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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 modi? d 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).
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INTRODUCTION S ELECTIVE
harmonic elimination (SHE) is a long-established method of generating pulsewidth modulation (PWM) with low baseband distortion –. 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 , 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 . Each solution has different frequency content above the baseband, which provides options for ? attening the high-frequency spectrum for noise suppression or optimizing ef? iency. 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 . SHE is normally a two-step digital process. First, the switching angles are calculated of? ine, 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 ? st step because of its computational dif? culty. One possibility is to replace the Fourier series formulation with another orthonormal set based on Walsh functions –. 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 . In –, 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 forCommunication, the NationalScienceFoundation under Contract NSF 02-24829, the Electric Power Networks Ef? ciency, and the Security (EPNES) Program in cooperation with the Of? ce 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:[email protected]edu). Digital Object Identi? er 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  where mirror surplus harmonics are used. This involves solving multilevel elimination by considering reduced harmonic elimination waveforms in each switching level.
In , a general-harmonic-families elimination concept simpli? es a transcendental system to an algebraic functional problem by zeroing entire harmonic families. Faster and more complete methods have also been researched. In , an optimal PWM problem is solved by converting to a single univariate polynomial using Newton identities, Pade approximation theory, and symmetric function properties, which . If a few can be solved with algorithms that scale as O solutions are desired, prediction of initial guess values allows rapid convergence of Newton iteration .
Genetic algorithms can be used to speed the solution , . An approach that guarantees all solutions ? t a narrowly posed SHE problem transforms to a multivariate polynomial system – through trigonometric identities  and solves with resultant polynomial theory. Another approach – that obtains all solutions to a narrowly-posed problem uses homotopy and continuation theory. Reference  points out the exponentially growing nature of the problem and proposes the “ simulated annealing” method as a way to rapidly design the waveform for optimizing distortion and switching loss.
Another optimization-based approach is given in  and , where harmonics are minimized through an objective function to obtain good overall harmonic performance. There have been several multilevel and approximate real-time methods proposed; these are beyond the scope here but discussed brie? y in . This manuscript proposes an alternative real-time SHE method based on modulation. A modi? ed triangle carrier is identi? ed that is compared to an ordinary sine wave. In place of the conventional of? ine solution of switching angles, the process simpli? s to generation and comparison of the carrier and sine modulation, which can be done in minimal time without convergence or precision concerns. The method does not require an initial guess. In contrast to other SHE methods, the method does not restrict the switching frequency to an integer multiple of the fundamental. The underlying idea was proposed in  but has been re? ned here to identify speci? c carrier requirements that exactly eliminate harmonics and improve performance in deeper modulation. The method involves a function of modulation depth that is derived from simulation and curve ? ting. In thisrespect, it has some similarity to  and , in which approximate switching angles are calculated and ? tted to simple functions for cases of both low-( 0. 8 p. u. ) and high-modulation depth. It is interesting that the proposed approach connects modulation to a harmonic elimination process.
Fig. 2. Direct calculation of the phase modulation function at various modulation depths with ? rst through 177th harmonics controlled. i? cation is common in other PWM work, as in switching frequency randomization intended to reduce high-frequency components. A detailed review is outside the scope, but one discussion is given in . The proposed technique is not a variation of random-frequency carriers. Instead, the carrier waveform is modi? ed in a speci? c and deterministic way to bring about a certain effect. The proposed method is readily implemented in real time.
The switching signals themselves can be generated by analog comparison, while the modi? ed carrier is generated with fast digital calculation and digital-to-analog conversion. Hardware demonstration is provided here. An approximate, low-cost implementation based on present-day hardware is given in , but further re? nement is needed for precise elimination.
SIGNAL DEFINITIONS AND SIMULATED RESULTS
Consider a quasi-triangular waveform to be used as the carrier signal in a PWM implementation. In principle, the frequency and phase can be modulated.
To represent this, consider a triangular carrier function written as (1) where is the base switching frequency, is a phase-mod0, (1) reulation signal, and is a static phase shift. For duces to an ordinary triangle wave based on conventional quadrant de? nitions of the inverse cosine function. The modulating where signal will be represented as is the depth of modulation. The pulsewidth-modulated signal, , is 1 if and 1, otherwise. In , a phase modulation function is considered, where is the desired output fundamental fre, but dequency. This was shown to approach SHE at low 0. To determine a better phase-modulagrades above tion function, the pattern of switching angles that occurs was investigated. Fig. 1 shows the phase modulation values needed for various with harmonics 1–109 conversus angle trolled. Fig. 2 shows the same with harmonics 1–177 controlled. Many other sets of controlled harmonics were tested with similar results. The pattern looks much like a shockwave pattern that can be modeled with the Bessel–Fubini equation from nonlinear acoustics  (2) where is a Bessel function of the ? rst kind. The natural is in? ity in principle, but for calculation purposes number 15 or higher is usually suf? cient, as discussed below. The and have been determined by curve functions ? tting as (3) 1. and (4), shown at the bottom of the page, where 0 Fig. 3 shows a closeup view of a PWM waveform generated as in (2). Nineteen harmonics are with a carrier that uses 0. 95. The waveform is compared controlled with a (high) to one generated with conventional elimination by numerical solution of nonlinear equations. As can be seen, the switching edges match well. Fig. 4 shows a full-period time waveform and a magnitude 11.
With spectrum [fast Fourier transform (FFT)] for this switching frequency ratio, the method eliminates harmonics two through ten (even harmonics are zero by symmetry). The 2 and the modulation depth carrier phase shift is set to 1. The spectrum con? rms the desired elimination. is 0. This value Fig. 5 shows the same study except with also achieves satisfactory baseband performance, but with a different pulse pattern. The pattern provides slight differences in higher-order harmonics. For example, the 11th and 13th harto . monics vary 2%–3% in magnitude as is varied from (4) 338
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Fig. 3. Conventional harmonic elimination waveform and proposed PWM 0. 95, harmonics controlled through the 19th). waveform (m = Fig. 5. Pulse waveform p, message signal m, and magnitude of pulse waveform 1, and ‘ 0. spectrum for ! =! 11, m = = = Fig. 4. Pulse waveform p, message signal m, and magnitude of pulse waveform spectrum for ! =! 11, m 1, and ‘ = 2. = = = In these cases, all baseband harmonics are eliminated. In three-phase systems, triplen harmonics may cancel in the currents automatically if neutral current does not ? w. Therefore it is not always necessary to eliminate them by design in the SHE process. Modulation-based harmonic elimination excluding triplen harmonics is similar in many respects to the case here. However, the phase-modulation functions resemble piecewise polynomials rather than the shockwave form of Figs. 1 and 2. This is discussed in detail in . The speed of calculating these waveforms is dictated by , the number of terms to keep in the series (2), and , the number of discrete points used to approximate the waveforms. A personal computer (1. 86-GHz Intel M Processor with 1. -GB RAM) running MATLAB on Windows XP was used to carry out the calculations. First, a modi? ed triangle wave was ap100 000 points per cycle, the modulation proximated with 1, and a frequency ratio of 19 was used. depth was set to was varied from ? ve to 35. Over this range, the The number quality of solution was acceptable and the average calculation time varied from 0. 327 to 0. 915 s. Next, the same conditions 35 and was varied from 10 000 were used with except to 200 000. The average calculation time varied almost linearly from 0. 149 to 1. 78 s with no signi? cant difference in the resulting spectrum.
Finally, with held constant at 100 000, the frequency ratio was varied from seven to 51. The average calculation time was consistently near 0. 92 s. This is expected since the number of harmonics eliminated has no scaling effect in (2). However, for larger frequency ratios, larger may be needed for precision. In summary, it is recommended that be set to at least 1, 000 the frequency ratio and set to at least 15. In any case, with present-day personal computers the solution can be calculated in less than 1 s (typically) without iteration, divergence, or need for an initial estimate, and reduced versions can be computed in less than 200 ms.
Notice that this time interval need not cause trouble with real-time implementations. The carrier only needs to be recomputed with the modulation signal changes. In applications such as uninterruptible supplies, this is infrequent. In motor-drive applications, a response time of 200 ms to a command change may be acceptable as is. Alternatively, a look-up table can store some of the relevant terms to speed up the process dramatically. Dedicated DSP Please de? ne DSPalgorithms will be much faster than PC computations based on MATLAB.
To show that the proposed technique satisfactorily eliminates harmonics, the modi? d carrier was programmed into a function generator. The output provided a carrier signal in a conventional sine-triangle process. Three examples are shown below to reveal a range of interesting conditions. Fig. 6 shows the resulting waveforms for a high-depth case 0, and 0. 95. The with nineteen harmonics eliminated, and are shown at frequency ratio is 21: 1. The signals the top, followed by the PWM waveform and the FFT spectrum. From the spectrum it can be seen that the desired harmonic-free baseband spectrum is achieved. In the next example, the phase 2.
The unexpected result was that the spectrum shift is was insensitive to , as shown in comparison to Fig. 7. The desired spectrum occurs despite the difference in carriers. The resulting PWM waveforms at various values of may not offer obvious advantages, but it is noteworthy that they are not the same as conventionally computed SHE waveforms and would not be achievable with conventional SHE solution techniques. As another example, it is shown that the carrier base fre, need not be an odd multiple of . In Fig. 8, the frequency, IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY 2007 339 Fig. . Experimental modulation-based SHE with ! ‘ 0. = =! = 21, m = 0. 95, Fig. 9. Experimental modulation-based SHE with ! ‘ 0. = =! = 13. 5, m = 0. 95, Fig. 7. Experimental modulation-based SHE with ! ‘ = 2. = =! = 21, m = 0. 95, Fig. 10. Experimental modulation-based SHE with ! ‘ 0. = =! = 50, m = 0. 95, The last example, shown in Fig. 10, applies to a case where a high number of harmonics is eliminated (50 1 ratio) effectively, which is much higher than typically are reported in the literature. IV. CONCLUSION A method for calculating and implementing SHE switching angles was proposed and demonstrated.
The method is based on modulation rather than solution of nonlinear equations or numerical optimization. The approach is based on a modi? ed carrier waveform that can be calculated based on concise functions requiring only depth of modulation as input. It rapidly calculates the desired switching waveforms while avoiding iteration and initial estimates. Calculation time is insensitive to the switching frequency ratio so elimination of many harmonics is straightforward. It is conceivable the technique could be realized with low-cost microcontrollers for real-time implementation.
Once the carrier is computed, a conventional carrier-modulator comparison process produces switching instants in real time.
F. G. Turnbull, “ Selected harmonic reduction in static dc-ac inverters,” IEEE Trans. Commun. Electron. , vol. CE-83, pp. 374–378, Jul. 1964. Fig. 8. Experimental modulation-based SHE with ‘ 0. = ! =! = 20, m = 1. 0, quency ratio is adjusted to be 20: 1, with 0, and now 1. 0. The same nineteen harmonics are eliminated, but now the switching frequency is 5% lower. Intervals during which the carrier waveform is not triangular can be seen in the ? gure. As shown in Fig. , the frequency ratio can also be a half-in0. 95 and 0. teger. In this case, the ratio is 13. 5: 1, 340
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