Direct synchronized PWM techniques with linear control ... - IEEE Xplore

Direct Synchronized PWM Techniques with Linear Control Functions for Adjustable Speed Drives Valentin Oleschuk

*,**

Frede Blaabjerg **

*

Power Engineering Institute of the Academy of Sciences of Moldova 5 Academy Str., MD-2028, Kishinau REPUBLIC OF MOLDOVA

Institute of Energy Technology Aalborg University Pontoppidanstraede 101, DK-9220, Aalborg East DENMARK

II. PROPERTIES OF THE PROPOSED PWM TECHNIQUES

Abstract: Novel techniques of direct synchronous pulsewidth modulation (PWM) for three-phase voltage source inverters with basic continuous and discontinuous schemes of PWM are considered in this paper. They are based on a new strategy of feedforward PWM with quasi-linear control dependencies, providing synchronization of the voltage waveforms and linear voltage control during the whole range including the zone of overmodulation. Algebraic and trigonometric versions of the method proposed, based on vector approach for determination of the pulse patterns, are analyzed and compared. Both onestage and two-stage variants of synchronous PWM are considered. The determination of the basic parameters of the output voltage of the inverter with low switching frequency are presented. Simulations give the behavior of the proposed techniques.

Generalized properties of these new PWM techniques are presented in Table I, and also compared with conventional voltage space vector modulation. They are based on the representation of pulse-widths of the signals of the inverter as a function of the fundamental frequency and the duration of the switching period. Special control signals are formed stepby-step in the clock-points of period in the output voltage of the inverter, providing a continuous adjustment and quarterwave symmetry of the voltage wave-form with smooth pulses ratio changing. Computing of the switching time intervals is here based on simple algebraic or trigonometric functions, which are easy to implement in real time.

I. INTRODUCTION

TABLE I. BASIC PARAMETERS OF PWM SCHEMES

Control of the majority of power converters is based on principle of modulation of pulse signals. Parameters and characteristics of power conversion systems are in dependence of the PWM methods and techniques used. It is known that asynchronous PWM control modes of power converters result in sub-harmonics that are very undesirable in most applications [1]-[3]. Synchronization of the output voltage of the wide-spread voltage source drive inverters (Fig. 1) is an important problem, especially for high power drive systems and for systems working in the zone of higher fundamental frequencies. This paper presents the results of analysis and comparison of algebraic and trigonometric versions of direct synchronous pulsewidth modulation for three-phase inverter [4], based on a vector approach with quasi-linear control functions and applied to the basic continuous and discontinuous schemes of voltage space vector modulation. +

V dc

a

b

c

3(010)

IM

7(111)

5(001)

_

Control (modulation) parameter Current and max parameter Modulation index m Duration of subcycles Centre of the k-signal

Switch-on durations

Proposed strategy and scheme of modulation Current & maximum fundamental frequency F and Fm

V / Vm

F / Fm

T

τ

α k (angles/degr.)

τ (k − 1) (sec)

Tak = mT [sin(600 −

β k = 1.1mτ [1 −

Trigonometric PWM

β k = 1.1mτ ×

α k ) + sin α k ]

A(k − 1)τF ]

cos[(k − 1)τ ]

tak = mT sin α k

γ k = β i − k +1[0.5 − 6(i − k )τF ]

γ k = β i − k +1[0.5 − 0.9 tan(i − k )τ ]

tbk = mT sin(600 −

βk − γ k

βk − γ k

αk ) Switch-off states (zero voltage)

1(100)

Special parameters providing synchronization of the process of PWM

6(101)

t0 k = T − t ak − tbk

λk = τ − β k λ' (clock-point notches)

β " (signals, the next to λ' ) Fi (boundary frequencies, where

λ' ⇒ 0 and β " ⇒ 0 )

Fig. 1. Structure of three-phase voltage source inverter with induction motor IM, and its output voltage vectors.

0-7803-7404-5/02/$17.00 (c) 2002 IEEE

Conventional schemes of vector PWM Current & max voltage V and Vm

Algebraic PWM

2(110)

0(000) 4(011)

**

76

switching sequence

III. PECULIARITIES OF TYPICAL PWM SCHEMES Fig. 2 presents the typical switching state sequences of a three-phase inverter inside a diapason 00-1200. It illustrates schematically, with a simplified representation of the duration of signals, some basic versions of PWM, which are used typically in adjustable speed drive systems. In Fig. 2, the conventional designation for state sequences for the switches of the phases abc of the inverter (see Fig. 1) is used: 1 – 100; 2 – 110; 3 – 010; 4 –011; 5 – 001; 6 – 101; 7 – 111; 0 – 000 (‘1’ - switch-on state, ‘0’ – switch-off state). It is necessary to note that this form of presentation, together with an approach described later, is very convenient for understanding the process of PWM in voltage source inverters, in order to understand similarities and differences between the main PWM methods and techniques. Fig. 2a shows the switching state sequences for the most popular (conventional) version of voltage space vector modulation (CPWM) [3],[5]-[6]. It is a typical scheme of continuous PWM, because the modulation principle is based on continuous operation of all switches of the inverter during every switching period (sub-cycle). This is the basic sequence for further analysis of different PWM techniques. The active switching states are situated in the centres of sub-cycles, shown by the arches below the sequences. Zero vector states (0 and 7) are changing step-by-step after every sub-cycle. The sequence of switchings is here: -0-1-2-7-2-1-0-. Fig. 3 shows more in details synchronous CPWM scheme for a quarter-period of the output voltage of the inverter. The upper curve is here the switching state sequence, then control signals for the cathode switches of the phases a, b and c. The lower curve in Fig. 3 shows the corresponding quarterwave of line-to-line output voltage of the inverter. Signals βj represent the total switch-on durations during the switching

4 γ1 γ2

γ3

γ4

3 2 β5 λ5

β4 λ4

β3 λ3

β2 λ2

β1

β2

β3

β4

β5

λ1

phase a

phase b

phase c Vab

Fig. 3. Control and output signals for quarter-period of three-phase inverter with continuous PWM (CPWM).

period (sub-cycle) τ , signals γ k are generated on the boundaries of the corresponding β . Widths of notches λ k represent the duration of zero state sequences. Fig. 2b and Fig. 2c present two versions of discontinuous PWM with an asymmetrical principle of generation of control and output signals of the inverter [5],[6]. All algorithms of discontinuous PWM, presented in Fig. 2b – Fig. 2e, are characterized by the same zero states (0 or 7) during the corresponding 600 intervals. Both the first variant, presented in Fig. 2b (DPWM0 in [6]), and the second one (Fig. 2c, DPWM2 in [6]), in opposition to conventional continuous CPWM (Fig. 2a), are organized by the symbolic artificial mutual movement of the positions of active switching states to the borders of sub-cycles, with corresponding mutual junction of the main part of the pulses. Here is the prevalence of active (and zero) switching states with doubled width in comparison with typical continuous PWM. The switching sequence is: -0-1-2-1-0-.. inside 00- 600 for the version of PWM, presented in Fig. 2b, and for variant, presented in Fig. 2c, it is: -7-2-1-2-7-. Fig. 2d shows a symmetrical discontinuous PWM algorithm, which is one of the best regarding spectral composition of the output voltage of the inverter in the zone of the middle and higher fundamental frequencies of a drive system (Method 4 in [5], DPWM3 in [6]). Here is the mutual junction of basic switching sequences (see Fig. 2a) on the both halves of 600 intervals around central sequence, situated in the centre of clock intervals. Initial switching sequence is here –0-1-2-1-0- in the first half of the cycle, and -7-2-1-2-7in the second half of the cycle. Fig. 4 illustrates in details the synchronous version of DPWM3, presenting the corresponding switching state sequences and control signals for quarter-period of line-to-line output voltage of the inverter. Fig. 2e illustrates an other popular symmetrical version of discontinuous PWM ([5]-[6], DPWM1 in [6]). The switching sequences have in this case a mirror symmetry in respect to the center of 600 intervals regarding previous scheme of

Fig. 2. Switching state sequences for typical PWM schemes: a) CPWM; b) DPWM0; c) DPWM2; d) DPWM3; e) DPWM1.

77

switching sequence

4

γ2

γ1

specific determination of the pulse patterns. Special signals λ ' ( λ 5 for CPWM, λ 4 for DPWM1 and DPWM3 in Figs. 3-

γ3

3 2 β4 λ4

β2

β3 λ3

λ2

β1

β2

β3

β4

5) (with the neighbouring β " ( β 5 for CPWM, β 4 for DPWM1 and DPWM3)) are formed in the clock-points (00,600,1200..) of the output curve. They are reduced simultaneously till close to zero value at the boundary frequencies Fi , providing a continuous adjustment of voltage

λ1

phase a

phase b

with smooth pulses ratio changing. Fi is calculated in a general form as a function of width of sub-cycles τ in accordance with (1), and the next Fi −1 - from (2). Index i is equal to the number of notches inside a half of 600 intervals and is determined from (3), where fractional number is rounded off to the nearest higher integer:

phase c Vab

Fig. 4. Control and output signals for quarter-period of three-phase inverter with discontinuous PWM (DPWM3).

switching sequence

modulation (Fig. 2d). So the basic switching sequence is here -7-2-1-2-7- in the first half of the first cycle, and -0-1-2-1-0in the second half of 600 interval. Fig. 5 presents the synchronous DPWM1 scheme more in details. All five presented schemes of PWM are characterized by the most rational sequences of switchings, providing minimum number of switch commutations (and switching losses) in the inverter. In particular, every switch commutation is executed in all cases here between two switching sequences most close to each other: from 1 to 2, from 2 to 1, from 2 to 3, from 3 to 4, etc. Zero switching sequences are selected in accordance with this rule too: 0 sequence is selected before/after 1, 3 and 5. Zero sequence 7 is selected in this case before/after 2, 4, 6. As an example, in accordance with this rule for the PWM scheme presented in Fig. 2e, the basic sequence of switchings here is -7-2-1-2-7-21-0-1-2-1-0-3-2-3-0-3-2-7-2-3-2- inside the 00– 1200 interval. 4

γ1

β4 λ4

λ3

λ2

(2)

i = (1 / 6 F + K 1τ ) / 2τ ,

(3)

βj β1

2 β2

Fi −1 = 1 /[6(2i − K 2 )τ ]

the central β 1 signal can be written as

3 β3

(1)

where K1=1, K2=3 for CPWM, K1=1.5, K2=3.5 for DPWM1 and DPWM3. The determination of the widths of active switching states and pulses of the output voltage of inverters for drive applications is reasonable to execute on the base of classical voltage space vector modulation [3]. In accordance with this principle, due to the fact, that in the proposed algorithm the β 1 signal, which is formed in the centres of 600 clockintervals (Fig. 2), has the maximum width regarding other β -signals, the ratio of durations of the corresponding β j to

γ3

γ2

Fi = 1 /[6(2i − K 1 )τ ]

β1

β2

β3

β4

= sin(60 0 − α j ) + sin(α j ) .

(4)

The relative widths of the γ -components of total active switching states, which have less duration regarding ( β − γ ) component in Figs. 3-5, are equal in this case to:

λ1

phase a

γk

phase b

β i − k +1

phase c

=

sin(α i − k +1 ) sin(60 − α i − k +1 ) + sin(α i − k +1 ) 0

.

(5)

Fig. 6 and Fig. 7 present the variation of the function in (4) and (5), and they are the basics for the determination of widths of control and output signals of the inverter for both continuous and discontinuous synchronized PWM. In order to avoid trigonometric calculations, a piece-wise approximation (curves 1 and 2 in Fig. 6) of the function (4) can be done for all presented algorithms of PWM, and also a simple linearization of the required γ -dependencies (Fig. 7, k=j here) can be done, which is close to the sine function (5).

Vab

Fig. 5. Control and output signals for quarter-period of three-phase inverter with discontinuous PWM (DPWM1).

IV. BASIC CONTROL FUNCTIONS In accordance with the basic properties presented in Table I the proposed PWM methodology is characterized by a

78

(

)

λ i = λ ' = τ − β " K ov1 K s ,

(11)

β 1 = 1.1τm (m – modulation index) until Fov1 = 0.907 Fm , and β 1 = τ after Fov1 ; - K s = [1 − ( F − Fi ) /( Fi −1 − Fi )] - linear coefficient of synchronization; - coefficient of overmodulation K ov1 = 1 until Fov1 , and K ov1 = [1 − ( F − Fov1 ) /( Fov 2 − Fov1 )] (linear function) between Fov1 and Fov 2 = 0.952 Fm ; - coefficient of overmodulation K ov 2 = 1 until Fov 2 , and K ov 2 = [1 − ( F − Fov 2 ) /( Fm − Fov 2 )] (linear function) in the zone between Fov 2 and Fm ; - K3=0.25 for DPWM1 and DPWM3, and K3=0 for CPWM. Fig. 8 presents the general algorithm to calculate the pulse patterns of the inverter output voltage for all presented versions of the direct synchronized PWM based on either trigonometric or algebraic control functions. where:

angle from the beginning of clock-intervals, degrees Fig. 6. Variation of the function for total switch-on durations.

B. Algebraic Synchronous PWM Eqs. (12)-(18) present the basic set of algebraic control functions, based on more precise four-stage piece-wise approximation of (4) (curve 2 in Fig.6) and linearization of (5). It describes the parameters of modulated line voltage in absolute value (seconds) as a function of the fundamental frequency F for scalar control mode of drive system:

angle from the beginning of clock-intervals, degrees Fig. 7. Relative widths of the

γ

-signals from their position.

for j=2,...i/4:

β j = β 1 [1 − 0.43( j − K 3 − 1)τFK ov1 ]

A. Trigonometric Synchronous PWM The implementation of algorithms of direct synchronous PWM can be based on either algebraic or trigonometric dependencies. Eqs. (6)-(11) present a trigonometric set of the main control functions, based on a simple transformation of (4) and (5), for determination of the parameters for control and output signals of the inverter in absolute value (seconds) for scalar control mode of the drive system during the whole range including the zone of overmodulation:

(12)

for j=i/4+1,..i/2: F

τ i Fi −1 and F i

for j=2,…i-1:

β j = β 1 cos[( j − 1 − K 3 )τK ov1 ]

(6)

β i = β " = β 1 cos[(i − K 3 − 1)τK ov1 ]K s

(7)

γ j = β i − j +1{0.5 − 0.87tg[(i − j − K 3 )τ ]}K ov 2

(8)

γ 1 = β " {0.5 − 0.87tg[(i − K 3 − 2)τ + ( β i −1 + β i + λi −1 ) / 2]}K s K ov 2

(9)

β1 βk

β" λ′

λk

γ1

γk

Direct synthesis of the waveform

λ j = τ − ( β j + β j +1 ) / 2

(10)

Fig. 8. Algorithm of calculation of control parameters

79

β j = β 1 [1 − (1.2( j − K 3 − 1)τF − 0.02) K ov1 ] ;

(13)

for j=i/2+1,..3i/4:

β j = β 1 [1 − (2.02( j − K 3 − 1)τF − 0.05) K ov1 ]

(14)

for j=3i/4+1,..i-1:

β j = β 1 [1 − (2.78( j − K 3 − 1)τF − 0.1) K ov1 ]

(15)

β i = β " = β 1 [1 − (2.78(i − K 3 − 1)τF − 0.1) K ov1 ]K s

(16)

γ j = β i − j +1 [0.5 − 6(i − j − K 3 )τF ]K ov 2

(17)

γ 1 = 3β " (λ ' + β " ) FK ov 2 K s ,

(18)

and (10)-(11) are used here too for determination of the duration of notches. Fig. 9 illustrates the variation of parameters of line-to-line output voltage of the inverter from the initial frequency 2.5 Hz of the scalar control mode until the zone of overmodulation near the maximum fundamental frequency of the drive system Fm = 60 Hz. The speed ratio (diapason) is D = 24 for this variant. As an example, here is the non-linear dependence on the duration of switching periods τ from F :

Fi ' = 1 /[6(2i − 1.5)τ ]

(22)

τ = Fm /[6 F ( DFm + Fm − DF )] .

Fi" = 1 /[6(2i − 2.5)τ ] .

(23)

Fig. 9. Variation of control parameters for DPWM1 and DPWM3 (a), dependence of the number of pulses in half-wave of output voltage (b).

(19)

The boundary frequencies are calculated in accordance with (20) for DPWM1 and DPWM3 in this case: Fi = Fm ( D − 2i + 2.5) / D .

Algorithm of two-stage synchronous PWM is characterised by two control sub-zones, which are changing each other step-by-step during adjustment of the fundamental frequency. In the first of these sub-zones, when Fi" > F ≥ Fi ' , the voltage waveforms for discontinuous PWM are like the presented in Fig. 4 and Fig. 5, and the basic set of control functions is the same too, but here are another value of the coefficient of synchronization, which is equal to

(20)

Fig. 9a shows the width variation of τ , β 1 , λ ' , β " . Here is a synchronous quasi-linear variation of λ ' and β " , which are decreased simultaneously till close to zero width at the boundary frequencies. Fig. 9b presents the number of pulses in half-wave of the output voltage of the inverter, which is changing by eight pulses after every boundary frequency for discontinuous one-stage versions of synchronized PWM.

K s1 = [1 − ( F − Fi ' ) /( Fi" − Fi ' )] for determination of λ' and γ 1 , and

K s' 1 = [1 − 0.5( F − Fi ' ) /( Fi" − Fi ' )]

C. Two-Stage Scheme of Synchronous PWM

(25)

for determination of the β " signal. So here is a linear

In order to provide more smooth variation of the number of pulses in half-wave of the output voltage of the inverter, a two-stage scheme of synchronous PWM can be used. Its algorithm is based on two threshold (boundary) frequencies Fi ' and Fi" , determined for discontinuous PWM versions as a function of the number of notches i (21) inside a half of 600 clock-intervals in accordance with (22)-(23): i = (1 / 6 F + 0.5τ ) / 2τ

(24)

decrease of duration of the β " , and then the junction of two equal halves into one signal β ' , situated on the edges of 600 clock-intervals, at the Fi" boundary frequencies. Control in the second sub-zones, when Fi ' > F ≥ Fi"+1 , is based on the frame set of functions, described in previous parts. It is also characterized by the linear variation of the

(21)

80

β ' -parameter in accordance with (26), whose width is close to zero at the next boundary frequency. Here is also smooth quasi-linear variation of the notches λ" (27), neighboring with the β ' -signal. The coefficient of synchronization Ks2 is determined from (28) in these control sub-zones: β ' = β i +1 = 0.433β 1 K s 2

(26)

λ" = λi +1 = 1 / 12 F − (i − 1.25)τ − β i / 2 − β ' / 2

(27)

K s 2 = [1 − ( F − Fi"+1 ) /( Fi ' − Fi"+1 )] .

(28)

V. SPECTRAL ANALYSIS OF THE OUTPUT VOLTAGE In order to compare the modulation methods including comparison of synchronous and asynchronous PWM versions, a comparative analysis of the spectra of the line-toline output voltage of the inverter has been executed based on computer simulation. Weighted total harmonic distortion (WTHD) factor, computing in accordance with (29), is used for determination of its quality: n

WTHD = (1 / V1 ) ∑ (Vi / i ) 2 .

(29)

i =2

Fig. 10 shows the variation of the basic control signals for a two-stage scheme in discontinuous PWM until the zone of overmodulation. The parameters of the control process are the same as in the example presented in the previous part. Fig. 10a illustrates the variation of the clock-points signals λ' and β ' which are changing each other step-by-step on the boundary frequencies. Fig. 10b presents variation of the signals λ" and β " which are the next with the corresponding clock-point signal. Fig. 10c shows the pulse number variation N in the half-wave of the inverter output voltage. It is characterized by more flexible smooth pulses ratio changing compared with the ratio changing of one-stage scheme, presented in Fig. 9b. Here is non-uniform step-by-step changing the number of pulses (by 6 pulses, by 2, by 6, and so on).

Fig. 11 and Fig. 12 show the variation of the average WTHD for different analysed modulation techniques. The voltage harmonics have been determined in (29) until the double value of the switching frequency by means of an FFT algorithm of the MATLAB software package. Fig. 11 presents a comparison of the WTHD factor between two versions of synchronized continuous scheme of PWM (CPWM), based on algebraic and trigonometric control functions, and the conventional asynchronous version of continuous PWM [3]. WTHD has been determined as a function of the ratio Fs / F between the switching frequency Fs and the fundamental frequency F of the inverter, where WTHD (%) 2.5 2.0

1

2 1.5

3

1.0 0.5

FS /F 0

20

40

60

80

100

Fig. 11. WTHD of line voltage for asynchronous CPWM (1), and for synchronized algebraic (2) and trigonometric (3) CPWM.

Fig. 12. WTHD versus modulation index for the synchronized CPWM (1), DPWM1 (2) and DPWM3 (3).

Fig. 10. Variation of the clock-point signals (a), signals neighbouring with the clock-point signals (b), and number of pulses in half-wave of voltage (c).

81

F = 30 Hz, modulation index m = 0.6. The determination of WTHD for the asynchronous control mode of the conventional continuous PWM is based on the approach, which is close to the described in [7]. The presented results show advantage of trigonometric synchronous PWM before others, and also some advantage of algebraic synchronous PWM compared to the asynchronous until the frequency ratio is equal to 70. Fig. 12 shows the variation of the WTHD factor of line-toline output voltage of the inverter from the modulation index m (until the zone of overmodulation) for the analyzed algebraic continuous and discontinuous variants of synchronized PWM, for scalar control mode of the system and for three values of the average switching frequency Fs, equal to 1.1, 2.2 and 4.4 kHz. Curves 1 correspond to CPWM, curves 2 – to DPWM1, and curves 3 – to PWM3. The presented results prove a fact of better performance of continuous PWM techniques in the zone of the middle fundamental frequencies of the drive system. Then, at the higher fundamental frequencies of the system, discontinuous synchronized schemes of PWM are more preferable. It has also proved the fact of better performance of synchronized version of DPWM3 before DPWM1. Analysis and comparison of the average WTHD for onestage and two-stage schemes of synchronous PWM show practical identity of this factor for the both schemes.

Novel techniques of direct synchronous pulsewidth modulation, based on both trigonometric and algebraic approach with linear and quasi-linear control functions, are applied to basic continuous and discontinuous schemes of voltage space vector modulation. It provides synchronization of the voltage waveforms of the inverter and smooth pulses ratio changing during the whole control range including the zone of overmodulation. The proposed PWM techniques belong to the class of direct modulation methods, so its implementation can be based on a typical digital control system for direct pulsewidth modulation, like described in [8]. Due to quarter-wave symmetry of voltage waveforms the spectrum of the output voltage of the inverter does not contain even harmonics and combined harmonics (subharmonics), that is especially important for high power systems and for systems with low switching frequency of power devices. These techniques of synchronous PWM can also be disseminated to other continuous and discontinuous, symmetrical and asymmetrical schemes of modulation, and also to different topologies of power converters, in particular, to multilevel power converters and to modular converters consisted from standard inverter modules [9].

VI. CONTROL IN THE ZONE OF OVERMODULATION

ACKNOWLEDGMENT

The method described allows to provide linear control of the fundamental output voltage of the inverter during the whole control range, including the zone of overmodulation. Control during overmodulation is based on the same set of control functions, by the use of two special linear coefficients of overmodulation Kov1 and Kov2 (see part IV of the paper), providing a linear voltage control between two threshold Fov1 = 0.907 Fm fundamental frequencies and Fov 2 = 0.952 Fm , determined from [3], and the maximum fundamental frequency Fm. Fig. 13 shows the variation of the magnitude of the fundamental voltage versus modulation index in the zone of overmodulation for algebraic synchronous DPWM3 ( Fs = 4.4 kHz), characterised by a high level of the linearity.

This research has been supported in part by the Danish Technical Research Council in the range of the project of the NATO Fellowship Program in Denmark (supervisor Dr. F.Blaabjerg).

CONCLUSION

REFERENCES [1] [2] [3] [4]

[5]

[6]

[7]

[8]

[9] Fig. 13. Fundamental voltage versus modulation index.

82

N. Mohan, T.M. Undeland, and W.P. Robbins, Power Electronics, John Willey & Sons, 1995. B.K. Bose, Modern Power Electronics and AC Drives, Prentice Hall, 2001. J. Holtz, ”Pulsewidth modulation for electronic power conversion,” IEEE Proc., vol.82, n.8, 1994, pp.1194-1213. V. Oleschuk and B.K. Bose, “Fast algorithms of modulation of inverter output voltage for induction motor drive,” Proc. of IEEE Intern. Symp. on Industrial Electronics (ISIE’1999), pp.801-805. J.W. Kolar, H. Ertl and F.C. Zach, “Influence of the modulation method on the conduction and switching losses of a PWM converter system,” IEEE Trans. on Ind. Appl., vol.IA-27, no.6, 1991, pp.10631075. A.M. Hava, R.J. Kerkman and T. Lipo, “A high-performance generalized discontinuous PWM algorithm,” IEEE Trans. on Ind. Appl., vol.IA-34, no.5, 1998, pp.1059-1071. T. O’Gorman, “Discrete Fourier Transform harmonic analysis of digitally-generated PWM waveforms which are distored by switch dead time,” Proc. of IEEE-IAS’2000 Conf., pp.2197-2204. D.-W. Chung, J.-S. Kim and S.-K. Sul, “Unified voltage modulation technique for real-time three-phase power conversion,” IEEE Trans. on Ind. Appl., vol.IA-34, no.2, 1998, pp.374-380. R. Teodorescu, F. Blaabjerg, J.K. Pedersen, P. Enjeti and E. Cengelci, “Space Vector Modulation Applied to Modular Multilevel Converters”, Proc. of PCIM’99, pp.363-368.