---
title: "Fundamental and 3rd harmonic of 180° inverter output"
subject: "Power Electronics"
chapter: 5
key: "semester_7/power_electronics:inverter-waveform-fundamental-third"
source: https://www.bctengineer.com/q/semester_7/power_electronics/inverter-waveform-fundamental-third
site: BCT Engineering
---

# Fundamental and 3rd harmonic of 180° inverter output

Power Electronics · Chapter 5: Inverter

- Asked in: 2079 Bhadra, 8 marks
- Video: [Lec 46(b) Fourier Series Expression of Three Phase Inverter | 180 Mode | Power Electronics](https://www.youtube.com/watch?v=K3-nz86hxV0) — Gatematic by Sohail Sir
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/power_electronics/inverter-waveform-fundamental-third
- Whole chapter: https://www.bctengineer.com/md/semester_7/power_electronics/chapter-5

## Question

Calculate the rms value of fundamental and 3rd harmonics component of output voltage for 180° conduction mode of three phase inverter shown in figure below. [Figure: phase voltage Van vs ωt — 200 V from 0 to π/3, 400 V from π/3 to 2π/3, 200 V from 2π/3 to π, then −200 V from π to 4π/3, −400 V from 4π/3 to 5π/3, −200 V from 5π/3 to 2π, repeating]

## Answer

The given phase voltage (200 V, 400 V, 200 V steps of 60°) is the six-step wave of a 180° conduction inverter with $V_{dc}/3 = 200$ V, so $V_{dc} = 600$ V. It is odd with half-wave symmetry, so only odd sine terms exist.

#### Fourier coefficient
$$
\begin{aligned}
b_n &= \frac{2}{\pi}\int_0^{\pi}v_{an}\sin n\theta\,d\theta \\
&= \frac{2}{n\pi}\Big[200\big(1-\cos\tfrac{n\pi}{3}\big) + 400\big(\cos\tfrac{n\pi}{3}-\cos\tfrac{2n\pi}{3}\big) + 200\big(\cos\tfrac{2n\pi}{3}-\cos n\pi\big)\Big] \\
&= \frac{400}{n\pi}\Big[2 + \cos\tfrac{n\pi}{3} - \cos\tfrac{2n\pi}{3}\Big] \quad (n\ \text{odd})
\end{aligned}
$$

#### Fundamental ($n = 1$)
$$
\begin{aligned}
b_1 &= \frac{400}{\pi}\big[2 + 0.5 - (-0.5)\big] = \frac{1200}{\pi} = 381.97\ \text{V (peak)} \\
V_{1,rms} &= \frac{381.97}{\sqrt2} = 270.09\ \text{V}
\end{aligned}
$$

Check with the standard result: $\hat V_{1} = \dfrac{2V_{dc}}{\pi} = \dfrac{2\times600}{\pi} = 381.97$ V.

#### Third harmonic ($n = 3$)
$\cos\pi = -1$, $\cos 2\pi = 1$:

$$
b_3 = \frac{400}{3\pi}\big[2 + (-1) - 1\big] = 0, \qquad V_{3,rms} = 0
$$

The third harmonic (and every triplen harmonic) is absent from the phase voltage of a 180° inverter with a balanced star load, because triplen components of the three pole voltages are in phase and appear only in the neutral voltage $v_{nO}$.

(For reference: overall rms $= \sqrt{(200^2+400^2+200^2)/3} = 282.84$ V; next harmonic $b_5 = 76.39$ V peak.)

**Answer:** $V_{1,rms} = 270.09$ V; $V_{3,rms} = 0$.

## Check yourself

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

1. Which relation fixes Vdc for the given 200 V and 400 V steps?
   A. Steps are Vdc/2 and Vdc
   B. Steps are Vdc/3 and 2Vdc/3
   C. Steps are Vdc/6 and Vdc/3
   D. Steps are Vdc and 2Vdc
   Answer: B. A 180° inverter has V/3 and 2V/3 phase levels, so V/3 = 200 V gives Vdc = 600 V. (Key point: Why Vdc = 600 V)

2. What is the rms fundamental of the phase voltage?
   A. 381.97 V
   B. 282.84 V
   C. 270.09 V
   D. 200 V
   Answer: C. Peak = 2Vdc/pi = 381.97 V; rms = 381.97/sqrt2 = 270.09 V. (Key point: Fundamental rms)

3. What is the rms third harmonic of the phase voltage?
   A. 90.0 V
   B. 127.3 V
   C. 76.39 V
   D. 0 V
   Answer: D. b_3 = (400/3pi)[2 + cos pi - cos 2pi] = 0. (Key point: Third harmonic)

4. Why are triplen harmonics absent in the phase voltage of a balanced star load?
   A. The load filters them
   B. They appear only in the neutral voltage since they are in phase in all three poles
   C. The 180° mode has a zero-state
   D. They cancel in the dc source
   Answer: B. Triplen components of the pole voltages are in phase, so they sit in v_nO and not across the phases. (Key point: Why triplens vanish)

## Related in Power Electronics

- [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
- [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
- [Three-phase six-step (180° conduction) inverter](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Athree-phase-inverter-180-six-step) — Power Electronics, chapter 5, asked 4×
- [RMS and fundamental of stepped inverter waveform](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Astepped-waveform-rms-fundamental) — Power Electronics, chapter 5, asked 3×
- [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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