How Sinusoidal PWM Synthesizes Three-Phase AC Waveforms

This article explains how Sinusoidal Pulse-Width Modulation (SPWM) converts a constant direct current (DC) bus voltage into balanced three-phase alternating current (AC) waveforms within inverter electronics. It provides a straightforward breakdown of the six-switch bridge topology, the interaction between high-frequency carrier signals and reference sine waves, the mechanics of pulse width variation, and the final harmonic filtering process required to generate smooth AC power.

The Three-Phase Inverter Bridge Topology

To convert DC to three-phase AC, an inverter relies on a constant DC voltage source—known as the DC bus—and a switching network. The standard topology uses six power semiconductor switches, typically Insulated Gate Bipolar Transistors (IGBTs) or MOSFETs, arranged into three parallel legs. Each leg corresponds to one of the three AC output phases (Phase A, Phase B, and Phase C).

Each phase leg consists of an upper switch connected to the positive DC bus rail and a lower switch connected to the negative rail. The output for each phase is tapped from the midpoint between the upper and lower switches. To prevent a catastrophic short circuit across the DC bus (known as "shoot-through"), the two switches in any given leg are driven complementarily; when the upper switch is on, the lower switch must be off, with a brief "dead time" inserted between transitions.

Reference and Carrier Signal Comparison

SPWM controls the exact timing of these six switches by comparing two distinct types of signals inside a microcontroller or digital signal processor:

  1. The Reference Signals (Modulating Waves): Three pure sine waves are generated at the desired fundamental output frequency (such as 50 Hz or 60 Hz). To create a balanced three-phase output, these three sine waves are identical in amplitude and frequency but are shifted in phase by 120 degrees relative to one another (0°, 120°, and 240°).
  2. The Carrier Signal: A single high-frequency triangular wave, often operating between 2 kHz and 20 kHz, is shared across all three phases.

Generating the Variable Width Pulses

The core mechanism of SPWM is the continuous comparison of the three reference sine waves against the single shared triangular carrier wave. Each phase leg has its own comparator logic:

  • When the instantaneous voltage of a phase's reference sine wave is greater than the triangular carrier wave, the upper switch for that phase is turned ON (and the lower switch is turned OFF), connecting the output phase to the positive DC rail.
  • When the instantaneous voltage of the reference sine wave is less than the triangular carrier wave, the upper switch is turned OFF (and the lower switch is turned ON), connecting the output phase to the negative DC rail.

Because the reference wave is sinusoidal, the resulting output is a train of constant-amplitude voltage pulses whose widths vary sinusoidally. Near the peak of the sine wave, the reference stays above the carrier for longer periods, producing very wide positive pulses. Near the zero-crossing of the sine wave, the widths of the positive and negative pulses are nearly equal, resulting in an average voltage of zero.

Synthesizing the Balanced Line-to-Line Voltage

The raw output at each phase midpoint is a series of rectangular voltage pulses switching rapidly between the positive and negative DC bus levels. However, three-phase loads (like induction motors) are driven by the line-to-line voltage—the voltage difference between any two phases (\(V_{AB}\), \(V_{BC}\), \(V_{CA}\)).

Because the three reference sine waves are displaced by 120 degrees, the switching duty cycles of Phases A, B, and C are also offset by 120 degrees. When subtracting the pulsed voltage of Phase B from Phase A to find \(V_{AB}\), the common-mode carrier frequencies partially cancel out, and a three-level stepped PWM waveform emerges (switching between \(+V_{DC}\), \(0\), and \(-V_{DC}\)). The fundamental component of this line-to-line pulse train is a balanced sine wave shifted by 30 degrees from the phase references, with an amplitude controlled by the Modulation Index (the ratio of the reference wave peak to the carrier wave peak).

Harmonic Filtering into Smooth AC

While the voltage exiting the inverter is pulsed, the resulting current becomes a smooth sinusoid. The SPWM process intentionally pushes voltage harmonics up to the high frequency of the triangular carrier wave and its multiples.

When connected to an inductive load, such as a motor winding or a dedicated LC output filter, the inductance naturally resists rapid changes in current. The load acts as a low-pass filter, blocking the high-frequency carrier pulses and responding only to the moving average of the voltage. Consequently, three balanced, smooth, sinusoidal AC currents separated by 120 degrees flow through the load, completing the synthesis of three-phase AC power from a static DC bus.