Mbhe Final

8/3/2019 Mbhe Final

1/5

336 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

AbstractA modulation-based method for generating pulsewaveforms with selective harmonic elimination is proposed. Har-monic elimination, traditionally digital, is shown to be achievableby comparison of a sine wave with modified triangle carrier. Themethod can be used to calculate easily and quickly the desiredwaveform without solution of coupled transcendental equations.

Index TermsPulsewidth modulation (PWM), selective har-monic elimination (SHE).

I. INTRODUCTION

S

ELECTIVE harmonic elimination (SHE) is a long-estab-lished method of generating pulsewidth modulation (PWM)

with low baseband distortion [1][6]. Originally, it was usefulmainly for inverters with naturally low switching frequency dueto high power level or slow switching devices. Conventionalsine-triangle PWM essentially eliminates baseband harmonicsfor frequency ratios of about 10:1 or greater [7], so it is arguablethat SHE is unnecessary. However, recently SHE has receivednew attention for several reasons. First, digital implementationhas become common. Second, it has been shown that there aremany solutions to the SHE problem that were previously un-known [8]. Each solution has different frequency content abovethe baseband, which provides options for flattening the high-fre-quency spectrum for noise suppression or optimizing efficiency.Third, some applications, despite the availability of high-speedswitches, have low switching-to-fundamental ratios. One ex-ample is high-speed motor drives, useful for reducing mass inapplications like electric vehicles [9].

SHE is normally a two-step digital process. First, theswitching angles are calculated offline, for several depths ofmodulation, by solving many nonlinear equations simulta-neously. Second, these angles are stored in a look-up tableto be read in real time. Much prior work has focused on thefirst step because of its computational difficulty. One possi-bility is to replace the Fourier series formulation with anotherorthonormal set based on Walsh functions [10][12]. Theresulting equations are more tractable due to the similarities

between the rectangular Walsh function and the desired wave-form. Another orthonormal set approach based on block-pulsefunctions is presented in [13]. In [14][20], it is observed that

Manuscript received August 2, 2006; revised September 11, 2006. This workwas supported by the Grainger Center for Electric Machines and Electrome-chanics, the Motorola Center for Communication, the National Science Foun-dation under Contract NSF 02-24829, the Electric Power Networks Efficiency,and the Security (EPNES) Program in cooperation with the Office of Naval Re-search. Recommended for publication by Associate Editor J. Espinoza.

J. R. Wells is with P. C. Krause and Associates, Hentschel Center, WestLafayette, IN 47906 USA.

X. Geng, P. L. Chapman, P. T. Krein, and B. M. Nee are with the GraingerCenter for Electric Machines and Electromechanics, University of Illinois atUrbana-Champaign, Urbana, IL 61801 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TPEL.2006.888910

the switching angles obtained traditionally can be representedas regular-sampled PWM where two phase-shifted modulatingwaves and a pulse position modulation technique achievenear-ideal elimination. Another approximate method is posedby [21] where mirror surplus harmonics are used. This involvessolving multilevel elimination by considering reduced har-monic elimination waveforms in each switching level. In [22],a general-harmonic-families elimination concept simplifies atranscendental system to an algebraic functional problem byzeroing entire harmonic families.

Faster and more complete methods have also been researched.In [23], an optimal PWM problem is solved by converting to asingle univariate polynomial using Newton identities, Pad ap-

proximation theory, and symmetric function properties, whichcan be solved with algorithms that scale as O . If a fewsolutions are desired, prediction of initial guess values allowsrapid convergence of Newton iteration [24]. Genetic algorithmscan be used to speed the solution [25], [26]. An approach thatguarantees all solutions fit a narrowly posed SHE problem trans-forms to a multivariate polynomial system [27][30] throughtrigonometric identities [31] and solves with resultant polyno-mial theory. Another approach [32][34] that obtains all solu-tions to a narrowly-posed problem uses homotopy and continua-tion theory. Reference [35] points out the exponentially growingnature of the problem and proposes the simulated annealingmethod as a way to rapidly design the waveform for optimizing

distortion and switching loss. Another optimization-based ap-proach is givenin [36] and [37], where harmonics are minimizedthrough an objective function to obtain good overall harmonicperformance. There have been several multilevel and approxi-mate real-time methods proposed; these are beyond the scopehere but discussed briefly in [38].

This manuscript proposes an alternative real-time SHEmethod based on modulation. A modified triangle carrier isidentified that is compared to an ordinary sine wave. In placeof the conventional offline solution of switching angles, theprocess simplifies to generation and comparison of the carrierand sine modulation, which can be done in minimal time

without convergence or precision concerns. The method doesnot require an initial guess. In contrast to other SHE methods,the method does not restrict the switching frequency to aninteger multiple of the fundamental. The underlying idea wasproposed in [39] but has been refined here to identify specificcarrier requirements that exactly eliminate harmonics and im-prove performance in deeper modulation. The method involvesa function of modulation depth that is derived from simulationand curve fitting. In this respect, it has some similarity to [15]and [16], in which approximate switching angles are calculatedand fitted to simple functions for cases of both low-( 0.8 p.u.)and high-modulation depth.

It is interesting that the proposed approach connects modula-

tion to a harmonic elimination process. Carrier waveform mod-

0885-8993/$25.00 2007 IEEE

8/3/2019 Mbhe Final

2/5

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY 2007 337

Fig. 1. Direct calculation of the phase modulation function at various modula-tion depths with first through 109th harmonics controlled.

ification is common in other PWM work, as in switching fre-quency randomization intended to reduce high-frequency com-

ponents. A detailed review is outside the scope, but one discus-sion is given in [40]. The proposed technique is not a variation

of random-frequency carriers. Instead, the carrier waveform is

modified in a specific and deterministic way to bring about acertain effect.

The proposed method is readily implemented in real time.

The switching signals themselves can be generated by analog

comparison, while the modified carrier is generated with fastdigital calculation and digital-to-analog conversion. Hardware

demonstration is provided here. An approximate, low-cost im-

plementation based on present-day hardware is given in [41],

but further refinement is needed for precise elimination.

II. 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 trian-

gular carrier function written as

(1)

where isthe baseswitching frequency, isa phase-mod-

ulation signal, and is a static phase shift. For 0, (1) re-

duces to an ordinary triangle wave based on conventional quad-

rant definitions of the inverse cosine function. The modulatingsignal will be represented as where

is the depth of modulation. The pulsewidth-modulated signal,

, is 1 if and 1, otherwise.

In [39], a phase modulation function 2

is considered, where is the desired output fundamental fre-

quency. This was shown to approach SHE at low , but de-

grades above 0.8. To determine a better phase-modula-

tion function, the pattern of switching angles that occurs was

investigated. Fig. 1 shows the phase modulation values needed

Fig. 2. Direct calculation of the phase modulation function at various modula-tion depths with first through 177th harmonics controlled.

versus angle for various with harmonics 1109 con-trolled. Fig. 2 shows the same with harmonics 1177 controlled.

Many other sets of controlled harmonics were tested with sim-ilar results. The pattern looks much like a shockwave pattern

that can be modeled with the BesselFubini equation from non-linear acoustics [42]

(2)

where is a Bessel function of the first kind. The naturalnumber is infinity in principle, but for calculation purposes

15 or higher is usually sufficient, as discussed below. Thefunctions and have been determined by curve

fitting as

(3)

and (4), shown at the bottom of the page, where 0 1.

Fig. 3 shows a closeup view of a PWM waveform generated

with a carrier that uses as in (2). Nineteen harmonics are

controlled with a (high) 0.95. The waveform is compared

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

spectrum [fast Fourier transform (FFT)] for 11. With

this switching frequency ratio, the method eliminates harmonics

two through ten (even harmonics are zero by symmetry). The

carrier phase shift is set to 2 and the modulation depth

is 1. The spectrum confirms the desired elimination.Fig. 5 shows the same study except with 0. This value

also achieves satisfactory baseband performance, but with a dif-

ferent pulse pattern. The pattern provides slight differences in

higher-order harmonics. For example, the 11th and 13th har-

monics vary 2%3% in magnitude as is varied from to .

(4)

8/3/2019 Mbhe Final

3/5


Fig. 3. Conventional harmonic elimination waveform and proposed PWMwaveform ( m = 0.95, harmonics controlled through the 19th).

Fig. 4. Pulsewaveformp

, message signalm

, andmagnitudeof 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 cur-

rents automatically if neutral current does not flow. Therefore itis 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 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 1.5-GB

RAM) running MATLAB on Windows XP was used to carry

out the calculations. First, a modified triangle wave was ap-proximated with 100 000 points per cycle, the modulation

depth was set to 1, and a frequency ratio of 19 was used.

The number was varied from five to 35. Over this range, thequality of solution was acceptable and the average calculation

time varied from 0.327 to 0.915 s. Next, the same conditions

were used with except 35 and was varied from 10000

to 200 000. The average calculation time varied almost linearlyfrom 0.149 to 1.78 s with no significant difference in the

Fig. 5. Pulsewaveform p , message signal m , andmagnitudeof pulse waveformspectrum for ! = ! = 11, m = 1, and ' = 0.

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 re-

duced 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-driveapplications, 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 define DSPalgorithms will be muchfaster than PC computations based on MATLAB.

III. EXPERIMENTAL EXAMPLES

To show that the proposed technique satisfactorily eliminates

harmonics, the modified carrier was programmed into a functiongenerator. 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 casewithnineteenharmonics eliminated, 0,and 0.95.The

frequency ratio is 21:1. The signals and are shown at

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

shift is 2. The unexpected result was that the spectrum

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-quency, , neednot bean oddmultiple of . In Fig. 8, the fre-

8/3/2019 Mbhe Final

4/5

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 22, NO. 1, JANUARY 2007 339

Fig. 6. Experimental modulation-based SHE with ! = ! = 21, m = 0.95,' = 0.

Fig. 7. Experimental modulation-based SHE with! = ! =

21,m =

0.95,' = = 2.


20,m =

1.0,

' =

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 figure.

As shown in Fig. 9, the frequency ratio can also be a half-in-teger. In this case, the ratio is 13.5:1, 0.95 and 0.


13.5,m =

0.95,

' = 0.

Fig. 10. Experimental modulation-based SHE with ! = ! = 50, m = 0.95,' =

0.

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 nu-merical optimization. The approach is based on a modified car-rier 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 straight-

forward. 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 com-

parison 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.

8/3/2019 Mbhe Final

5/5


[2] H. S. Patel and R. G. Hoft, Generalized techniques of harmonic elim-ination and voltage control in thyristor inverters: part I-harmonic elim-ination, IEEE Trans. Ind. Appl., vol. IA-9, no. 3, pp. 310317, May/Jun. 1973.

[3] , Generalized techniques of harmonic elimination and voltagecontrol in thyristor inverters: part II-voltage control techniques, IEEETrans. Ind. Appl., vol. IA-10, no. 5, pp. 666673, Sep./Oct. 1974.

[4] I. J. Pitel, S. N. Talukdar, and P. Wood, Characterization of pro-

grammed-waveform pulsewidth modulation, IEEE Trans. Ind. Appl.,vol. IA-16, no. 5, pp. 707715, Sep./Oct. 1980.[5] , Characterization of programmed-waveform pulse-width mod-

ulation, in Proc. IEEE Ind. Appl. Soc. Annu. Meeting, 1979, pp.375382.

[6] P. N. Enjeti, P. D. Ziogas, and J. F. Lindsay, Programmed PWM tech-niques 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 Width Modulation for Power Con-

verters Principles and Practice. Hoboken, NJ: IEEE Press, 2003.[8] J. R. Wells, B. M. Nee, P. L. Chapman, and P. T. Krein, Selective har-

monic control: a general problem formulation and selected solutions,IEEE Trans. Power Electron., vol.20, no. 6,pp. 13371345, Nov. 2005.

[9] P. L. Chapman and P. T. Krein, Motor re-rating for traction appli-cationsfield weakening 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 elimi-

nation, in Proc. IEEE Appl. Power Electron.Conf., 1993,pp. 847853.[11] T.-J. Liang, R. M. OConnell, and R. G. Hoft, Inverter harmonic re-

duction using Walsh function harmonic elimination method, IEEETrans. Power Electron., vol. 12, no. 6, pp. 971982, Nov. 1997.

[12] F. Swift and A. Kamberis, A new Walsh domain technique of har-monic elimination and voltage control in pulse-width modulated in-verters, IEEE Trans. Power Electron., vol. 8, no. 2, pp. 170185, Apr.1993.

[13] J. Nazarzadeh, M. Razzaghi, and K. Y. Nikravesh, Harmonic elimi-nation in pulse-width modulated inverters using piecewise constant or-thogonal functions, Elect. Power Syst. Res., vol. 40, pp. 4549, 1997.

[14] S. R. Bowes and P. R. Clark, Simple microprocessor implementationof new regular-sampled harmonic elimination PWM techniques, inProc. IEEE Ind. Appl. Soc. Annu. Meeting, 1990, pp. 341347.

[15] , Transputer-based harmonic-elimination PWM control ofinverter drives, IEEE Trans. Ind. Appl., vol. 28, no. 1, pp. 7280,Jan./Feb. 1992.

[16] , Simplemicroprocessor implementation of new regular-sampledharmonic elimination PWM techniques, IEEE Trans. Ind. Appl., vol.28, no. 1, pp. 8995, Jan./Feb. 1992.

[17] , Regular-sampled harmonic-elimination PWM control ofinverter drives, IEEE Trans. Power Electron., vol. 10, no. 5, pp.521531, Sep. 1995.

[18] S. R. Bowes, Advanced regular-sampled PWM control techniques fordrives and static power converters,IEEE Trans. Ind. Electron., vol. 42,no. 4, pp. 367373, Aug. 1995.

[19] S. R. Bowes,S. Grewal, andD. Holliday, Single-phaseharmonic elim-ination PWM, Electron. Lett., vol. 36, pp. 106108, 2000.

[20] S. R. Bowes and S. Grewal, Novel harmonic elimination PWM con-trol strategies for three-phase PWM inverters using space vector tech-niques, Proc. Inst. Elect. Eng., vol. 146, pp. 495514, 1999.

[21] L. Li, D. Czarkowski, Y. Liu, and P. Pillay, Multilevel selectiveharmonic elimination PWM technique in series-connected voltage in-

verters, IEEE Trans. Ind. Appl., vol. 36, no. 1, pp. 160170, Jan./Feb.2000.

[22] P. Bolognesi and D. Casini, General harmonic families eliminationmethodology for static converters control, in Proc. Int. Conf. Power

Electron. Var. Speed Drives, 1998, pp. 8691.

[23] D. Czarkowski, D. V. Chudnovsky, G. V. Chudnovsky, and I. W.Selesnick, Solving the optimal PWM problem for single-phaseinverters, IEEE Trans. Circuits Syst. I, vol. 49, no. 4, pp. 465475,Apr. 2002.

[24] J. Sun and H. Grotstollen, Solving nonlinear equations for selectiveharmonic eliminated PWM using 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 flexible wayto generate

PWM-SHE switching patterns using genetic algorithm, in Proc. IEEEAppl. Power Electron. Conf., 2001, pp. 11301134.[26] B. Ozpineci, L. M. Tolbert, and J. N. Chiasson, Harmonic optimiza-

tion of multilevel converters using genetic algorithms, in Proc. IEEEPower Electron. Spec. Conf., 2004, pp. 39113916.

[27] J. Chiasson, L. Tolbert, K. McKenzie, and D. Zhong, Eliminating har-monics in a multilevel converter using resultant theory, in Proc. IEEEPower Electron. Spec. Conf., 2002, pp. 503508.

[28] J. N. Chiasson, L. M. Tolbert, K. J. McKenzie, and Z. Du, A completesolution to the harmonic elimination problem, IEEE Trans. Power

Electron., vol. 19, no. 2, pp. 491499, Mar. 2004.[29] J. Chiasson, L. M. Tolbert, K. McKenzie, and Z. Du, Elimination of

harmonics in a multilevel converterusingthe theoryof symmetricpoly-nomials and resultants, in Proc. IEEE Conf. Dec. Contr., 2003, pp.35073512.

[30] J. N. Chiasson, L. M. Tolbert, K. J. McKenzie, and Z. Du, Control ofa multilevel converter using resultant theory, IEEE Trans. Contr. Syst.

Technol., vol. 11, no. 3, pp. 345354, May 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, Sequential homotopy-based computation of multiple so-lutions for selected harmonic elimination in PWM inverters, IEEETrans. Circuits Syst. I, vol. 46, no. 5, pp. 586593, May 1999.

[33] J. Sun, S. Beineke, and H. Grotstollen, Optimal PWM based on real-time solution of harmonic elimination equations, IEEE Trans. Power

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, Computer-aided design and application of sinusoidal switching patterns, in Proc.

IEEE Workshop Comput. Power Electron., 1998, pp. 185191.[36] V. G. Agelidis, A. Balouktsis, and I. Balouktsis, On applying a min-

imization technique to the harmonic elimination PWM control: thebipolar waveform,IEEE Power Electron. Lett.,vol.2,no.2,pp.4144,Jun. 2004.

[37] V. G. Agelidis, A. Balouktsis, and C. Cosar, Multiple sets of solu-tions for harmonic elimination PWM bipolar waveforms: analysis andexperimental verification, IEEE Trans. Power Electron., vol. 21, no.2, pp. 415421, Mar. 2006.

[38] J. R. Wells, Generalized Selective Harmonic Control, Ph.D. disser-tation, Univ. Illinois, Urbana, 2006.

[39] P. T. Krein, B. M. Nee, and J. R. Wells, Harmonic eliminationswitching through modulation, in Proc. IEEE Workshop Comput.Power Electron., 2004, pp. 123126.

[40] A. M. Stankovic, G. C. Verghese, and D. J. Perreault, Analysis andsynthesis of randomized modulation schemes for power converters,

IEEE Trans. Power Electron., vol. 10, no. 6, pp. 680693, Nov. 1995.[41] M. J. Meinhart, Microcontroller Implementation of Modula-

tion-Based Selective Harmonics Elimination, M.S. thesis, Univ.Illinois, Urbana, 2006.

[42] B. Enflo and C. Hedberg , Theory of Nonlinear Acoustics in Fluids.Dordrecht, The Netherlands: Kluwer, 2002.

Documents

Mbhe Final