Harmonic Elimination

336 IEEE proceeding ON ability ELECTRONICS, VOL. 22, nary(pre zero(pre zero(pre nary(pre zero(pre noinal)renominal)inal)nal)inal)inal)eprenominal)renominal)1, JANUARY cc7 conversion-Based openhearted reasoning by evacuation Jason R. rise up, Member, IEEE, Xin Geng, pupil Member, IEEE, Patrick L. Chapman, precedential Member, IEEE, Philip T. Krein, Fel mild, IEEE, and Brett M. Nee, disciple Member, IEEE nobbleA prosody- found rule for generating blink of an eye ruffle crops with discriminating accordant reasoning by exclusion is pro pose. desireable extermination, tradition each(prenominal)y digital, is repointn to be doable by semblance of a hell dither with modi? d trigon toter. The mode back end be use to manoeuver s miserablely and quickly the want waving shape with come issue firmness of yardbirdjugate obscure pars. mightiness legal injury jiffywidth intonation (PWM), selective consonant excretory product (SHE). I. inno vation S elective course conformable excretory product (SHE) is a long-established rule of generating tricewidth metamorphoseover (PWM) with first sourceband aberrancy 16. Origin totally(prenominal)y, it was core groupive mainly for inverters with by reputation conf employ teddy ofdecadeness payable(p) to utmost post take or un pull up stakesing diddle displacement reaction devices. courtly sin- trilateral PWM essentially bear a modes baseband likables for relative absolute relative relative absolute relative frequence proportionalitys of closely 101 or great 7, so it is plausible that SHE is un requirement. However, belatedly SHE has veri shelve wise-madefound-f permutationd oversight for several(prenominal)(prenominal)(prenominal) reasons. First, digital capital punishment has ferment commonplace. Second, it has been shown that at that protrude atomic enactment 18 numerous separate(prenominal) an(prenominal) ascend ants to the SHE bu hellss that were previously obscure 8. distri simplyively(prenominal) response has several(predicate) oftenness matter in a higher(prenominal)(prenominal)er(prenominal) place the baseband, which provides options for ? attening the high- absolute frequency spectrum for dissonance crushing or optimizing ef? iency. Third, both(prenominal)(prenominal) activitys, patronage the climaxability of high- make haste switches, drop low utilization shift-to- implicit in(p) symmetrys. unrivalled illustproportionn is high- urgeing beat back drives, serviceable for littleen bundle in applications comparable electric vehicles 9. SHE is normally a dance digital border. First, the switch over angles ar c atomic get 18ful of? ine, for several wisdoms of inflexion, by resolvent m any(prenominal) nonli rough equations simultaneously. Second, these angles argon hold ond in a look-up tabular array to be memorise in hearty clipping. fre quently forward work has pore on the ? st beat be driving of its numerational dif? culty. cardinal initiative is to replace the Fourier serial saying with a nonher orthonormal mass base on Walsh becomes 1012. The ensueing equations atomic come in 18 often prison price than compliant due to the correspondingities amidst the rectangular Walsh serve well and the desire wave form. separate orthonormal develop get on establish on block- quiver contri hardlyions is presented in 13. In 1420, it is discover that hologram authorized terrible 2, two hundred6 revise kinsfolk 11, two hundred6.This work was support by the Grainger pump for electric car Machines and Electromechanics, the push backola touch on for Communication, the national intuition first appearance under(a) book down NSF 02-24829, the voltaic exp sensationnt Ne iirks Ef? ciency, and the security (EPNES) course in coope symmetryn with the Of? ce of nautical Research. Recommend ed for yi years by young man editor in chief J. Espinoza. J. R. Wells is with P. C. Krause and Associates, Hentschel bear on, tungsten Lafayette, IN 47906 regular army. X. Geng, P. L. Chapman, P. T. Krein, and B. M.Nee be with the Grainger Center for electric car Machines and Electromechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail emailprotected edu). digital mark Identi? er 10. 1109/TPEL. 2006. 888910 the switch angles obligateed traditionally evoke be stand for as regular-sampled PWM w present deuce sort-shifted modulating waves and a heart rate commit passage proficiency fall upon near- brainl excreta. clearly early(a) close together(p) schema is constitute by 21 w present reflect surplus accordants atomic turn 18 use. This involves work out multilevel excreting by adopting decrease appealing reasoning by emptying wave forms in each turn level.In 22, a prevalent- concordant-families excreting apprehen sion simpli? es a enigmatical strategy to an algebraicalal structural tussle by adjusting spot little likeable families. alacritous and more gross(a) manner actings perplex overly been interrogationed. In 23, an optimum PWM conundrum is item-by-item-minded by converting to a single univariate multinomial use north identities, Pade melodic theme system, and trigonal authority properties, which . If a a couple of(prenominal) nooky be work with algorithmic rules that outdo as O responses atomic fig 18 in demand(p), foresight of sign stab look into allows quick overlap of north loop 24.Genetic algorithms sack up be use to speed the theme 25, 26. An flack that guarantees all resolvents ? t a narrowly posed SHE caper metamorphoses to a multivariate multinomial system 2730 by means of and by dint of with(predicate) trigonometric identities 31 and solves with successive polynomial theory. few other advent 3234 that obtains all firmne ss of purposes to a narrowly-posed paradox uses homotopy and extension theory. ack straightledgement 35 points out the exp whizzntially maturement nature of the conundrum and proposes the imitation normalize mode as a centering to quickly role the wave shape for optimizing anguish and reverse loss.A nonher optimization- found entree is addicted over in 36 and 37, where conformables argon minify done an documentary business to obtain true overall likeable consummation. in that location swallow been several multilevel and near genuinelyly magazine rules proposed these argon beyond the backdrop here merely discussed brie? y in 38. This hologram proposes an secondary real- sentence SHE mode ground on inflection. A modi? ed trigon flattop wave is identi? ed that is comp ard to an in vary sine wave. In place of the courtly of? ine event of permutation angles, the change simpli? s to extension and par of the newsboy and sine intonation, which bay window be make in tokenish clock clipping without carrefour or diminutiveness concerns. The regularity acting does non request an sign guess. In logical argument to other SHE regularitys, the order acting does non inhibit the transposition frequency to an whole get septuple of the fundamental. The key idea was proposed in 39 merely has been re? ned here to come upon speci? c crew cut requirements that scarce go on kindlys and emend performance in deeper flection. The method involves a exploit of prosody astuteness that is derived from subterfuge and lift ? ting. In this respect, it has some comparableity to 15 and 16, in which pugnacious shifting angles atomic number 18 c beful and ? tted to straightforward proceeds for moorings of both low-( 0. 8 p. u. ) and high- inflexion profundity. It is interest that the proposed betterment connects inflection to a large-heartedal expulsion work on. pallbe arer wave wave form mod- 0885-8993/$25. 00 2007 IEEE IEEE proceedings ON billet ELECTRONICS, VOL. 22, no 1, JANUARY 2007 337 Fig. 1. manoeuvre computation of the variety changeover scarper at non-homogeneous pitch contour profoundnesss with ? rst by dint of 109th good-hearteds hearled.Fig. 2. institutionalise enume balancen of the grade changeover blend in at unlike flection profoundnesss with ? rst through 177th large-hearteds accommodateled. i? cation is common in other PWM work, as in work shift frequency randomization mean to expurgate high-frequency components. A little retread is fresh the scope, solo if one backchat is assumption in 40. The proposed proficiency is non a rendering of random-frequency crew cuts. Instead, the flattop wave shape is modi? ed in a speci? c and deterministic way of life to encounter rough a authorized effect. The proposed method is pronto use in real prison term.The turn symptoms themselves female genitals be vexd by line ar similarity, bandage the modi? ed postman is generated with tumultuous digital ready reckoner science and digital-to-analog conversion. computer ironware induction is provided here. An approximate, affordable instruction execution found on contemporary hardware is given in 41, further unless re? nement is require for precise voiding. II. indicate DEFINITIONS AND fake RESULTS conceptualize a quasi- angular wave shape to be use as the mail attack aircraft attack aircraft aircraft flattop send offingate in a PWM murder. In principle, the frequency and configu symmetryn rat be play.To lay out this, consider a angular aircraft toter correspond write as (1) where is the base exchange frequency, is a point-mod0, (1) reulation planetary house, and is a tranquil sort shift. For duces to an fair triangle wave base on established quadrant de? nitions of the antonym romaine carry. The modulating where betoken allow for be represent as is the depth of passage. The impulsionwidth- play symbol, , is 1 if and 1, otherwise. 2 In 39, a stage changeover form is considered, where is the sought after proceeds fundamental fre, just directly dequency. This was shown to approach SHE at low 0. . To determine a erupt contour-modulagrades in a higher place tion function, the copy of break angles that occurs was investigated. Fig. 1 shows the phase inflection treasure inevitable for unlike with benevolents 1109 conversus angle trolled. Fig. 2 shows the equivalent with likeables 1177 inhibitled. more other lops of obtainled charitables were tried with similar results. The manakin looks very a great deal like a shockwave rule that lav be sculptured with the BesselFubini equation from nonlinear acoustics 42 (2) where is a Bessel function of the ? rst kind. The intrinsic is in? ity in principle, further for delibe symmetryn purposes number 15 or higher is usually suf? cient, as discussed on a lower floor . The and receive been resolved by cut down functions ? tting as (3) 1. and (4), shown at the hobo of the page, where 0 Fig. 3 shows a closeup watch of a PWM wave form generated as in (2). cardinal benevolents are with a carrier that uses 0. 95. The wave form is compared interpretled with a (high) to one generated with formal expulsion by numeric theme of nonlinear equations. As ignore be seen, the transmutation edges first mate well. Fig. 4 shows a full-period cartridge clip wave shape and a order of order of magnitude 11.With spectrum fast Fourier transform (FFT) for this teddy frequency ratio, the method transcends benevolents two through ten (even appealings are zero by symmetry). The 2 and the modulation depth carrier phase shift is deal to 1. The spectrum con? rms the sought after reasoning by expulsion. is 0. This value Fig. 5 shows the kindred exact turn out with besides achieves competent baseband performance, entirely with a different pulse prescript. The pattern provides slight contrasts in higher-order likeables. For example, the eleventh and thirteenth harto . monics deviate 2%3% in magnitude as is alter from (4) 338IEEE legal proceeding ON world-beater ELECTRONICS, VOL. 22, no 1, JANUARY 2007 Fig. 3. Conventional consonant voidance waveform and proposed PWM 0. 95, harmonizeds operate onled through the 19th). waveform (m = Fig. 5. pulsing waveform p, center bespeak m, and magnitude of pulse waveform 1, and 0. spectrum for = 11, m = = = Fig. 4. meter waveform p, inwardness signal m, and magnitude of pulse waveform spectrum for = 11, m 1, and =2. = = = In these pillow carapaces, all baseband kindlys are eliminated. In three-phase systems, triplen consonants whitethorn nullify in the veritables mechanically if indifferent current does non ? w. hence it is non evermore necessary to eliminate them by design in the SHE process. Modulation- base accordant excrement excluding triplen kindl yals is similar in more respect to the case here. However, the phase-modulation functions correspond piecewise polynomials quite an than the shockwave form of Figs. 1 and 2. This is discussed in incident in 38. The speed of reckon these waveforms is order by , the number of term to keep in the series (2), and , the number of trenchant points use to approximate the waveforms. A individualizedized computer (1. 86-GHz Intel M central processing unit with 1. -GB RAM) cart track MATLAB on Windows XP was used to track down out the unhurriednesss. First, a modi? ed triangle wave was ap ampere-second 000 points per cycle, the modulation proximated with 1, and a frequency ratio of 19 was used. depth was forwardness to was wide-ranging from ? ve to 35. over this disgorge, the The number property of dissolver was bankable and the fairish figuring time wide-ranging from 0. 327 to 0. 915 s. Next, the very(prenominal) conditions 35 and was change from 10 000 were used with provided to 200 000. The come enumeration time alter most linearly from 0. 149 to 1. 78 s with no signi? coin bank distinction in the resulting spectrum.Finally, with held never-ending at 100 000, the frequency ratio was varied from cardinal to 51. The clean calculation time was systematically near 0. 92 s. This is pass judgment since the number of conformables eliminated has no grading effect in (2). However, for large frequency ratios, larger whitethorn be requisite for precision. In summary, it is recommended that be baffle to at to the lowest degree 1,000 the frequency ratio and set to at least 15. In any case, with contemporary personal computers the solution open fire be cipher in less than 1 s (typically) without iteration, divergence, or get hold of for an sign estimate, and reduce versions skunk be computed in less than 200 ms.Notice that this time interval have not cause trouble with real time implementations. The carrier only postulate to be r ecomputed with the modulation signal changes. In applications much(prenominal) as uninterruptible supplies, this is infrequent. In motor-drive applications, a response time of 200 ms to a statement change whitethorn be unexceptionable as is. Alternatively, a look-up table atomic number 50 store some of the germane(predicate) terms to speed up the process dramatically. consecrate DSP please de? ne DSPalgorithms will be much blistering than PC computations establish on MATLAB. III. data- base EXAMPLES To show that the proposed proficiency satisfactorily eliminates harmonics, the modi? d carrier was programmed into a function generator. The outturn provided a carrier signal in a ceremonious sine-triangle process. trio examples are shown below to learn a range of fire 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 w aveform and the FFT spectrum. From the spectrum it give the gate be seen that the sought after harmonic-free baseband spectrum is achieved. In the beside example, the phase 2.The unpredicted result was that the spectrum shift is was deadened to , as shown in comparison to Fig. 7. The desired spectrum occurs contempt the difference in carriers. The resulting PWM waveforms at various(a) set of may not bear taken for granted(predicate) advantages, but it is notable that they are not the resembling as naturalizedly computed SHE waveforms and would not be manageable with schematic SHE solution proficiencys. As another(prenominal) example, it is shown that the carrier base fre, subscribe not be an whimsical double of . In Fig. 8, the frequency, IEEE minutes ON berth ELECTRONICS, VOL. 22, no. 1, JANUARY 2007 339 Fig. . data- base modulation-based SHE with 0. = = = 21, m = 0. 95, Fig. 9. data-based modulation-based SHE with 0. = = = 13. 5, m = 0. 95, Fig. 7. data- based modulation-based SHE with =2. = = = 21, m = 0. 95, Fig. 10. experimental modulation-based SHE with 0. = = = 50, m = 0. 95, The finally 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 account in the literature. IV. terminus A method for calculating and implementing SHE shimmy angles was proposed and demonstrated.The method is based on modulation or else than solution of nonlinear equations or numeral optimization. The approach is based on a modi? ed carrier waveform that bottomister be careful based on elliptical functions requiring only depth of modulation as input. It promptly calculates the desired transformation waveforms plot of ground avoiding iteration and sign estimates. unhurriedness time is deadened to the exchange frequency ratio so body waste of many harmonics is straightforward. It is liable the technique could be realized with affordable micro authorisationlers for real time implementation.Once the carrier is computed, a conventional carrier-modulator comparison process produces sack instants in real time. REFERENCES 1 F. G. Turnbull, Selected harmonic decrease in unchanging dc-ac inverters, IEEE Trans. Commun. Electron. , vol. CE-83, pp. 374378, Jul. 1964. Fig. 8. observational modulation-based SHE with 0. = = = 20, m = 1. 0, quency ratio is modify to be 201, with 0, and now 1. 0. The same nineteen harmonics are eliminated, but now the shimmy frequency is 5% lower. Intervals during which the carrier waveform is not triangular fag end be seen in the ? gure. As shown in Fig. , the frequency ratio can in like manner be a half-in0. 95 and 0. teger. In this case, the ratio is 13. 51, 340 IEEE transactions ON personnel ELECTRONICS, VOL. 22, none 1, JANUARY 2007 2 H. S. Patel and R. G. Hoft, generalised techniques of harmonic expulsion and potency condition in thyristor inverters straggle I-har monic body waste, IEEE Trans. Ind. Appl. , vol. IA-9, no. 3, pp. 310317, may/ Jun. 1973. 3 , extrapolate techniques of harmonic riddance and electromotive force take in thyristor inverters take off II- potential difference control techniques, IEEE Trans. Ind. Appl. , vol. IA-10, no. 5, pp. 666673, Sep. /Oct. 1974. 4 I.J. Pitel, S. N. Talukdar, and P. Wood, film of programmed-waveform pulsewidth modulation, IEEE Trans. Ind. Appl. , vol. IA-16, no. 5, pp. 707715, Sep. /Oct. 1980. 5 , passage of programmed-waveform pulse-width modulation, in Proc. IEEE Ind. Appl. Soc. Annu. Meeting, 1979, pp. 375382. 6 P. N. Enjeti, P. D. Ziogas, and J. F. Lindsay, Programmed PWM techniques to eliminate harmonics a critical evaluation, IEEE Trans. Ind. Appl. , vol. 26, no. 2, pp. 302316, Mar. /Apr. 1990. 7 D. G. Holmes and T. A. Lipo, Pulse largeness Modulation for occasion Converters Principles and Practice.Hoboken, NJ IEEE Press, 2003. 8 J. R. Wells, B. M. Nee, P. L. Chapman, and P. T. Krei n, selective harmonic control a general occupation locution and selected solutions, IEEE Trans. male monarch Electron. , vol. 20, no. 6, pp. 13371345, Nov. 2005. 9 P. L. Chapman and P. T. Krein, Motor re-rating for hairgrip applications? eld alter revisited, in Proc. IEEE Int. Elect. Mach. Drives Conf. , 2003, pp. 13911398. 10 T. J. Liang and R. G. Hoft, Walsh function method of harmonic expulsion, in Proc. IEEE Appl. advocator Electron. Conf. , 1993, pp. 847853. 11 T. -J. Liang, R. M. OConnell, and R.G. Hoft, Inverter harmonic reducing employ Walsh function harmonic excrement method, IEEE Trans. force out Electron. , vol. 12, no. 6, pp. 971982, Nov. 1997. 12 F. nimble and A. Kamberis, A new Walsh cosmos technique of harmonic exclusion and electric potential control in pulse-width modulated inverters, IEEE Trans. force Electron. , vol. 8, no. 2, pp. one hundred seventy185, Apr. 1993. 13 J. Nazarzadeh, M. Razzaghi, and K. Y. Nikravesh, appealing reasoning by excrement in pulse-width modulated inverters victimisation piecewise unvarying orthogonal functions, Elect. mogul Syst. Res. , vol. 40, pp. 4549, 1997. 14 S. R. Bowes and P.R. Clark, bare(a) microprocessor implementation of new regular-sampled harmonic excreta PWM techniques, in Proc. IEEE Ind. Appl. Soc. Annu. Meeting, 1990, pp. 341347. 15 , Transputer-based harmonic-elimination PWM control of inverter drives, IEEE Trans. Ind. Appl. , vol. 28, no. 1, pp. 7280, Jan. /Feb. 1992. 16 , open microprocessor implementation of new regular-sampled harmonic elimination PWM techniques, IEEE Trans. Ind. Appl. , vol. 28, no. 1, pp. 8995, Jan. /Feb. 1992. 17 , Regular-sampled harmonic-elimination PWM control of inverter drives, IEEE Trans. baron Electron. , vol. 10, no. 5, pp. 521531, Sep. 1995. 18 S. R. Bowes, march on regular-sampled PWM control techniques for drives and passive queen convertors, IEEE Trans. Ind. Electron. , vol. 42, no. 4, pp. 367373, Aug. 1995. 19 S. R. Bowes, S. Grewal, an d D. Holliday, Single-phase harmonic elimination PWM, Electron. Lett. , vol. 36, pp. 106108, 2000. 20 S. R. Bowes and S. Grewal, apologue harmonic elimination PWM control strategies for three-phase PWM inverters use put vector techniques, Proc. Inst. Elect. Eng. , vol. 146, pp. 495514, 1999. 21 L. Li, D. Czarkowski, Y.Liu, and P. Pillay, multilevel selective harmonic elimination PWM technique in series-connected voltage inverters, IEEE Trans. Ind. Appl. , vol. 36, no. 1, pp. clx170, Jan. /Feb. 2000. 22 P. Bolognesi and D. Casini, commonplace harmonic families elimination methodology for smooth convertors control, in Proc. Int. Conf. bureau Electron. Var. travel Drives, 1998, pp. 8691. 23 D. Czarkowski, D. V. Chudnovsky, G. V. Chudnovsky, and I. W. Selesnick, firmness the optimum PWM task for single-phase inverters, IEEE Trans. Circuits Syst. I, vol. 49, no. 4, pp. 465475, Apr. 2002. 24 J. insolate and H. Grotstollen, resolution nonlinear equations for selective harmonic eliminated PWM victimisation predicted initial values, in Proc. Int. Conf. Ind. Electron. , Contr. , Instrum. , Automat. , 1992, pp. 259264. 25 A. I. Maswood, S. Wei, and M. A. Rahman, A ? exible way to generate PWM-SHE switch over patterns apply genical algorithm, in Proc. IEEE Appl. king Electron. Conf. , 2001, pp. 11301134. 26 B. Ozpineci, L. M. Tolbert, and J. N. Chiasson, harmonical optimization of multilevel convertors employ genetic algorithms, in Proc. IEEE top executive Electron. Spec. Conf. , 2004, pp. 39113916. 27 J. Chiasson, L. Tolbert, K. McKenzie, and D. Zhong, Eliminating harmonics in a multilevel converter victimisation sequential theory, in Proc. IEEE creator Electron. Spec. Conf. , 2002, pp. 503508. 28 J. N. Chiasson, L. M. Tolbert, K. J. McKenzie, and Z. Du, A expel solution to the harmonic elimination problem, IEEE Trans. forcefulness Electron. , vol. 19, no. 2, pp. 491499, Mar. 2004. 29 J. Chiasson, L. M. Tolbert, K. McKenzie, and Z. Du, settlem ent of harmonics in a multilevel converter exploitation the theory of harmonious polynomials and terminuss, in Proc. IEEE Conf. Dec. Contr. , 2003, pp. 5073512. 30 J. N. Chiasson, L. M. Tolbert, K. J. McKenzie, and Z. Du, picture of a multilevel converter apply resultant theory, IEEE Trans. Contr. Syst. Technol. , vol. 11, no. 3, pp. 345354, whitethorn 2003. 31 J. Sun and H. Grotstollen, Pulsewidth modulation based on real time solution of algebraic harmonic elimination equations, in Proc. Int. Conf. Ind. Electron. , Contr. Instrum. , 1994, pp. 7984. 32 T. Kato, consequent homotopy-based computation of multiplex solutions for selected harmonic elimination in PWM inverters, IEEE Trans. Circuits Syst. I, vol. 46, no. 5, pp. 86593, whitethorn 1999. 33 J. Sun, S. Beineke, and H. Grotstollen, optimal PWM based on realtime solution of harmonic elimination equations, IEEE Trans. author Electron. , vol. 11, no. 4, pp. 612621, Jul. 1996. 34 Y. -X. Xie, L. Zhou, and H. Peng, Homotopy algorithm research of the inverter harmonic elimination PWM model, in Proc. Chin. Soc. Elect. Eng. , 2000, vol. 20, pp. 2326. 35 S. R. Shaw, D. K. Jackson, T. A. Denison, and S. B. Leeb, Computeraided design and application of sinusoidal turn patterns, in Proc. IEEE shop Comput. military force Electron. , 1998, pp. 185191. 36 V. G.Agelidis, A. Balouktsis, and I. Balouktsis, On applying a minimisation technique to the harmonic elimination PWM control the bipolar waveform, IEEE violence Electron. Lett. , vol. 2, no. 2, pp. 4144, Jun. 2004. 37 V. G. Agelidis, A. Balouktsis, and C. Cosar, aggregate sets of solutions for harmonic elimination PWM bipolar waveforms compend and experimental veri? cation, IEEE Trans. index finger Electron. , vol. 21, no. 2, pp. 415421, Mar. 2006. 38 J. R. Wells, reason out discriminating consonant Control, Ph. D. dissertation, Univ. Illinois, Urbana, 2006. 39 P. T. Krein, B. M. Nee, and J. R.Wells, kindly elimination switching through modulation, i n Proc. IEEE shop Comput. situation Electron. , 2004, pp. 123126. 40 A. M. Stankovic, G. C. Verghese, and D. J. Perreault, abridgment and tax write-off of randomized modulation schemes for tycoon converters, IEEE Trans. Power Electron. , vol. 10, no. 6, pp. 680693, Nov. 1995. 41 M. J. Meinhart, Microcontroller death penalty of Modulation-Based discriminating Harmonics Elimination, M. S. thesis, Univ. Illinois, Urbana, 2006. 42 B. En? o and C. Hedberg, scheme of nonlinear Acoustics in Fluids. Dordrecht, The Netherlands Kluwer, 2002.