# Space Vector Modulation

In the 1980s there was a strong interest in microprocessor-controlled motor drives with digitally generated PWM waveforms rather than analog, to take advantage of the increased flexibility of software. A 1982 conference paper by Pfaff, Weschta, and Wick (later published in IEEE Transactions on Industry Applications) described creating switching waveforms utilizing a technique for electric machine analysis known as space vectors. The basic idea is that you analyze the state of a motor or generator by considering per-phase quantities together as a group, whether they are voltages, currents, or fluxes. This idea had been around since the early 1900s but was primarily developed by R.H. Park at General Electric, who published a 1929 paper on the so-called "two-reaction" (d-q) theory. The term "space vector" appears to be of European origin (specifically a 1959 textbook by two Hungarian professors, K. P. Kovacs and I. Racz), referring to the use of symmetrical components and Park's two-reaction theory, not just as abstract quantities used to analyze three-phase power, but as real physical quantities representing the spatial variation of currents or magnetic fluxes in electric motors or generators.

The technique used by Pfaff, Weschta, and Wick for generating switching waveforms is to treat three-phase PWM generation as a single task. Instead of generating three independent switching waveforms, treat the three-phase bridge as one unit that can generate eight different switching states: a three-dimensional Cartesian product of the two states on phase A, the two states on phase B, and the two states on phase C.

Two of these states, (A,B,C) = (0,0,0) and (1,1,1), represent zero instantaneous line-to-line voltages and are often called the "zero" or "null" vectors. The remaining six states represent nonzero vector voltages applied across the motor terminals.

Pfaff, Weschta and Wick described how to generate a desired average voltage from the three-phase bridge by using a weighted average of the three closest switching states (defining six "sectors", shown in Roman numerals I-VI in the figure above), and published a formula to calculate the on-time of each phase in order to achieve that desired average voltage. This paper did not distinguish between the two zero vectors but only specified the necessary zero vector time.

A 1986 conference paper by van der Broeck, Skudelny, and Stanke (later published in IEEE Transactions) described the "optimal" use of space vector modulation in PWM to reduce switching losses and current ripple. It's pretty simple and can be achieved by following two key rules:

- Time spent in zero vectors is divided equally between (0,0,0) and (1,1,1).
- Switching waveforms should be symmetric, utilizing center-aligned PWM.

The authors also described the PWM strategy in terms of switch on-time calculation in each of the six sectors. This paper appears to be the first mention of the now-familiar waveforms that are generally associated with space vector modulation:

The van der Broeck paper has been cited by over 600 papers about motor control and is the foundation for PWM in modern digital motor control.

## Nitpicking at van der Broeck et al.

In the motor control industry, Space Vector Modulation (SVM) refers to the technique outlined on this page. While waveforms are usually the right choice in most motor control applications, three notable outcomes are unfortunate.

The first is that the term "space vector modulation" is often misunderstood. SVM is a more general idea — referring to the practice of treating a multiphase bridge as a way of creating a unified output vector, and the ability to jump back and forth between vector states arbitrarily over time. SVM does not have to be fixed-frequency like PWM.

Second, the technique mentioned in the van der Broeck paper is more precisely called Conventional Space Vector PWM (CSVPWM); Conventional because it is the most common approach; Space Vector PWM (rather than SVM), because it is a fixed-frequency technique utilizing ordinary PWM generators, whether they are implemented using analog or digital circuitry.

Third, some examples of space vector modulation do not use fixed-frequency PWM. Both hysteresis-based modulation and delta modulation are found occasionally in high-voltage or high-power systems where switching speeds are lower, and minimizing switching losses is very important.