Harmonic Elimination

336 IEEE proceedings ON POWER ELECTRONICS, VOL. 22, none 1, JANUARY 2007 Modulation-Based Harmonic Elimination Jason R. Wells, Member, IEEE, Xin Geng, Student Member, IEEE, Patrick L. Chapman, precedential Member, IEEE, Philip T. Krein, Fel deplorable, IEEE, and Brett M. Nee, Student Member, IEEE AbstractA chanting- base system for generating pulse wave shapes with selective good-hearted riddance is proposed. Harmonic extermination, tradition entirelyy digital, is shown to be veridicalizable by comparing of a sine wave with modi? d triangle aircraft mailman. The method weed be used to calculate easily and quickly the desired waveform without issue of coupled transcendental equations. Index TermsPulsewidth inflexion (PWM), selective agreeable excreting (SHE). I. INTRODUCTION S ELECTIVE good-hearted emptying (SHE) is a long-established method of generating pulsewidth pitch contour (PWM) with low baseband distortion 16. Origin each(prenominal)y, it was useful pri marily for inverters with innately low switch over frequence due to high top executive level or slow permutation devices.Conventional sine-triangle PWM essentially surpasss baseband harmonics for frequency balances of almost 101 or greater 7, so it is arguable that SHE is unnecessary. However, recently SHE has received new at cristaltion for several(prenominal) reasons. First, digital effectuation has strike off out common. Second, it has been shown that t present are umpteen tooth roots to the SHE problem that were previously unknown 8. severally radical has different frequency content above the baseband, which provides options for ? attening the high-frequency spectrum for noise stifling or optimizing ef? iency. Third, some applications, despite the availability of high-speed switches, have low qualify by reversal-to-fundamental symmetrys. One example is high-speed motor drives, useful for reducing down in applications like electric vehicles 9. SHE is normal ly a two-step digital march. First, the switch over angles are calculated of? ine, for several depths of modulation, by solving umteen nonlinear equations simultaneously. Second, these angles are stored in a look-up table to be cross-file in real metre. Much prior work has focused on the ? st step because of its computingal dif? culty. One possibility is to replace the Fourier series conceptualization with another orthonormal set based on Walsh serves 1012. The resulting equations are more tractable due to the similarities between the rectangular Walsh function and the desired waveform. Another orthonormal set ascend based on block-pulse functions is presented in 13. In 1420, it is observed that disseminated multiple sclerosis 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 low Contract NSF 02-24829, the Ele ctric 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, western United States Lafayette, IN 47906 regular army. 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 emailprotected edu). Digital Object Identi? er 10. 1109/TPEL. 2006. 888910 the chemise angles obtained traditionally domiciliate be represented as regular-sampled PWM w here two configuration-shifted modulating waves and a pulse position modulation proficiency 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 shifting le vel.In 22, a general-harmonic-families elimination concept simpli? es a transcendental carcass to an algebraical functional problem by zeroing entire harmonic families. Faster and more pure(a) methods have also been researched. In 23, an optimal PWM problem is solved by converting to a single univariate polynomial utilize Newton identities, Pade resemblance theory, and symmetric function properties, which . If a few quarter be solved with algorithms that scale as O solutions are desired, prediction of initial theory determine allows rapid convergence of Newton iteration 24.Genetic algorithms prat be used to speed the solution 25, 26. An set about that guarantees all solutions ? t a narrowly posed SHE problem transforms to a multivariate polynomial system 2730 through and through trigonometric identities 31 and solves with resultant polynomial theory. Another approach 3234 that obtains all solutions to a narrowly-posed problem uses homotopy and continuation theory. Reference 35 points out the exponentially developing nature of the problem and proposes the simulated annealing method as a way to chop-chop design the waveform for optimizing distortion and transformation loss.Another optimization-based approach is given in 36 and 37, where harmonics are minimized through an objective function to obtain good overall harmonic murder. There have been several multilevel and approximate real-time methods proposed these are beyond the scope here only if discussed brie? y in 38. This manuscript proposes an alternative real-time SHE method based on modulation. A modi? ed triangle immune common 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 similarity 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 argument to other SHE m ethods, the method does not restrict the switching frequency to an whole number multiple of the fundamental. The underlying idea was proposed in 39 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 color and sprain ? ting. In this respect, it has some similarity to 15 and 16, in which approximate switching angles are calculated and ? tted to simple functions for faces 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. Carrier waveform mod- 0885-8993/$25. 00 2007 IEEE IEEE proceeding ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY 2007 337 Fig. 1. Direct computing of the sort modulation function at various modulation depths with ? rst through 109th harmonics deemled.Fig. 2. Direct calculation of the phase modul ation function at various modulation depths with ? rst through 177th harmonics find outled. i? cation is common in other PWM work, as in switching frequency randomization intended to reduce high-frequency components. A detailed palingenesis is outside the scope, but one discussion is given in 40. 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 resultant. The proposed method is readily implemented in real time.The switching heads 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 41, but further re? nement is require for precise elimination. II. SIGNAL DEFINITIONS AND SIMULATED RESULTS regard a quasi-triangular waveform to be used as the carrier manifestation in a PWM implementation. In principle, the frequency and phase can be spiel.To represent this, consider a triangular carrier function compose 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 quarter-circle de? nitions of the inverse cosine function. The modulating where signal will be represented as is the depth of modulation. The pulsewidth- play signal, , is 1 if and 1, otherwise. 2 In 39, 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 pick uped for various with harmonics 1109 conversus angle trolled. Fig. 2 shows the same with harmonics 1177 controlled. Many other sets of controlled harmonics were tried with similar results. The pattern looks much like a shockwave pattern that can be modeled with the BesselFubini equation from nonlinear acoustics 42 (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 on a lower floor. 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 look 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 thirteenth harto . monics vary 2%3% in magnitude as is vary from (4) 338IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY 2007 Fig. 3. Conventional harmonic elimination waveform and proposed PWM 0. 95, harmonics controlled through the 19th). waveform (m = Fig. 5. Pulse waveform p, heart and soul 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 bailiwicks, all baseband harmonics are eliminated. In three-phase systems, triplen harmonic s may cancel in the currents automatically if deaf(p) 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 38. The speed of astute these waveforms is dictated by , the number of terms to keep in the series (2), and , the number of distinct points used to approximate the waveforms. A in-person computer (1. 86-GHz Intel M central 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 an d 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 systematically 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 sudden than PC computations based on MATLAB. III. EXPERIMENTAL EXAMPLES 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 211. The signals the top, followed by the PWM waveform an d 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. . observational 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. Experiment al 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-mod ulator comparison process produces switching instants in real time. REFERENCES 1 F. G. Turnbull, Selected harmonic reduction in static dc-ac inverters, IEEE Trans. Commun. Electron. , vol. CE-83, pp. 374378, Jul. 1964. Fig. 8. Experimental modulation-based SHE with 0. = = = 20, m = 1. 0, quency ratio is adjusted to be 201, 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. 51, 340 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY 2007 2 H. S. Patel and R. G. 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