Pulse Width Modulation (PWM)

What is pulse-width modulation, anyway?

If you're working with a switching power converter for motor control, it probably has a set of half-bridge output stages. Each of these has two switches connected in series across a DC link that carries some voltage, VDC.

At any given instant, we have three choices for controlling the half bridge: we can pull the output phase low to the negative side of the DC link (let's call this the "0" state), we can pull the output phase high to the positive side of the DC link (let's call this the "1" state), and we can leave the output phase open-circuited. For the moment let's ignore this last choice, and any transients involved going between the 0 and 1 states. We can output 0 or 1, and switch back and forth any way we like over time.

Here's the 0 state:

and here's the 1 state:

PWM switches between the 0 and 1 state at a fixed frequency. In each period of time T, the switch is at the 1 state some fraction of the time, and at the 0 state the rest of the time. The average fraction of time is D (the "duty cycle"), and this creates an average voltage on the half-bridge output of ${V}_{out} = D \times {V}_{dc}$.

# A bit of history

It may help to get some historical perspective on PWM for motor control.

Early switching converters (1960s-1980s) for three-phase motors generally used an analog circuit technique for generating the PWM waveforms, called "sine-triangle" PWM. The idea is that if you want to generate a duty cycle with an average value that approximates a sine wave, create a triangle waveform and compare it to the desired output waveform (sine wave), using the results of an analog comparator to determine whether or not to be in the 0 or 1 states.

# Three-phase PWM, and zero-sequence voltage

Sine-triangle PWM will work for 3 sine waves that are 120° out of phase:

The maximum line-to-line voltage you can get in this case, without creating distorted waveforms, is $\frac{\sqrt{3}}{2}{V}_{dc}$, or 86.6% of the DC link voltage. But we can do better than this.

In 1974, K. G. King of Westinghouse Brake & Signal Co. Ltd. received U.S. Patent #3839667 on adding harmonics of the operating frequency to increase the line-to-line voltage of an inverter. In particular he described adding 3rd, 6th, 9th, etc. harmonics, also known as "triplens", to each of the phase voltages. This affects the line-to-ground voltage, but the line-to-line voltages are unaffected, because the triplen harmonics on each phase line up. The resulting line-to-ground voltages appear squashed (figure from this patent):

This general idea involves adding what is known as a zero-sequence component, namely the same voltage added identically to each phase. (The term "zero-sequence" is from C. L. Fortescue's 1918 paper on symmetrical components to analyze three-phase power systems: positive sequence represents equal-amplitude sine waves in the normal phase order, e.g. A leading B leading C; negative sequence represents equal-amplitude sine waves in the reverse phase order e.g. A leading C leading B; and zero sequence represents equal-amplitude sine waves that are aligned in phase. Fortescue showed that any set of 3 phase sinusoidal voltages can be expressed as the sum of a positive, negative, and zero-sequence component. Only the positive and negative sequence components affect the voltage measured between any two phases.)

In 1984, Houldsworth and Grant published a paper demonstrating that a 3rd-harmonic component, with 1/6 the amplitude of the fundamental, can be added to each phase, to produce waveforms that have line-to-line voltages reaching the full DC link voltage (15.5% higher than without any triplen harmonics).