1. Trei-Phase PWM Invertor Folosind 3-D Model de Modulare

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Analytical Solution to Harmonic Characteristics ofThree-Phase PWM Inverter Using 3-D Modulation ModelKUO-HSIUNG TSENG a , WEN-SHIOW KAO a , YUNG-FU CHENG a & SHIN-FU LIN aa Department of Electrical Engineering, National Taipei University of Technology, Taipei,Taiwan, R.O.C.Version of record first published: 24 Jun 2010.

To cite this article: KUO-HSIUNG TSENG, WEN-SHIOW KAO, YUNG-FU CHENG & SHIN-FU LIN (2004): Analytical Solution toHarmonic Characteristics of Three-Phase PWM Inverter Using 3-D Modulation Model, Electric Power Components and Systems,32:11, 1105-1120

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EMP 32(11) #17102

Electric Power Components and Systems, 32:11051120, 2004Copyright c Taylor & Francis Inc.ISSN: 1532-5008 print/1532-5016 onlineDOI: 10.1080/15325000490441381

Analytical Solution to Harmonic Characteristics ofThree-Phase PWM Inverter Using 3-D

Modulation Model

KUO-HSIUNG TSENGWEN-SHIOW KAOYUNG-FU CHENGSHIN-FU LINDepartment of Electrical EngineeringNational Taipei University of TechnologyTaipei, Taiwan, R.O.C.

The purpose of this article is to analyze the harmonic spectrum of the three-phase PWM inverter drives used in a power system. The analytical methodsare based on a 3-D modulation model and double Fourier series (DFS). Inthis article, with Matlab, we analyzed the harmonic model created by DFS,and used time-domain simulation tools, Pspice and Simulink, to construct themodel and simulation of the three-phase DC to AC PWM inverter found inthe system. Results show that the harmonics generated by the simulator toolsand that generated by DFS method are almost identical, and that the harmonicmodel constructed by means of the 3-D modulation model proposed is highlyaccurate.

Keywords 3-D modulation model, double Fourier series, three-phase PWMinverter

1. Introduction

Given a xed or variable DC voltage, a PWM inverter may generate single-phase orthree-phase AC voltage. However, the voltage or current it generates never appearsin the form of a sinusoidal wave. In other words, the output of an inverter isharmonic. The harmonics are greatly detrimental to the circuit itself, for example,aecting the power of the circuit, lowering the power factor, imposing a negativeeect on the modulation of work of the inverter circuit itself or other powerelectronic equipment, or interfering the other loads and communication circuitsof the system. On the other hand, owing to the fact that the output of the inverteris never a sinusoidal wave, a great part of the harmonic current generated byan inverter-driven inductive motor ends up in the inductive motor, giving rise tounexpected eects, such as the pulsation or additional loss of motor torque [1, 2].

Manuscript received in nal form on 24 September 2003.Address correspondence to Kuo-Hsiung Tseng, National Taipei University of Technol-

ogy, Department of Electrical Engineering, I, Section 3, Chung-Hsiao East Road, Taipei106, Taiwan. E-mail: [email protected]

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1106 K.-H. Tseng et al.

To evade the aforesaid harmonics-induced problems, it is necessary to analyze theharmonic characteristics of the PWM inverter, so as to predict the magnitudes andconstituents of the harmonics likely to be generated.

The characteristics of harmonics vary with the way a PWM is controlled [36].Studies on the harmonic characteristics of two-level PWM inverters used in a drivesystem have been widely discussed [79]. As shown in the above publications, a 2-Dmodulation model is employed in the analysis of the waveform generated by a two-level PWM inverter. It is impossible to illustrate directly the relationship betweena modulated wave, a carrier and a harmonic through a harmonic model created inthis manner. Accordingly, the harmonic model does not help reveal the harmoniccharacteristics of a PWM inverter precisely and completely whenever the carrier tomodulated wave modulation frequency ratio varies. To address the shortcomings ofthe aforesaid method, this article proposes a 3-D modulation model, and analyzesthe waveform and spectral characteristics of the three-phase PWM inverter used inthe system by means of double Fourier series.

This study consists of three parts: (1) Analyzing the time-domain waveformof the voltage and current output by the three-phase PWM inverter used in thesystem by a 3-D modulation model and double Fourier series; (2) acquiring theharmonic spectrum of the inverter by processing the time-domain waveform withfast Fourier transform (FFT) with a view to analyzing the harmonic characteristics;and (3) analyzing the actual inverter circuit by means of the time-domain simulationtools, Pspice and Simulink, and then comparing the ndings with that of theharmonic model created by double Fourier series with a view to verifying theprecision of the method proposed in this article.

2. Three-Phase PWM Inverter Circuit and Principle

Figure 1 shows the circuit of a three-phase PWM inverter. It is usually controlledin a bipolar manner. The PWM of phase A, B, and C is usually jointly controlledby a triangular wave carrier vtri , and the phase dierences between modulationsignals vmA, vmB , and vmC of the three phases are always 120. The power switchcomponents of phase A, B, and C are controlled in the same way. Suppose theswitch is an ideal component, and take phase A as an example. If vmA > vtri , acut-o signal is sent to upper bridge arm power switch S1, and a cut-o signal issent to lower bridge arm power switch S4, then the output voltage for phase A,vAO, is equal to Vd/2 with respect to the DC current hypothetical neutral point O.

Figure 1. Three-phase PWM inverter circuit.

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Characteristics of PWM Modulation 1107

Figure 2. Three-phase PWM inverter voltage switching waveforms.

If vmA < vtri , a cut-o signal is sent to S1, and a connection signal is sent to S4,then the output voltage for phase A, vAO, is equal to Vd/2. S1 and D1, as wellas S4 and D4, motion signals or states are always complementary. If a connectionsignal is sent to S1(S4), either S1(S4) or the diode D1(D4) continuous ow will beconnected; this will be determined by the direction and magnitude of the currentinitially found in electric inductive load, and it is similar to the condition whereinthe single-phase, two-level PWM bipolar circuit is controlled. The control methodsof phase B and C are the same as that of phase A.

Figure 2bc show the waveforms of vAO and vBO whose voltage waveform isa mere Vd. The waveform of linear voltage vAB is the dierence between vAO andvBO, that is, vAOvBO, as illustrated in Figure 2d. If switch component arms 1 and6 are connected, then vAB = Vd. If switch component arms 3 and 4 are connected,then vAB = Vd. If switch component arms 1 and 3, or switch component arms 4and 6, are connected, then vAB = 0. Hence, the linear voltage output by an inverterconsists of three kinds of voltages, that is, Vd and zero. The loaded phase voltagevAN is expressed in Eq. (1).

vAN = vAO vAO + vBO + vCO3 (1)

As shown in Figure 2e, vAN is composed of ve voltages, namely (2/3)Vd,(1/3)Vd, and 0.

3. Analyzing the Harmonic Characteristics of Three-PhasePWM Inverter

If a periodic waveform arises from the modulation of two or more periodic wave-forms, then the computation of its spectrum will become very complicated. It is

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normally achieved by means of a 3-D modulation model and double Fourier series.The 3-D modulation model was the rst put forth by W.R. Bennet in 1933. Thespectral analysis used in the model may be applied to any modulation wherein boththe carrier and the modulated wave are periodic functions of time [9, 10]. Since thePWM sampled waveform adopted by the three-phase PWM inverter conforms tothe requirements of the 3-D modulation model, the researcher in this article putforth the idea that the harmonic characteristics of a three-phase PWM invertershould be analyzed with the 3-D modulation model.

3.1. Harmonic Characteristics of Three-Phase PWM Inverter with3-D Modulation Model

A unipolar natural-sampling back-rim modulation pulse train generated by thethree-phase PWM inverter is depicted in Figure 3. Figure 3a illustrates the wayin which a double-rim modulation pulse train is generated by a 2-D model. The3-D modulation model for analyzing the pulse train spectrum adopts the waveformdepicted in Figure 3b. Figure 3c and d depict the waveform vAO and vBO of the back-rim modulation pulse trains in phase A and B, respectively. Figure 3e illustratesthe linear voltage vAB generated by the three-phase PWM inverter.

The waveform depicted in Figure 3a is compared with that shown in Figure 3b.One will discover that the PWM waveform produced by the 2-D modulation modelis actually identical to that generated by the 3-D modulation model. Regardingthe 2-D modulation model, the carrier vtri is compared with two modulated waves,that is, vmA and vmB , at a frequency modulation ratio of Mf = c/m, Mf > 1.The pulse signal that arises from the comparison between vmA and vtri controlsthe way the switch component of the PWM inverter switches in phase A. Simi-larly, the pulse signal that arises from the comparison between vmB and vtri con-trols the way the PWM inverter switches in phase B. The slope of the carriervaries every half a carrier period. The pulse train of a three-phase PWM inverter

Figure 3. PWM inverter modulation process illustrated by a 2-D model and a 3-D model.

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vAB may be acquired by extending the modulated wave and the carrier along thetime axis.

As shown in the 3-D modulation model depicted in Figure 3b, if the modulatedwave and the crossing point extend along -axis and -axis, the slope of theintersection wall that varies in accordance with the modulation ratio, Mf = c/m(hereinMf = 7), will not alter accordingly. The image of the wall may be producedin a way parallel to the -axis by drawing the intersection wall across the modulatedwave obliquely. The image as such helps produce a pulse train that repeats in the-axis every Mf carrier period or in the -axis every modulation period. All thepulse trains generated, in the -axis and the -axis, by the intersection wall andthe modulated wave depicted in Figure 3b contain modulation information [9, 10].

The aforesaid modulated wave is a sinusoidal wave, as PWM adopts naturalsampling. However, given the development of electrical and electronic technology,the modulation method actually applied to a system has already been replaced witha more advanced sampling method [56, 11], that is, an asymmetrical PWM sam-pling technique, as shown in Figure 4. The method involves altering the samplingfactor, , of the modulated wave; whenever varies, the point of the intersectionof the triangular carrier and the linear modulate wave segment lying between twoadjacent sampling points will vary, leading to dierent triggering times.

3.2. Analysis of Harmonic Characteristics of PWM Inverter by DFS

According to the reference [10] on DFS equations, whatever the pulse trains thatarises from a 3-D modulation process can be expressed in equation (2).

F (, ) =12Aoo +

n=1

[Aon cos(n) +Bon sin(n)]

+m=1

[Amo cos(m) +Bmo sin(m)]

+m=1

n=1

[Amn cos(m + n) +Bmn sin(m + n)]

(2)

where

Amn =122

20

20

F (, ) cos(m + n) d d

Bmn =122

20

20

F (, ) sin(m + n) d d

The DC of the pulse train accounts for the rst term of Eq. (2). The harmonicsof the modulated wave accounts for the second term of Eq. (2). The combinedmodulation of the carrier and the modulated wave accounts for the last term ofEq. (2). The spectrum of the pulse train is acquired by replacing the and ofthe series equations with time functions ct and mt. The spectral analysis of thewaveforms of the three-phase PWM inverter by means of DFS and asymmetricalsampling technique.

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(a)

(b)

Figure 4. The 3-D modulation model of the asymmetrical sampling of a three-phasePWM inverter. (a) plane and (b) plane.

To simplify the analysis of the harmonics of the switching of the three-phasePWM inverter switch components, the following hypotheses are adopted in thetheoretical inference process: (1) The DC chain voltage of a PWM inverter doesnot comprise any ripple; (2) the switching switch of a PWM inverter has idealoperation characteristics; and (3) the switching triggering signal does not containany dead time. Take the circuit depicted in Figure 1 as an example. If the input

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voltage of the three-phase PWM inverter is an ideal DC voltage source, then the hthharmonic current output by the PWM inverter in phase A is expressed in Eq. (3).

i1(h) =

vAB(h)3

v1(h)R1 + jXL1(h)

(3)

where h is the harmonic number.Figure 4a depicts the asymmetrical PWM sampling structure. Figure 4a shows

the double rim employed to produce the rst pulse, that is, the rectangular coor-dinates (i.e., 1 and 2) of the point of intersection between the sampling prole = /Mf and the non-sinusoidal wall. Such rectangular coordinates are exactlythe rectangular coordinates of the point of intersection between the sine wall andthe sampling prole = [ (X 1)]/Mf (the solid line) that has alreadybeen converted, as illustrated in Figure 4b. Therefore, the pulse train and the spec-trum that arise from the two aforesaid modulation processes are the same. The factthat it is easier to gure out the double Fourier series coecient on the planethan on the plane is proved by the following explanation.

As shown in Figure 4b, the period for the carrier and the modulated wave of the3-D modulation model is 2, and the modulated wave has to perform an anti-phaseshift of 180 every half period () of the carrier whenever the carrier undergoes aperiod. Figure 4b also shows that the carrier has a maximum amplitude of /2, andthat the amplitude modulation ratio of the PWM sampling is M1 = 2a/ becausethe curve of the modulated wave is expressed as a sin. Suppose the frequencymodulation ratio of the PWM sampling is Mf = 7, or in other words, fourteenmodulated wave walls intersect a carrier wall. The point, 1, of the intersectionbetween the rst modulated wave (X = 1) and the carrier slant cross-section isworked out in Figure 4b. The intersection point, 1, is exactly the front rim of therst pulse train of vAO. Thus, 1 can be expressed with Eq. (4).

1 =

2 a sin =

2 MI

2sinAB

=

2 MI

2sin[(1 )1/Mf ]

(4)

As shown in Figure 4b, the straight line OC is plotted, as point O (0, 0) andpoint C (, (1 )/Mf ) are well-identied.

= (/Mf ) (5)With the above-mentioned linear conversion, Eq. (4) may be converted into a

simpler sinusoidal waveform.

1 =

2 MI

2sin, 0 < 1 < (6)

where denotes the sampling prole that results from linear conversion.For the same reason, the point of intersection, 2, of the intersection between

the second modulated wave (X = 2) and the carrier slant cross-section is exactlythe back rim of the second pulse train of vAO. Thus, 2 can be expressed withEq. (7).

2 =32

+MI2

sin, 0 < 2 < (7)

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By substituting Eqs. (6) and (7) into Eq. (2), it is possible to gure outthe voltage, vAO, of the phase-A, three-phase, two-level PWM inverter, which isexpressed with Eq. (8).

vAO =12+

n=1,3,5,...

2Jn(npMI/2)np

sin(nmt np2

)

+

m=1,3,5,...

2J0(mMI/2)m

(1)m+12 sin(mct 2

)

+

m=1,3,...

n=2,4,...

2Jn[(m+ np)MI/2](m+ np)

sin(m

2

)

cos[(mc + mn)t

((m+

np

2

))]

+

m=2,4,...

n=1,3,...


cos(m

2

)

sin[(mc + mn)t

((m+

np

2

))]

(8)

In Eq. (8), is the sampling factor between 0 and 1, p is the reciprocal of thecarrier to modulated wave frequency ratio c/m,MI is a modulation index, andJn is the nth Bessel function.

To gure out the PWM inverter voltage, vBO, in phase B, it is necessary toprocess the modulation signal of vAO with a phase shift of 120. In other words, inEq. (7), nmt is replaced by n(mt+ 2/3). Thus, vBO is expressed with Eq. (9).

VBO =12+

n=1,3,5,...

2Jn(npMI/2)np

sin[n

(mt+

23

) np

2

]

+

m=1,3,5,...

2J0(mMI/2)m

(1)m+12 sin(mct 2

)

+

m=1,3,...

n=2,4,...


sin(m

2

)

cos[[mct+ n

(mt+

23

)]((

m+np

2

))]

+

m=2,4,...

n=1,3,...


cos(m

2

)

sin[[mct+ n

(mt+

23

)]((m+

np

2

))]

(9)

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Hence, the linear voltage output by a set of three-phase, two-level PWM invert-ers, vAB , is gure out. In other words, vAB = vAO vBO can be expressed withEq. (10).

vAB =

n=1,3,5,...

4Jn(npMI/2)np

sin(n3

)cos

(nmt np2 +

n

3

)

+

m=1,3,5,...

n=2,4,...


sin(m

2

)sin(n3

)

sin[(mc + mn)t

((m+

np

2 n

3

))]

+

m=2,4,6,...

n=1,3,...


cos(m

2

)sin(n3

)

cos[(mc + mn)t

((m+

np

2 n

3

))]

(10)

Hence, the phase-A loaded harmonic current, i1, can be expressed as follows:

i1 ={vAB3 v1

}/R1 + jXL1

=

{[ n=1,3,5,...

4Jn(npMI/2)np

sin(n3

)cos

(nmt np2 +

n

3

)

+

m=1,3,5,...

n=2,4,...


sin(m

2

)sin(n3

)

sin[(mMf + n)mt

((m+

np

2 n

3

))]

+

m=2,4,6,...

n=1,3,...


cos(m

2

)sin(n3

)

cos[(mMf + n)mt

((m+

np

2 n

3

))]] /

3 V1}

1R1 + jXL1

(11)

4. Time-Domain Simulations and Spectral Analysis

In this article, the researcher adopts the hypothesis that the load functions with aconstant power for the sake of solution and simulation. The questions are solvedwith Matlab, by means of the harmonic mathematic model, expression Eq. (11),created by DFS. The model and simulation of the system are constructed the time-domain simulator tools, Pspice and Simulink. Eventually, the ndings of simulationare compared with the mathematical model created by DFS for the sake of analysis,with a view to verifying its precision.

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(a) (b) (c)

Figure 5. Symbolic representation of various components of Pspice. (a) An AC switch;(b) the switching diode; and (c) a three-phase of a power transistor PWM controller.

Figure 6. The circuit of a three-phase PWM inverter simulated by Pspice.

4.1. Constructing Three-Phase PWM Inverter Used in the Systemwith Pspice

All the components of the system are roughly divided into three types, namely theinput voltage (DC), the three-phase PWM inverter and the load. The PWM inverteris composed of a power switch and a counter diode, as shown in Figures 5a and b.Figure 5c shows a three-phase PWM controller. Given the aforesaid components,the circuit model diagram of the three-phase PWM inverter used in the system canbe constructed, as shown in Figure 6.

4.2. Constructing Three-Phase PWM Inverter Used in the Systemwith Simulink

In this section, the construction of the system-based inverter circuit by Simulink(Power System Blockset) is discussed. The models of the power switch and the

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Figure 7. The symbolic representation of the power switch.

Figure 8. The symbolic representation of the counter diode.

Figure 9. The three-phase PWM control block diagram.

counter diode of the PWM inverter are borrowed from the component library ofSimulink. Figures 7 and 8 show the symbolic representation of the power switchand the counter diode, respectively.

Regarding the controller for the three-phase PWM inverter, the triggeringsignal for the switch can be devised with the concepts mentioned above in Section 2.Its control block diagram is depicted in Figure 9.

The equivalent circuit of the three-phase PWM inverter used in the system canbe created by means of the components constructed with Simulink. Figure 10 showsthe circuit of the three-phase PWM inverter constructed by means of Simulink.

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Figure 10. The Simulink circuit structure diagram for the three-phase PWM inverter.

Table 1System simulation parameters

Three-phase DC input voltage: Vd = 313.97 VThree-phase induction motor: V1 = 105.73 12.36 V,

f = 60 Hz, R = 2 , L = 10 mHThree-phase PWM inverter: Mf = 7, MI = 0.75, = 0

4.3. Discussion of Simulation Results

The aforesaid simulation method and the system model of the PWM invertercreated above are analyzed. Table 1 shows the parameters adopted by the system,wherein it is supposed that the load functions with a constant power.

Figure 11a and b show the ndings acquired by Pspice simulation and Simulinksimulation, respectively. Figure 11c shows the ndings of the simulationbyDFS.Theirspectral characteristics are depicted in Figure 12a, b, and c, respectively. Regardingthese threemethods, thecurrentwaveformsandharmonic spectrumsat the load-endofthe three-phase PWM inverter are simulated, while unipolar natural sampling back-rim modulation is taking place. As indicated by the ndings about the spectrumsdepicted in Figure 12a, b, and c, although the measurements of the three types ofsimulation dier slightly, the dierence is insignicant. We treat the Simulink-basedsimulated values as the standard so as to conduct the analysis, wherein Ih denotesthe current of the nth harmonic, and I1 denotes the current of the basic harmonic.As for the 3rd, 5th, 7th, and 9th harmonics, a relative low Ih/I1 ratio is adoptedin Pspice simulation and a relative low Ih/I1 ratio is adopted in DFS problem-solving process, though the discrepancy is just as great as 10%, approximately.

According to the aforesaid ndings of comparison, the main spectrum workedout by DFS put forth in this article approximates closely the main spectrumsworked out by Pspice simulation and Simulink simulation, and the computationis fast and precise. However, the DC link of the Pspice simulation and Simulinksimulation is not DC in nature as assumed by the mathematical model created inthis article. Instead, it is even harmonic. In addition, its switch switching can be

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(a)

(b)

(c)

Figure 11. The time-domain waveforms of the current output by the three-phase PWMinverter. (a) Using Pspice simulation; (b) using Simulink simulation; and (c) using DFSfor solution.

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(a)

(b)

(c)

Figure 12. The harmonic spectrum of the current. (a) Using Pspice simulation; (b) usingSimulink simulation; and (c) using DFS for solution.

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delayed. These two phenomena contribute to the low order harmonics (the 3rd,5th and 7th harmonics, etc.). Furthermore, there are atypical harmonics, becauseSimulink switch model is not an ideal component at all.

Based on study in this article using 3-D modulation model to analysis theharmonic spectrum of three-phase PWM inverter, the authors suggest that consid-erations, such as the dead time of switch components, the fact that the power sourcecomprises ripples, and the phenomenon wherein the load comprises even harmonics,should be incorporated into the establishment of a harmonic mathematical modelin further study, so that a harmonic model that is more accurate can be established.Regarding further study, it is important to take into account the dynamic load ofan operating system so as to gure out the dynamic output power of the inverter.As a result, the dynamic harmonics of the system is monitored, and it is possibleto supply the harmonic source to the load so as to conduct harmonic analysis, witha view to providing the data about the harmonic characteristics of an operatingsystem and thus making the harmonic model put forth in this article more useful.

5. Conclusions

This article involved using a PWM inverter as the major source for supplyingharmonics to the load, and studied the structure of a three-phase PWM inverterand its principle of motion, with a view to creating a harmonic mathematicalmodel for the three-phase PWM inverter used in the system and then analyzingits harmonic characteristics, by means of a 3-D modulation model and the doubleFourier series. As a result, the precision of the harmonic mathematical model putforth was proved by the ndings about the three-phase PWM inverter harmoniccharacteristics simulated by the time-domain simulation tools, Pspice and Simulink.The conclusions of this research are as follows:

(1) In view of the combined modulation relation between a modulated waveand a carrier, the harmonic voltage mathematical model of the three-phasePWM inverter is constructed by means of the 3-D modulation model andthe double Fourier series. In this way, the harmonic current input to theload by the three-phase PWM inverter is gured out accurately.

(2) Construct the circuit of the three-phase PWM inverter with a simulationtool. Simulate the harmonic mathematical model and verify its precision,so as to help identify the harmonic characteristics of the system while it isoperating in practice.

(3) The harmonic characteristics of the three-phase PWM inverter constructedby DFS are distributed around Mf , and the harmonics appear at thefrequency modulation ratio, Mf , and the adjoining rim. Hence, harmonicnumber, h, may be expressed with the following

h = j(Mf )Kwhere

if j is an odd number, then K is an even number;if j is an even number, then K is an odd number.

Hence, the distribution of the harmonic source is swiftly assessed in accor-dance with the variation of the frequency modulation ratio, Mf , for thesake of prevention and prohibition.

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References

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[2] J. Arrillaga, B. C. Smith, N. R. Watson, and A. R. Wood, Power System HarmonicsAnalysis, New York: John Wiley & Sons, 1997.

[3] G. Carrara, S. Gardella, M. Marchesoni, R. Salutari, and G. Sciutto, A new mul-tilevel PWM method: A theoretical analysis, IEEE Trans. on Power Electronics,vol. 7, no. 3, pp. 497505, July 1992.

[4] G. Franzo, M. Mazzucchelli, and L. Puglisi, Analysis of PWM techniques usinguniform sampling in variable-speed electrical drives with large speed range, IEEETrans. on Industry Applications, vol. IA-21, no. 4, pp. 996974, July 1985.

[5] P. H. Mellor, An adjustable sinusoidal PWM sampling process for AC drives,EPE93 Fifth European Conference on Power Electronics and Applications, vol. 4,pp. 231236, 1993.

[6] I. D. Mosely, P. H. Mellor, and C. M. Bingham, Digital sampling process for audioclass D, pulse width modulated power ampliers, Electronics Letters, vol. 28, no. 1,pp. 5658, Jan. 1992.

[7] S. R. Bowes and B. M. Bird, Novel approach to the analysis and synthesis ofmodulation processes in power converters, Proc. IEE, vol. 122, no. 5, pp. 507513,March 1975.

[8] S. R. Bowes and A. Midoun, Suboptimal switching strategies for microprocessor-controlled PWM inverter drives, IEE Proc., vol. 132, pt. B, no. 3, May 1985.

[9] J. Shen, J. A. Taug, and A. D. Mansell, Analytical solution to harmonic character-istics of traction PWM converters, IEE Proc.-Electr. Power Appl., vol. 144, no. 2,pp. 158168, March 1997.

[10] H. S. Black, Modulation Theory, Princeton, New Jersey: Van Nostrand, 1953.[11] P. H. Mellor, S. P. Leigh, and B. M. G. Cheetham, Reduction of spectral distortion

in class D ampliers by an enhanced pulse width modulation sampling process, IEEProceedings G, vol. 138, no. 4, pp. 441448, Aug. 1991.

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1. Trei-Phase PWM Invertor Folosind 3-D Model de Modulare

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