string(81) ‘ switching frequency randomization intended to lessen high-frequency components\. ‘
336 IEEE TRANSACTIONS ABOUT POWER CONSUMER ELECTRONICS, VOL. twenty two, NO . 1, JANUARY 2007 Modulation-Based Harmonic Elimination Jason R.
Wells, Member, IEEE, Xin Geng, Student Affiliate, IEEE, Tanker L. Chapman, Senior Affiliate, IEEE, Philip T. Krein, Fellow, IEEE, and Omfattande M. Nee, Student Affiliate, IEEE Abstract—A modulation-based means for generating heartbeat waveforms with selective harmonic elimination is usually proposed. Harmonic elimination, customarily digital, can be shown to be feasible by comparison of any sine trend with modi? d triangle carrier. The process can be used to compute easily and quickly the specified waveform with out solution of coupled transcendental equations. Index Terms—Pulsewidth modulation (PWM), picky harmonic reduction (SHE). I. INTRODUCTION T ELECTIVE harmonic elimination (SHE) is a long-established method of creating pulsewidth modulation (PWM) with low baseband distortion [1]–[6]. Originally, it was valuable mainly for inverters with naturally low switching frequency because of high power or slower switching equipment.
Conventional sine-triangle PWM essentially eliminates baseband harmonics intended for frequency percentages of about 15: 1 or greater [7], it is therefore arguable that SHE is unneeded. However , lately SHE has received new interest for several factors. First, digital implementation is becoming common. Second, it has been proven that there are various solutions to the SHE difficulty that were previously unknown [8]. Every single solution offers different rate of recurrence content over a baseband, which offers options to get? attening the high-frequency variety for noise suppression or optimizing ef? iency. Third, some applications, despite the accessibility to high-speed fuses, have low switching-to-fundamental percentages. One example can be high-speed motor unit drives, useful for reducing mass in applications like electric powered vehicles [9]. She actually is normally a two-step digital process. 1st, the turning angles happen to be calculated of? ine, for several depths of modulation, simply by solving various nonlinear equations simultaneously. Second, these aspects are kept in a look-up table to become read instantly. Much before work provides focused on the? st stage because of its computational dif? culty. One probability is to change the Fourier series ingredients with one other orthonormal arranged based on Walsh functions [10]–[12]. The resulting equations are more tractable due to the commonalities between the square Walsh function and the ideal waveform. One more orthonormal collection approach based on block-pulse functions is presented in [13]. In [14]–[20], it is noticed that Manuscript received August two, 2006, modified September 14, 2006.
This work was supported by the Grainger Center for Electrical Machines and Electromechanics, the Motorola Middle for Communication, the Nationwide Science Basis under Deal NSF 02-24829, the Energy Networks Ef? ciency, plus the Security (EPNES) Program in co-operation with the Of? ce of Naval Analysis. Recommended to get publication by simply Associate Publisher J. Espinoza. J. L. Wells is to use P. C. Krause and Associates, Hentschel Center, West Lafayette, IN 47906 UNITED STATES. X. Geng, P. D. Chapman, L. T. Krein, and N. M.
Pas du tout are together with the Grainger Centre for Electric Machines and Electromechanics, School of The state of illinois at Urbana-Champaign, Urbana, IL 61801 UNITED STATES (e-mail: [email, protected] edu). Digital Target Identi? er 10. 1109/TPEL. 2006. 888910 the moving over angles attained traditionally may be represented because regular-sampled PULSE WIDTH MODULATION where two phase-shifted modulating waves and a “pulse position modulation” technique attain near-ideal elimination. Another estimated method is carried by [21] in which mirror excessive harmonics are used. This involves solving multilevel elimination by considering reduced harmonic elimination waveforms in each switching level.
In [22], a general-harmonic-families eradication concept simpli? es a transcendental system to an algebraic functional problem by zeroing entire harmonic families. More quickly and more total methods are also researched. In [23], an optimum PWM issue is solved by converting to a single univariate polynomial using Newton identities, Pade estimation theory, and symmetric function properties, which in turn. If a handful of can be solved with methods that range as To solutions will be desired, prediction of preliminary guess values allows speedy convergence of Newton version [24].
Genetic methods can be used to rate the solution [25], [26]. An approach that assures all alternatives? t a narrowly posed SHE difficulty transforms to a multivariate polynomial system [27]–[30] through trigonometric identities [31] and solves with resultant polynomial theory. Another procedure [32]–[34] that obtains almost all solutions to a narrowly-posed difficulty uses homotopy and extension theory. Reference [35] points out the significantly growing characteristics of the issue and offers the “simulated annealing” technique as a way to quickly design the waveform to get optimizing bias and switching loss.
One other optimization-based procedure is given in [36] and [37], where harmonics are minimized through an objective function to obtain good general harmonic overall performance. There have been a number of multilevel and approximate real-time methods proposed, these are further than the scope here yet discussed brie? y in [38]. This manuscript proposes an alternative real-time THE GIRL method depending on modulation. A modi? education triangle transporter is identi? ed that is compared to a common sine trend. In place of the standard of? ine solution of switching perspectives, the process simpli? s to generation and comparison of the carrier and sine modulation, which can be required for minimal time without convergence or precision concerns. The strategy does not require an initial guess. In contrast to other SHE methods, the method would not restrict the switching regularity to an integer multiple with the fundamental. The underlying thought was recommended in [39] but have been re? ned here to spot speci? c carrier requirements that exactly eliminate harmonics and boost performance in deeper modulation. The method consists of a function of modulation depth that is created from simulation and curve? rollator walker. In this respect, it includes some likeness to [15] and [16], through which approximate moving over angles are calculated and? tted to simple features for circumstances of both equally low-( 0. 8 l. u. ) and high-modulation depth. It truly is interesting the fact that proposed strategy connects modulation to a harmonic elimination procedure. Carrier waveform mod- 0885-8993/$25. 00 © 2007 IEEE IEEE DEALS ON POWER ELECTRONICS, VOLUME. 22, NUMBER 1, JANUARY 2007 337 Fig. 1 . Direct computation of the period modulation function at various modulation absolute depths with? rst through 109th harmonics controlled.
Fig. installment payments on your Direct calculations of the period modulation function at different modulation absolute depths with? rst through 177th harmonics manipulated. i? cation is common consist of PWM operate, as in transitioning frequency randomization intended to reduce high-frequency elements.
The transitioning signals themselves can be made by analog comparison, even though the modi? male impotence carrier is generated with fast digital calculation and digital-to-analog transformation. Hardware demonstration is offered here. An approximate, low-cost setup based on present-day hardware is given in [41], although further lso are? nement is needed for correct elimination. II. SIGNAL EXPLANATIONS AND SIMULATED RESULTS Think about a quasi-triangular waveform to be applied as the carrier signal in a PULSE WIDTH MODULATION implementation. In principle, the frequency and phase can be modulated.
To represent this, think about a triangular transporter function written as (1) where is definitely the base transitioning frequency, is known as a phase-mod0, (1) reulation signal, and is a static period shift. To get duces to an ordinary triangle wave based upon conventional sector de? nitions of the inverse cosine function. The modulating where signal will be displayed as is the depth of modulation. The pulsewidth-modulated transmission, is one particular if and 1, otherwise. 2 In [39], a phase modulation function is considered, where is the ideal output important fre, nevertheless dequency. This was shown to way SHE in low zero.. To determine a better phase-modulagrades over tion function, the design of switching angles that occurs was looked at. Fig. 1 shows the phase modulation values necessary for various with harmonics 1–109 conversus position trolled. Fig. 2 displays the same with harmonics 1–177 controlled. Various other sets of controlled harmonics were examined with similar results. The routine looks very much like a shockwave pattern that can be modeled with the Bessel–Fubini formula from non-linear acoustics [42] (2) exactly where is a Bessel function with the? rst kind. The normal is in? ity in basic principle, but for calculation purposes amount 15 or higher is usually suf? cient, while discussed beneath. The and have been determined by curve functions? tting as (3) 1 . and (4), displayed at the bottom with the page, exactly where 0 Fig. 3 displays a closeup view of a PWM waveform generated as with (2). 19 harmonics happen to be with a transporter that uses 0. 96. The waveform is in contrast controlled using a (high) to just one generated with conventional eradication by statistical solution of nonlinear equations. As can be observed, the switching edges meet well. Fig. 4 displays a full-period time waveform and a magnitude 11.
With spectrum [fast Fourier change (FFT)] for this moving over frequency ratio, the method gets rid of harmonics two through ten (even harmonics are absolutely no by symmetry). The 2 and the modulation depth carrier period shift is placed to 1. The spectrum que incluye? rms the specified elimination. is definitely 0. This kind of value Fig. 5 reveals the same examine except with also achieves satisfactory baseband performance, but with a different heart beat pattern. The pattern delivers slight variations in higher-order harmonics. For example , the 11th and 13th saciado. monics differ 2%–3% in magnitude as varied coming from (4) 338
IEEE ORDERS ON ELECTRICAL POWER ELECTRONICS, VOL. 22, NUMBER 1, JANUARY 2007 Fig. 3. Standard harmonic elimination waveform and proposed PULSE WIDTH MODULATION 0. ninety five, harmonics managed through the 19th). waveform (m = Fig. 5. Pulse waveform l, message signal m, and magnitude of pulse waveform 1, and , zero. spectrum to get! =! 10, m = = sama dengan Fig. 4. Pulse waveform p, meaning signal meters, and degree of heart beat waveform spectrum for! sama dengan! 11, meters 1, and , =2. = sama dengan = In these cases, all baseband harmonics are eliminated. In three-phase systems, triplen harmonics may end in the currents automatically if neutral current does not? watts. Therefore it is not always necessary to get rid of them simply by design inside the SHE procedure. Modulation-based harmonic elimination excluding triplen harmonics is similar in many respects to the case here. However , the phase-modulation functions look like piecewise polynomials rather than the shockwave form of Figs. 1 and 2 . This can be discussed in more detail in [38]. The speed of calculating these waveforms is determined by, the quantity of terms to keep in the series (2), and, the number of under the radar points accustomed to approximate the waveforms. An individual computer (1. 86-GHz Intel M Processor with 1 . -GB RAM) running MATLAB on Or windows 7 was used to handle the computations. First, a modi? male impotence triangle wave was ap100 000 items per pattern, the modulation proximated with 1, and a frequency ratio of 19 utilized. depth was set to was varied by? ve to 35. More than this selection, the The quantity quality of solution was acceptable plus the average calculations time different from zero. 327 to 0. 915 s. Next, the same circumstances 35 and was varied from 12 000 were used with except to 2 hundred 000. The average calculation period varied practically linearly via 0. 149 to 1. 80 s without having signi? cannot difference in the resulting range.
Finally, with held constant at 95 000, the frequency ratio was different from eight to 51. The average computation time was constantly near zero. 92 t. This is anticipated since the range of harmonics removed has no your own effect in (2). However , for greater frequency percentages, larger may be needed for precision. In summary, it is recommended that be set to at least 1, 1000 the consistency ratio and set to at least 12-15. In any case, with present-day pcs the solution could be calculated in under 1 t (typically) devoid of iteration, divergence, or requirement of an initial estimate, and decreased versions can be computed in under 200 ms.
Notice that this time interval do not need to cause difficulty with current implementations. The carrier just needs to be recomputed with the modulation signal improvements. In applications such as uninterruptible supplies, this is infrequent. In motor-drive applications, a response time of 200 ms to a command change might be acceptable being. Alternatively, a look-up desk can retail outlet some of the relevant terms to speed up the procedure dramatically. Devoted DSP Please de? nenni DSPalgorithms will be much faster than PC calculations based on MATLAB. III. EXPERIMENTAL EXAMPLES To demonstrate that the recommended technique satisfactorily eliminates harmonics, the modi? d transporter was developed into a function generator. The outcome provided the flagship signal within a conventional sine-triangle process. Three examples happen to be shown listed below to reveal a variety of interesting conditions. Fig. 6 reveals the causing waveforms for the high-depth case 0, and 0. 95. The with nineteen harmonics eliminated, and therefore are shown by frequency proportion is 21: 1 . The signals the best, followed by the PWM waveform and the FFT spectrum. From the spectrum it can be seen the fact that desired harmonic-free baseband spectrum is obtained. In the next case in point, the period 2 .
The unexpected effect was that the spectrum switch is was insensitive to, as displayed in comparison to Fig. 7. The desired spectrum happens despite the big difference in companies. The producing PWM waveforms at several values of may not offer obvious advantages, but it is noteworthy that they can be not the same as conventionally computed THE LADY waveforms and would not end up being achievable with conventional SHE solution techniques. As another case, it is shown the fact that carrier bottom fre, do not need to be a strange multiple of. In Fig. 8, the frequency, IEEE TRANSACTIONS IN POWER GADGETS, VOL. twenty-two, NO . 1, JANUARY 3 years ago 339 Fig.. Experimental modulation-based SHE with! , 0. = =! = twenty one, m = 0. 95, Fig. on the lookout for. Experimental modulation-based SHE with! , 0. = =! = 13. 5, meters = 0. 95, Fig. 7. Trial and error modulation-based THE LADY with! , =2. = =! sama dengan 21, m = zero. 95, Fig. 10. Experimental modulation-based SHE with! , 0. = =! = 50, m = zero. 95, The past example, shown in Fig. 10, applies to a case where a high number of harmonics is usually eliminated (50 1 ratio) effectively, which is much higher than typically will be reported in the literature. IV. CONCLUSION A way for establishing and putting into action SHE switching angles was proposed and demonstrated.
The method is based on modulation rather than option of non-linear equations or numerical optimization. The approach is based on a modi? education carrier waveform that can be worked out based on concise functions requiring only depth of modulation as suggestions. It speedily calculates the desired switching waveforms while staying away from iteration and initial quotes. Calculation time is insensitive to the moving over frequency rate so eradication of many harmonics is straightforward. It is conceivable the technique could possibly be realized with low-cost microcontrollers for real-time implementation.
As soon as the carrier is usually computed, a conventional carrier-modulator assessment process makes switching instants in real time. REFERRALS [1] Farreneheit. G. Turnbull, “Selected harmonic reduction in stationary dc-ac inverters, ” IEEE Trans. Petit. Electron., volume. CE-83, pp. 374–378, Jul. 1964. Fig. 8. Experimental modulation-based SHE with , 0. =! =! sama dengan 20, m = 1 ) 0, quency ratio is adjusted to become 20: you, with 0, and now 1 ) 0. Similar nineteen harmonics are taken away, but now the switching consistency is 5% lower. Time periods during which the carrier waveform is certainly not triangular can be seen in the? gure. As demonstrated in Fig., the rate of recurrence ratio can also be a half-in0. 95 and 0. teger. In this case, the ratio is 13. five: 1, 340 IEEE DEALS ON ELECTRICITY ELECTRONICS, VOL. 22, NO . 1, JANUARY 2007 [2] H. H. Patel and R. G. Hoft, “Generalized techniques of harmonic reduction and voltage control in thyristor inverters: part I-harmonic elimination, ” IEEE Trans. Ind. Appl., vol. IA-9, no . several, pp. 310–317, May/ Jun. 1973. [3] ——, “Generalized techniques of harmonic eradication and ac electricity control in thyristor inverters: part II-voltage control techniques, ” IEEE Trans. Ind. Appl., volume. IA-10, number 5, pp. 666–673, Sep. /Oct. 1974. [4] We.
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