---
title: "Average and RMS of rectified wave"
subject: "Basic Electrical Engineering"
chapter: 5
key: "semester_1/basic_electrical_engineering:average-rms-rectified-wave-10v"
source: https://www.bctengineer.com/q/semester_1/basic_electrical_engineering/average-rms-rectified-wave-10v
site: BCT Engineering
---

# Average and RMS of rectified wave

Basic Electrical Engineering · Chapter 5: Alternating Quantities

- Asked in: 2073 Shrawan, 4 marks
- Video: [How to calculate the RMS and Average value of Half wave Rectifier and Full wave rectifier](https://www.youtube.com/watch?v=A2SMI31EgMA) — ALL ABOUT ELECTRONICS
- 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-rectified-wave-10v
- Whole chapter: https://www.bctengineer.com/md/semester_1/basic_electrical_engineering/chapter-5

## Question

Calculate the (i) average value and (ii) RMS value of voltage wave shown in figure below: [Figure: half-wave pulses of sinusoidal shape with peak 10 V, positive half cycles from 0 to π and from 2π to 3π with gaps in between, period T = 2π.]

## Short answer

The half-wave pulse train has period 2π, peak 10 V. Average over full period: Vav = 10/π ≈ 3.18 V. RMS value: Vrms = Vm/2 = 5 V.

## Answer

From the figure: sinusoidal half-wave pulses of peak Vm = 10 V, one in 0 to π, the next in 2π to 3π. The voltage is zero from π to 2π. Time period T = 2π.

#### i) Average value over one period

Vav = (1/2π) ∫₀^π 10 sin θ dθ = (10/2π) [−cos θ]₀^π = (10/2π)(2) = 10/π

**Vav = 3.18 V**

(If the average is taken only over the conducting half-cycle 0 to π, it is 2Vm/π = 6.37 V. For the periodic wave shown, with T = 2π, the average is 3.18 V.)

#### ii) RMS value

Vrms² = (1/2π) ∫₀^π (10 sin θ)² dθ = (100/2π) ∫₀^π (1 − cos 2θ)/2 dθ

= (100/2π) × (π/2) = 25

**Vrms = √25 = 5 V** (that is, Vm/2)

**Answer: Vav = 3.18 V, Vrms = 5 V.**

## Check yourself

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

1. Vav over period 2 pi?
   A. 10 V
   B. 5 V
   C. 6.37 V
   D. 3.18 V
   Answer: D. Divide by 2 pi because of the zero half. (Key point: Average)

2. Vrms?
   A. 10 V
   B. 7.07 V
   C. 5 V
   D. 3.18 V
   Answer: C. Vm/2. (Key point: RMS)

3. If averaged only on 0 to pi?
   A. 0
   B. Vm/pi
   C. 2Vm/pi = 6.37 V
   D. Vm/2
   Answer: C. The mean over the conducting half only. (Key point: Averaging pitfall)

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