---
title: "Average and RMS of full-wave rectified wave"
subject: "Basic Electrical Engineering"
chapter: 5
key: "semester_1/basic_electrical_engineering:average-rms-full-wave-rectified"
source: https://www.bctengineer.com/q/semester_1/basic_electrical_engineering/average-rms-full-wave-rectified
site: BCT Engineering
---

# Average and RMS of full-wave rectified wave

Basic Electrical Engineering · Chapter 5: Alternating Quantities

- Asked in: 2072 Chaitra, 4 marks
- Video: [Parameters of Full Wave Rectifier (Average & RMS DC voltage/Current, PIV, Form & Ripple Factor)](https://www.youtube.com/watch?v=IsNvwBSgkpk) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/basic_electrical_engineering/average-rms-full-wave-rectified
- Whole chapter: https://www.bctengineer.com/md/semester_1/basic_electrical_engineering/chapter-5

## Question

Calculate the average and rms value of full-wave rectified sine wave as shown below. [Figure: full-wave rectified sine pulses of peak Em; the first hump from 0 to π, the second hump from π and so on; marked at π/4, π, 5π/4.]

## Short answer

Full-wave rectified sine wave: average = 2Em/π ≈ 0.637 Em, RMS = Em/√2 ≈ 0.707 Em. RMS equals original sine wave RMS.

## Answer

A full-wave rectified sine wave is e = Em |sin θ|, with period π (each hump lasts 0 to π and the next begins at π). Take one hump, 0 to π.

#### Average value
Eav = (1/π) ∫₀^π Em sin θ dθ = (Em/π) [−cos θ]₀^π = (Em/π)(2)

**Eav = 2Em/π = 0.637 Em**

#### RMS value
Erms² = (1/π) ∫₀^π Em² sin²θ dθ = (Em²/π) ∫₀^π (1 − cos 2θ)/2 dθ = (Em²/π)(π/2) = Em²/2

**Erms = Em/√2 = 0.707 Em**

The rms value of a full-wave rectified wave equals that of the original sine wave. The form factor is 0.707/0.637 = 1.11 and the peak factor is √2 = 1.414.

## Check yourself

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

1. Period?
   A. pi
   B. pi/2
   C. 4pi
   D. 2pi
   Answer: A. Humps repeat each pi. (Key point: Period)

2. Eav?
   A. 2Em/pi
   B. 0.707Em
   C. Em/pi
   D. Em/2
   Answer: A. Integrate sin. (Key point: Average)

3. Erms?
   A. Em/sqrt2
   B. 2Em/pi
   C. Em
   D. Em/2
   Answer: A. Same as sine. (Key point: RMS)

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