Simple Analytical and Graphical Tools for Carrier Based PWM Methods Ahmet M. Hava Russel J. Kerkman Thomas A. Lipo University of Wisconsin-Madison Rockwell Automation-Allen Bradley 1415 Engineering Drive 6400 W. Enterprise Drive Madison, WI 53706-1691 Mequon, WI 53092-0760 e-mail: hava@cae.wisc.edu rjkerkman@meq1.ra.rockwell.com lipo@engr.wisc.edu — This paper provides analytical and graphical tools for the study, performance evaluation, and design of the modern carrier based PWM methods which are widely employed in PWM-VSI drives. Simple techniques for generating the modulation waves of the high performance PWM methods are described. The two most important modulator char- acteristics, the current ripple and the switching losses are analytically modeled. The graphical illustration of these often complex multivari- able functions accelerate the learning process and help understand the microscopic (per carrier cycle) and macroscopic (per fundamental cycle) behavior of all the modern PWM methods. The analytical formulas and graphics are valuable educational tools. They also aid the design and implementation of the high performance PWM methods. I. INTRODUCTION Voltage Source Inverters (VSIs) are utilized in AC motor drive, util- ity interface, and Uninterruptible Power Supply (UPS) applications as means for DC AC electric energy conversion. Shown in Figure 1, the classical VSI generates a low frequency output voltage with con- trollable magnitude and frequency by programming high frequency voltage pulses. Of the various pulse programming methods, the car- rier based Pulse Width Modulation (PWM) methods are the preferred approach in most applications due to the low harmonic distortion wave- form characteristics with well defined harmonic spectrum, the fixed switching frequency, and implementation simplicity. Sa+ Sb+ Sc+ Sa- Sb- Sc- ∼ L R n o V dc 2 V dc 2 + - + - a b c b v c v i a i b i c a e e b e c + + + R R L L I in ∼ ∼ a v Fig. 1. Circuit diagram of a PWM-VSI drive connected to an R-L-E type load. IEEE Power Electronics Specialists Conference St. Louis, Missouri, June 22-27, 1997 Volume 2, pp. 1462-1471 Carrier based PWM methods employ the “per carrier cycle volt- second balance” principle to program a desirable inverter output volt- age waveform. Two main implementation techniques exist: the tri- angle intersection technique and the direct digital technique. In the triangle intersection technique, for example in the Sinusoidal PWM (SPWM) method [1], as shown in Figure 2, the reference modulation wave is compared with a triangular carrier wave and the intersections define the switching instants. As illustrated in the spacevector diagram in Figure 3, the time length of the inverter states in the direct digital technique are precalculated for each carrier cycle by employing space vector theory and the voltage pulses are directly programmed [2, 3]. With the volt-second balance principle being quite simple, a variety of PWM methods have appeared in the technical literature; each method results from a unique placement of the voltage pulses in isolated neutral type loads. ω e t 0 1 V dc 2 V dc 2 - π 2π ω e t V tri S a+ V* a 0 2π π Fig. 2. Triangle intersection PWM phase “a” modulation and switching signals. V 1 V 2 V 3 V 4 V 5 V 6 (100) (110) (010) (011) (001) (101) dc V (2/3) ω e t R=1 R=2 R=3 R=4 R=5 R=6 V * 1 t T s V 1 t T s 2 V 2 ω e t = 0 Re Im (000) (111) V 0 V 7 Fig. 3. The space vector diagram illustrates the direct digital implementation principle. The upper switch states are shown in the bracket as (Sa+, Sb+, Sc+) and “1” is “on” state while “0” corresponds to “off” state. In most three phase AC motor drive and utility interface applications the neutral point is isolated and no neutral current path exists. In such applications in the triangle intersection implementations any zero se- quence signal can be injected to the reference modulation waves [4, 5]. The n-o potential in Figure 1 which will be symbolized with 0 can be freely varied. This degree of freedom is illustrated with the generalized signal diagram of Figure 4. A properly selected zero sequence signal can extend the volt-second linearity range of SPWM. Furthermore, it can improve the waveform quality and reduce the switching losses sig- nificantly. Recognizing these properties, many researchers have been