--- title: "Three-phase sinusoidal PWM inverter" subject: "Power Electronics" chapter: 5 key: "semester_7/power_electronics:three-phase-spwm-inverter" source: https://www.bctengineer.com/q/semester_7/power_electronics/three-phase-spwm-inverter site: BCT Engineering --- # Three-phase sinusoidal PWM inverter Power Electronics · Chapter 5: Inverter - Asked in: 2081 Baishakh, 8 marks; 2080 Baishakh, 8 marks; 2076 Asoj, 8 marks; 2075 Chaitra, 8 marks; 2071 Chaitra, 8 marks (asked 5 times) - Video: [SPWM (Sinusoidal PWM) Three Phase Inverters](https://www.youtube.com/watch?v=NXAAhArHang) — Ozan Keysan - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/power_electronics/three-phase-spwm-inverter - Whole chapter: https://www.bctengineer.com/md/semester_7/power_electronics/chapter-5 ## Question Explain the operation of a three phase sinusoidal PWM inverter with neat circuit diagram and associated waveforms. ## Short answer A three-phase SPWM inverter compares three 120°-shifted sinusoidal references with a high-frequency triangular carrier; each leg switches when ref > carrier, producing PWM line voltages. Fundamental line‑to‑line RMS voltage is 0.612·ma·Vdc (ma ≤ 1), enabling variable voltage/frequency control with low harmonics. ## Answer A three-phase sinusoidal PWM (SPWM) inverter is a six-switch bridge whose switching instants are fixed by comparing three sinusoidal reference waves (120° apart) with a common high-frequency triangular carrier. The output line voltages are pulse trains whose fundamental is sinusoidal and adjustable in both magnitude and frequency. #### Circuit ```text +----+--------+--------+ | | | | Vdc/2 S1 D1 S3 D3 S5 D5 --- | | | | O +-a +--b +--c | --- | | | | Vdc/2 S4 D4 S6 D6 S2 D2 | | | | +----+--------+--------+ a, b, c -> 3-phase load ``` Each leg (S1–S4, S3–S6, S5–S2) is switched in a complementary way (with a small dead time). Diodes carry reactive current back to the source. #### Principle of operation 1. Three references $v_{ra}, v_{rb}, v_{rc}$ of amplitude $A_r$ and frequency $f_r$ (desired output frequency), displaced by 120°. 2. One triangular carrier $v_c$ of amplitude $A_c$ and frequency $f_c \gg f_r$. 3. Leg a: if $v_{ra} > v_c$, S1 is ON and $v_{aO} = +V_{dc}/2$; if $v_{ra} < v_c$, S4 is ON and $v_{aO} = -V_{dc}/2$. Legs b and c work the same way with $v_{rb}$, $v_{rc}$. 4. Line voltage $v_{ab} = v_{aO} - v_{bO}$ takes values $+V_{dc}$, 0, $-V_{dc}$. ```text vc /\/\/\/\/\/\/\/\/\/\/\/\/ carrier vra .-''-. .-''- / \ / ref a ---'--------\------/-------- '-..-' vaO +Vdc/2 |-| |--| |---| |--| -Vdc/2 | |_| |_| |_| (wide pulses near +peak of vra) vab +Vdc |||| |||| 0 ------------------- -Vdc |||| |||| ``` #### Key quantities - **Amplitude modulation index** $m_a = A_r/A_c$. For $m_a \le 1$ (linear range): $$ \begin{aligned} \hat V_{aO1} &= m_a\frac{V_{dc}}{2} \\ V_{L1,rms} &= \frac{\sqrt3}{2\sqrt2}\,m_a V_{dc} = 0.612\,m_a V_{dc} \end{aligned} $$ - **Frequency modulation ratio** $m_f = f_c/f_r$. It is chosen odd and a multiple of 3, so that the three phases have identical waveforms and triplen carrier harmonics cancel in the line voltages. - Output frequency is changed by changing $f_r$; output voltage by changing $m_a$. #### Harmonics - Harmonics appear as side-bands around $m_f$, $2m_f$, $3m_f$, … (e.g. $m_f \pm 2$, $2m_f \pm 1$). - Low-order harmonics are almost absent, so a small filter (or the load inductance itself) gives near-sinusoidal current. - **Overmodulation** ($m_a > 1$): output rises less than linearly and low-order harmonics (5th, 7th) appear; at very large $m_a$ the inverter becomes a six-step (square-wave) inverter. #### Advantages, drawback and uses - Voltage and frequency control within the inverter (no variable dc link needed). - Low harmonic distortion, small filter. - Drawback: higher switching losses than six-step operation. - Used in variable-speed induction motor drives (V/f control), UPS and grid-tie inverters. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How are switching instants determined? A. Comparing three sinusoidal references 120° apart with a common triangular carrier B. Comparing a dc level with a sawtooth in each phase C. Fixed 180° conduction of six switches D. Comparing line voltages with zero Answer: A. Each leg switches when its reference crosses the carrier. (Key point: Gating principle) 2. In leg a, if vra > vc, which happens? A. S4 ON and vaO = +Vdc/2 B. S1 ON and vaO = +Vdc/2 C. S1 ON and vaO = -Vdc/2 D. Both S1 and S4 ON Answer: B. Complementary switching: S1 for reference above carrier, S4 otherwise. (Key point: Leg switching) 3. The fundamental phase voltage peak (linear range) is: A. ma Vdc B. Vdc/(2 ma) C. ma Vdc/2, where ma = Ar/Ac D. Vdc/2 regardless of ma Answer: C. Output varies linearly with ma for ma <= 1. (Key point: Modulation index) 4. Why is mf chosen odd and a multiple of 3? A. To raise the fundamental B. To remove the diodes C. To avoid dead time D. Phases have identical waveforms and triplen carrier harmonics cancel in line voltages Answer: D. This keeps the three phases symmetric. (Key point: Choice of mf) ## Related in Power Electronics - [Three-phase SPWM inverter switching instants](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Aspwm-switching-instants) — Power Electronics, chapter 5, asked 3× - [Single-phase PWM inverter rms and Fourier series](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Asingle-phase-pwm-inverter-fourier) — 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× - [Three-phase bridge inverter, 120° conduction](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Athree-phase-inverter-120-conduction) — Power Electronics, chapter 5 - [PWM half-bridge inverter switching instants](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Apwm-inverter-switching-instants) — Power Electronics, chapter 5 ## Related in other subjects - [CMOS inverter operation](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Acmos-inverter) — Digital Logic, chapter 10 - [Chopper definition; single-phase half bridge inverter](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Achopper-half-bridge-inverter) — Advanced Electronics, chapter 7 - [CMOS inverter / NOT gate operation](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Acmos-inverter-operation) — Basic Electronics Engineering, chapter 3, asked 4× - [Envelope detector circuit and waveforms](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenvelope-detector-circuit) — Communication Systems, chapter 2