---
title: "AM numerical: 10 sin1000πt on 50 sin200000πt"
subject: "Data Communication"
chapter: 5
key: "semester_5/data_communication:am-numeric-10sin1000pi-800"
source: https://www.bctengineer.com/q/semester_5/data_communication/am-numeric-10sin1000pi-800
site: BCT Engineering
---

# AM numerical: 10 sin1000πt on 50 sin200000πt

Data Communication · Chapter 5: Data Encoding and Modulation

- Asked in: 2073 Chaitra, 2+2+2+2 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/am-numeric-10sin1000pi-800
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-5

## Question

An audio signal of 10sin1000πt is used for AM with a carrier of 50sin200000πt. Calculate: i) Modulation index ii) Required bandwidth iii) Total power using load resistance of 800 ohm iv) Efficiency of AM

## Short answer

μ=Am/Ac=0.2. Bandwidth=2fm=1 kHz. Pc=Ac²/(2R)=50²/1600=1.56 W; total power Pt=Pc(1+μ²/2)≈1.59 W. Efficiency η=μ²/(2+μ²)×100≈1.96%.

## Answer

Message: Am = 10 V, ωm = 1000π, so fm = 500 Hz.
Carrier: Ac = 50 V, ωc = 200000π, so fc = 100 kHz. R = 800 Ω.

#### i) Modulation index
μ = Am/Ac = 10/50 = **0.2**

#### ii) Bandwidth
BW = 2fm = 2 × 500 = **1 kHz**

#### iii) Total power
Pc = Ac²/(2R) = 2500 / 1600 = 1.5625 W

Pt = Pc (1 + μ²/2) = 1.5625 × 1.02 = **1.594 W**

#### iv) Efficiency
η = (sideband power / total power) × 100 = μ² / (2 + μ²) × 100 = 0.04/2.04 × 100 = **1.96 %**

(Sideband power = 0.03125 W.)

## Check yourself

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

1. Pc with R = 800 Ω:
   A. 1.5625 W
   B. 3.125 W
   C. 0.78 W
   D. 1.594 W
   Answer: A. 2500/1600. (Key point: Pc)

2. Pt:
   A. 1.5625 W
   B. 1.594 W
   C. 3.19 W
   D. 0.03 W
   Answer: B. 1.5625 x 1.02. (Key point: Pt)

3. η = μ²/(2+μ²):
   A. 20 %
   B. 4 %
   C. 1.96 %
   D. 33 %
   Answer: C. 0.04/2.04. (Key point: Efficiency)

4. Sideband power:
   A. 1.5625 W
   B. 0.3 W
   C. 0.6 W
   D. 0.03125 W
   Answer: D. Pc x 0.02. (Key point: Sideband)

## Related in Data Communication

- [AM numerical: 10 sin1000πt on 50 sin 2π×10⁵t](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aam-numeric-10sin1000pi) — Data Communication, chapter 5
- [AM numerical: 20 sin2π500t on 60 sin2π10⁶t](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aam-numeric-20sin500) — Data Communication, chapter 5
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