---
title: "RMS and fundamental of stepped inverter waveform"
subject: "Power Electronics"
chapter: 5
key: "semester_7/power_electronics:stepped-waveform-rms-fundamental"
source: https://www.bctengineer.com/q/semester_7/power_electronics/stepped-waveform-rms-fundamental
site: BCT Engineering
---

# RMS and fundamental of stepped inverter waveform

Power Electronics · Chapter 5: Inverter

- Asked in: 2076 Chaitra, 8 marks; 2075 Asoj, 8 marks; 2070 Chaitra, 8 marks (asked 3 times)
- Video: [3-Phase dc-to-ac inverter operating in square wave mode tutorial solution - Power Electronics](https://www.youtube.com/watch?v=mvejrueAqZQ) — Nûr Khan Juhoor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/power_electronics/stepped-waveform-rms-fundamental
- Whole chapter: https://www.bctengineer.com/md/semester_7/power_electronics/chapter-5

## Question

Figure shows the waveform of output voltage (per phase) of three phase inverter. Calculate RMS value and peak value of fundamental component of the output voltage. [Figure: per-phase output voltage Vo of a three-phase inverter vs ωt — +200 V from 0° to 60°, +400 V from 60° to 120°, +200 V from 120° to 180°, then −200 V from 180° to 240°, −400 V from 240° to 300°, −200 V from 300° to 360°]

## Short answer

The six-step wave has 200 V and 400 V steps each 60°. RMS voltage = sqrt[(200^2*2 + 400^2)/3] = 282.84 V. Peak fundamental from Fourier expansion: V1m = (2/π)*600 = 381.97 V.

## Answer

The given wave (steps of 200 V and 400 V, each 60° wide) is the per-phase voltage of a 180°-conduction three-phase inverter with $V_{dc}/3 = 200$ V, i.e. $V_{dc} = 600$ V.

#### RMS value
Each 60° block occupies $\pi/3$ rad. Using the half cycle (the negative half is a mirror image):

$$
\begin{aligned}
V_{rms} &= \left[\frac{1}{\pi}\left(200^2\cdot\frac{\pi}{3} + 400^2\cdot\frac{\pi}{3} + 200^2\cdot\frac{\pi}{3}\right)\right]^{1/2} \\
&= \left[\frac{40000 + 160000 + 40000}{3}\right]^{1/2} = \sqrt{80000} \\
&= 282.84\ \text{V}
\end{aligned}
$$

(Check: $\dfrac{\sqrt2}{3}V_{dc} = 0.4714\times600 = 282.84$ V.)

#### Peak value of the fundamental
The wave has half-wave and quarter-wave (odd) symmetry, so $a_n = 0$ and

$$
\begin{aligned}
b_n &= \frac{2}{\pi}\int_0^{\pi}v(\theta)\sin n\theta\,d\theta \\
&= \frac{2}{n\pi}\Big[200\big(1 - \cos\tfrac{n\pi}{3}\big) + 400\big(\cos\tfrac{n\pi}{3} - \cos\tfrac{2n\pi}{3}\big) + 200\big(\cos\tfrac{2n\pi}{3} - \cos n\pi\big)\Big]
\end{aligned}
$$

For $n = 1$: $\cos 60^\circ = 0.5$, $\cos 120^\circ = -0.5$, $\cos 180^\circ = -1$:

$$
\begin{aligned}
b_1 &= \frac{2}{\pi}\big[200(0.5) + 400(1) + 200(0.5)\big] = \frac{2}{\pi}\times600 \\
&= 381.97\ \text{V}
\end{aligned}
$$

(Check: $\dfrac{2V_{dc}}{\pi} = \dfrac{2\times600}{\pi} = 381.97$ V.) RMS of fundamental $= 381.97/\sqrt2 = 270.09$ V.

The same formula gives $b_3 = 0$ (no triplen harmonics), $b_5 = 76.39$ V, $b_7 = 54.57$ V.

**Answer:** $V_{rms} = 282.84$ V; peak fundamental $V_{1m} = 381.97$ V (270.09 V rms).

## Check yourself

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

1. The steps are 200 V, 400 V, 200 V of a 180° three-phase inverter phase voltage. What is Vdc?
   A. 400 V
   B. 1200 V
   C. 200 V
   D. 600 V
   Answer: D. Phase voltage levels are Vdc/3 and 2Vdc/3, so Vdc/3 = 200 V gives Vdc = 600 V. (Key point: Identify Vdc)

2. What is the rms value of this phase voltage?
   A. 282.84 V
   B. 400 V
   C. 300 V
   D. 381.97 V
   Answer: A. sqrt((200^2 + 400^2 + 200^2)/3) = sqrt(80000) = 282.84 V. (Key point: RMS value)

3. What is the peak of the fundamental component?
   A. 282.84 V
   B. 270.09 V
   C. 381.97 V
   D. 763.9 V
   Answer: C. b1 = 2Vdc/pi = 2(600)/pi = 381.97 V. (Key point: Peak fundamental)

4. Why are only sine terms with odd n present in the Fourier series?
   A. The wave has odd and half-wave symmetry
   B. The wave is unipolar
   C. The wave is a pure square wave
   D. The wave has a dc offset
   Answer: A. Odd and half-wave symmetry make a_n = 0 and kill even harmonics. (Key point: Symmetry)

## Related in Power Electronics

- [180° three-phase inverter and fundamental phase voltage](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Ainverter-180-fundamental-per-phase) — Power Electronics, chapter 5
- [Fundamental and 3rd harmonic of 180° inverter output](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Ainverter-waveform-fundamental-third) — Power Electronics, chapter 5
- [Fundamental and 3rd harmonic of three-phase inverter output](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Ainverter-waveform-vdc-fundamental-third) — Power Electronics, chapter 5
- [Three-phase bridge inverter 180° and 120° numerical](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Athree-phase-inverter-180-120-numerical) — Power Electronics, chapter 5
- [Single-phase square-wave inverter rms and fundamental](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Asquare-wave-inverter-rms-fundamental) — Power Electronics, chapter 5
- [Single-phase inverter fundamental output voltage](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Asingle-phase-inverter-r-fundamental) — Power Electronics, chapter 5

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