IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY 2007

Modulation-Based Harmonic Elimination

Jason R. Wells, Member, IEEE, Xin Geng, Student Member, IEEE, Patrick L. Chapman, Senior Member, IEEE, Philip T. Krein, Fellow, IEEE, and Brett M. Nee, Student Member, IEEE

Abstract—A modulation-based method for generating pulse waveforms with selective harmonic elimination is proposed. Harmonic elimination, traditionally digital, is shown to be achievable by comparison of a sine wave with modiﬁed triangle carrier. The method can be used to calculate easily and quickly the desired waveform without solution of coupled transcendental equations. Index Terms—Pulsewidth modulation (PWM), selective harmonic elimination (SHE).

I. INTRODUCTION

S

ELECTIVE harmonic elimination (SHE) is a long-established method of generating pulsewidth modulation (PWM) with low baseband distortion [1]–[6]. Originally, it was useful mainly for inverters with naturally low switching frequency due to high power level or slow switching devices. Conventional sine-triangle PWM essentially eliminates baseband harmonics for frequency ratios of about 10:1 or greater [7], so it is arguable that SHE is unnecessary. However, recently SHE has received new attention for several reasons. First, digital implementation has become common. Second, it has been shown that there are many solutions to the SHE problem that were previously unknown [8]. Each solution has different frequency content above the baseband, which provides options for ﬂattening the high-frequency spectrum for noise suppression or optimizing efﬁciency. Third, some applications, despite the availability of high-speed switches, have low switching-to-fundamental ratios. One example is high-speed motor drives, useful for reducing mass in applications like electric vehicles [9]. SHE is normally a two-step digital process. First, the switching angles are calculated ofﬂine, for several depths of modulation, by solving many nonlinear equations simultaneously. Second, these angles are stored in a look-up table to be read in real time. Much prior work has focused on the ﬁrst step because of its computational difﬁculty. One possibility is to replace the Fourier series formulation with another orthonormal set based on Walsh functions [10]–[12]. The resulting equations are more tractable due to the similarities between the rectangular Walsh function and the desired waveform. Another orthonormal set approach based on block-pulse functions is presented in [13]. In [14]–[20], it is observed that

Manuscript received August 2, 2006; revised September 11, 2006. This work was supported by the Grainger Center for Electric Machines and Electromechanics, the Motorola Center for Communication, the National Science Foundation under Contract NSF 02-24829, the Electric Power Networks Efﬁciency, and the Security (EPNES) Program in cooperation with the Ofﬁce of Naval Research. Recommended for publication by Associate Editor J. Espinoza. J. R. Wells is with P. C. Krause and Associates, Hentschel Center, West Lafayette, IN 47906 USA. X. Geng, P. L. Chapman, P. T. Krein, and B. M. Nee are with the Grainger Center for Electric Machines and Electromechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: plchapma@uiuc.edu). Digital Object Identiﬁer 10.1109/TPEL.2006.888910

the switching angles obtained traditionally can be represented as regular-sampled PWM where two phase-shifted modulating waves and a “pulse position modulation” technique achieve near-ideal elimination. Another approximate method is posed by [21] where mirror surplus harmonics are used. This involves solving multilevel elimination by considering reduced harmonic elimination waveforms in each switching level. In [22], a general-harmonic-families elimination concept simpliﬁes a transcendental system to an algebraic functional problem by zeroing entire harmonic families. Faster and more complete methods have...