-FC) using a non-linear device (diode or transistor in its non-linear region) whose output contains a squared term, followed by a band-pass filter. #### Block diagram ```text m(t) -->(+)-v1->[Non-linear]-v2->[ BPF ]--> s(t) ^ [ device ] fc, 2W AM | Ac cos wct ``` #### Analysis The sum of message and carrier is applied to the device: $$ v_1(t)=A_c\cos\omega_ct+m(t) $$ The device characteristic (up to second order) is $$ v_2(t)=a_1v_1(t)+a_2v_1^2(t) $$ Substituting: $$ \begin{aligned} v_2(t)&=a_1A_c\cos\omega_ct+a_1m(t)+a_2\left[A_c^2\cos^2\omega_ct+2A_cm(t)\cos\omega_ct+m^2(t)\right]\\ &=\underbrace{a_1A_c\left[1+\frac{2a_2}{a_1}m(t)\right]\cos\omega_ct}_{\text{wanted AM}}+\underbrace{a_1m(t)+a_2m^2(t)+\frac{a_2A_c^2}{2}+\frac{a_2A_c^2}{2}\cos2\omega_ct}_{\text{unwanted}} \end{aligned} $$ A band-pass filter centred at $f_c$ with bandwidth $2W$ keeps only the first term: $$ s(t)=a_1A_c\left[1+k_am(t)\right]\cos\omega_ct,\qquad k_a=\frac{2a_2}{a_1} $$ This is exactly a DSB-AM (full carrier) signal with amplitude sensitivity $k_a$. Hence a square-law device plus a BPF works as an AM modulator. #### Spectrum at the input of the BPF For a message of bandwidth $W$, the components of $v_2(t)$ are: | Term | Frequency range | |---|---| | $a_2A_c^2/2$ (DC) | $f=0$ | | $a_1m(t)$ | $0$ to $W$ | | $a_2m^2(t)$ | $0$ to $2W$ | | AM: carrier + sidebands | $f_c-W$ to $f_c+W$ | | $\cos2\omega_ct$ | $2f_c$ | ```text |V2(f)| BPF passband | +-------+ |^ | ^ | ||\_ | /|\ | ^ || \__ | / | \ | | +|-----\-------+/--+--\+--------------+--> f 0 W 2W fc-W fc fc+W 2fc m(t) m^2(t) AM 2fc ``` To separate the AM band from $m^2(t)$ without overlap, we need $f_c-W>2W$, i.e. $f_c>3W$. #### Basic characteristics of AM 1. Carrier amplitude varies linearly with the message; envelope is $A_c[1+k_am(t)]$. 2. Spectrum has a **carrier plus two sidebands** (USB and LSB), each a copy of the message spectrum. 3. **Bandwidth** $B=2W$. 4. **Modulation index** $\mu=k_a\lvert m(t)\rvert_{max}\le1$ to avoid over-modulation. 5. **Power:** $P_t=P_c\left(1+\frac{\mu^2}{2}\right)$; maximum efficiency only 33.3% at $\mu=1$ (most power is in the carrier). 6. Can be demodulated by a simple **envelope detector**, making receivers cheap. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What produces the AM term in a square-law modulator? A. The a1 m(t) term B. The cross term 2 a2 Ac m(t) cos wct from the squared sum C. The m^2(t) term D. The cos 2wct term Answer: B. Squaring Ac cos wct + m(t) yields 2Ac m cos wct, which combined with a1 Ac cos wct gives the AM wave. (Key point: Square-law modulator) 2. What is the BPF's job? A. Pass the band around fc of width 2W, removing DC, m, m^2 and 2fc terms B. Pass the baseband message C. Pass 2fc D. Amplify all terms Answer: A. The BPF centred at fc with bandwidth 2W keeps only the AM term. (Key point: Role of BPF) 3. The amplitude sensitivity of the resulting AM is: A. ka = a1/a2 B. ka = 2a2/a1 C. ka = a2 D. ka = a1 a2 Answer: B. Output is a1Ac[1 + (2a2/a1) m(t)]cos wct. (Key point: Amplitude sensitivity) 4. Maximum efficiency of standard AM is: A. 100% B. 50% C. 33.3% at mu = 1 D. 66.7% Answer: C. eta = mu^2/(2+mu^2) = 1/3 at mu = 1, as most power is in the carrier. (Key point: AM efficiency) ## Related in Communication Systems - [AM modulation index; square law modulator](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-index-square-law-modulator) — Communication Systems, chapter 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 - [Define AM; switching modulator for DSB-FC](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adefine-am-switching-modulator) — Communication Systems, chapter 2 - [Ring modulator for DSB-SC; characteristics of DSB-SC](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aring-modulator-dsbsc-characteristics) — Communication Systems, chapter 2 - [DSB-AM vs SSB; ring modulator for DSB-SC](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbam-ssb-compare-ring-modulator) — Communication Systems, chapter 2