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

# Average and RMS of trapezoidal wave

Basic Electrical Engineering · Chapter 5: Alternating Quantities

- Asked in: 2074 Chaitra, 4 marks
- Video: [Simple steps to drive the RMS value of a trapezoidal waveform](https://www.youtube.com/watch?v=6E27OxgFJ_0) — Engineering Tutorials by Er. Udit Mamodiya
- 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-trapezoidal-wave
- Whole chapter: https://www.bctengineer.com/md/semester_1/basic_electrical_engineering/chapter-5

## Question

Calculate the average (half period) value and rms value of the waveform shown below. [Figure: trapezoidal wave of peak Vm; rises linearly from 0 at θ = 0 to Vm at π/6, stays at Vm until 5π/6, falls linearly to 0 at π; negative half cycle is the mirror image up to 2π.]

## Short answer

For the trapezoidal wave, the half-cycle average is found by area: Vav = (5/6)Vm = 0.833 Vm. The RMS value, computed from squared integral, is Vrms = (√7/3)Vm = 0.882 Vm.

## Answer

The waveform rises linearly from 0 to Vm over 0 to π/6, stays at Vm from π/6 to 5π/6, and falls linearly to 0 at π. The negative half is a mirror image, so we calculate over the half period 0 to π.

#### Average (half-period) value

Vav = (1/π) ∫₀^π v dθ

The area has three parts:
- Rising ramp: ½ × (π/6) × Vm = πVm/12
- Flat top: Vm × (5π/6 − π/6) = 2πVm/3
- Falling ramp: πVm/12

Total area = πVm (1/12 + 8/12 + 1/12) = 10πVm/12 = 5πVm/6

**Vav = (1/π)(5πVm/6) = 5Vm/6 = 0.833 Vm**

#### RMS value

Vrms² = (1/π) ∫₀^π v² dθ

- Rising ramp: v = Vm θ/(π/6) = 6Vmθ/π, so ∫₀^{π/6} (6Vmθ/π)² dθ = (36Vm²/π²) × (π/6)³/3 = πVm²/18
- Falling ramp: the same, πVm²/18
- Flat top: Vm² × 2π/3 = 12πVm²/18

Sum = πVm² (1 + 1 + 12)/18 = 14πVm²/18 = 7πVm²/9

Vrms² = (1/π)(7πVm²/9) = 7Vm²/9

**Vrms = (√7 / 3) Vm = 0.882 Vm**

## Check yourself

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

1. Vav?
   A. 5Vm/6
   B. Vm/2
   C. 2Vm/pi
   D. Vm
   Answer: A. Area 5 pi Vm/6 over pi. (Key point: Average)

2. Vrms?
   A. sqrt7/3 Vm = 0.882 Vm
   B. Vm/sqrt3
   C. 0.707 Vm
   D. 0.833 Vm
   Answer: A. Vrms^2 = 7Vm^2/9. (Key point: RMS)

3. Why half period?
   A. Negative half is mirror image
   B. Wave is triangular
   C. Integral is zero
   D. Wave is DC
   Answer: A. Symmetry. (Key point: Half period)

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