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Hindawi Publishing CorporationAdvances in Power ElectronicsVolume 2011, Article ID 581306, 9 pagesdoi:10.1155/2011/581306

Research Article

Simulation of Processes in Dual Three-Phase System on the Baseof Four Inverters with Synchronized Modulation

Valentin Oleschuk,1 Gabriele Grandi,2 and Padmanaban Sanjeevikumar2

1 Institute of Power Engineering, Academy of Sciences of Moldova, 2028 Chishinev, Moldova2 Department of Electrical Engineering, University of Bologna, 40136 Bologna, Italy

Correspondence should be addressed to Valentin Oleschuk, [email protected]

Received 30 June 2011; Accepted 15 September 2011

Academic Editor: Francesco Profumo

Copyright © 2011 Valentin Oleschuk et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Novel method of space-vector-based pulse-width modulation (PWM) has been disseminated for synchronous control of fourinverters feeding six-phase drive on the base of asymmetrical induction motor which has two sets of windings spatially shifted by30 electrical degrees. Basic schemes of synchronized PWM, applied for control of four separate voltage-source inverters, allow bothcontinuous phase voltages synchronization in the system and required power sharing between DC sources. Detailed MATLAB-based simulations show a behavior of six-phase system with continuous and discontinuous versions of synchronized PWM.

1. Introduction

Multiphase and, in particular, six-phase induction motordrives are a subject of increasing interest in the last yearsdue to some advantages compared with conventional three-phase drives [1–5]. One of the most perspective applicationsof six-phase induction motor drives lies in the field of highpower/high current systems (ship propulsion, locomotive,electrical vehicles, etc.), which are characterized by lowswitching frequency of power switches.

Recently, novel four-inverter-based topology of dualthree-phase drive system with an increased power rating hasbeen proposed, allowing quadrupling the power capabilityof a single inverter with given voltage and current rating[6]. Figure 1 shows this system structure, consisting of twogroups of two inverters, supplying the open-end windingsof asymmetrical dual three-phase motor. The inductionmachine has in this case two sets of winding spatially shiftedby 30 electrical degrees.

For electrical drives with increased power and/or currentit is necessary to synchronize the output voltage waveformsof converters for elimination of undesirable subharmonics ofvoltage and current [7–9]. In order to avoid asynchronism ofstandard versions of space-vector modulation, novel methodof synchronized space-vector PWM has been proposed anddeveloped with application to different topologies of power

electronic converters, electric drives, and renewable energysystems [10–17].

So, this paper presents results of detailed MATLAB-basedsimulation of processes in asymmetrical six-phase drivesystem on the base of four cascaded voltage-source inverterscontrolled by algorithms of synchronized space-vector mod-ulation.

2. Peculiarities of the Method ofSynchronized Space-Vector PWM

In order to avoid asynchronism of conventional space-vectormodulation, novel method of synchronized PWM [10, 11]can be used for control of each inverter in the six-phase (dualthree-phase) drive system.

Figures 2 and 3 present basic switching state sequencesof standard three-phase voltage-source inverter inside theinterval 0◦–90◦. They illustrate schematically continuous(Figure 2) and discontinuous (Figure 3) versions of synchro-nized space-vector PWM.

The upper traces in Figures 2 and 3 are switchingstate sequences (in accordance with conventional designation[10]) and then control signals for the cathode switches ofthe phases a, b, c (x, y, z) of each inverter. The lower tracesin Figures 2 and 3 show the corresponding quarter wave of

2 Advances in Power Electronics

Adc1

Adc2

Adc3

Adc4

INV1

INV1

INV2

INV2

INV3

INV3

INV4

INV4

30◦

cs-axis

zs-axis

ys-axis

bs-axis

ar-axis

xs-axis

as-axisias

π

6

π

6

bs brcr

zrar

y2x2

z2

y1x1 z1

+

+

a2

c2

−

−

+

−

+

+

−

−+

−

{1} {2}

α-axis

β-axis

0

0

0

0

Aas

b2

b1a1

c1

xryr

ys

zs

asxsixs

Axs

Vdc1

Vdc2

Vdc3

Vdc4

cs

ϑr

Figure 1: Topology of dual three-phase system on the base of four inverters (the first inverter group INV1+INV2 and the second invertergroup INV3+ INV4) supplying open-end windings of asymmetrical six-phase induction motor with two sets of winding spatially shifted by30 electrical degrees.

γ1 γ2 γ3 γ4

β1β2 β2β3 β3β4 β4β5 β5

λ1λ2λ3λ4λ5

2

3

4

Vab

Phase c

Phase b

Phase a

Swit

chin

g se

quen

ce

Figure 2: Switching state sequence, pole voltages, and line voltageVab of inverter with continuous synchronized PWM.

the line output voltage of inverters. Signals β j represent totalswitch-on durations during switching sub interval τ, andsignals γk are generated on the borders (Figure 2) or in thecenters (Figure 3) of the corresponding β. Widths of notchesλk represent duration of zero sequences (notches) [10, 11].

One of the basic ideas of the method of synchronizedPWM is in continuous synchronization of the positions of

γ1 γ2 γ3

β1β2 β2β3 β3β4 β4

λ1λ2λ3λ4

2

3

4

Vab

Phase c

Phase b

Phase a

Swit

chin

g se

quen

ce

Figure 3: Switching state sequence, pole voltages, and line voltageVab of inverter with discontinuous synchronized PWM.

all central signals β1 in the centres of the 60◦-clock intervals,and then in symmetrical generation of all other active β-signals around the centres of the 60◦-clock intervals. Inparticular, special signals λ′ (λ5 in Figure 2, λ4 in Figure 3)with the neighboring β′′ (β5 in Figure 2, β4 in Figure 3) areformed in the clock points (0◦, 60◦, 120◦) of the outputcurve of inverters with synchronized PWM for this purpose.

Advances in Power Electronics 3

Table 1: Basic parameters of PWM methods.

Control (modulation) parameterConventional schemes of

vector PWMProposed method of modulation

Operating and max parameterOperating & max voltage

V and VmOperating & maximum fundamental frequency F and Fm

Modulation index m V/Vm F/Fm

Duration of subcycles T τ

Center of the k-signal αk (angles/degr.) τ(k − 1) (sec)

Switch-on durations

Tαk = 1.1mT[sin (60◦ −αk) + sinαk]

Algebraic PWM Trigonometric PWM

tαk = 1.1mT sinαkβk = β1[1− A× (k − 1)τFKOν1] βk = β1 × cos[(k − 1)τKOν1]

γk = βi−k+1[0.5− 6(i− k)τF]KOν2 γk = βi−k+1[0.5− 0.9m(i− k) τ] KOν2

tbk = 1.1mT×sin(60◦−αk) βk − γk βk − γk

Switch-off states (zero voltage) t0k = T − tαk − tbk λk = τ − βkSpecial parameters providingsynchronization of the process ofPWM

β′′ = β1[1−A×(K−1)τFKOν1]Ks β′′ = β1 × cos[(k − 1)τKOν1]Ks

λ′ = (τ − β′′)× KOν1Ks λ′ = (τ − β′′)× KOν1Ks

They are reduced simultaneously until close to zero valueat the boundary frequencies between control subzones. So,this novel PWM strategy provides continuous symmetricaloutput voltage control of inverters.

Table 1 presents generalized properties and basic controlcorrelations for the proposed method of synchronized PWM[10]. It is also compared here with conventional asynchro-nous space-vector modulation.

3. Synchronized PWM in Dual Three-PhaseSystem on the Base of Four Inverters

Control of four three-phase inverters supplying asymmet-rical six-phase induction motor has some specific peculiar-ities.

In particular, these inverters are grouped into twogroups with two cascaded inverters in each group, and eachinverter group is connected with the corresponding open-end windings of dual three-phase induction motor. Syn-chronous symmetrical control of the output voltage of eachinverter of each inverter group in accordance with algorithmsof synchronized PWM provides synchronous symmetri-cal regulation of voltage in the corresponding inductionmachine phase windings. Rational phase shifting betweenoutput voltage waveforms of the two inverters in eachinverter group is equal in this case to one half of the switchinginterval (subcycle) τ [18].

In the case when the two DC-link voltage sources havethe same voltage, the resulting voltage space vectors areequal to the space-vector patterns of conventional three-levelinverter [3, 6, 18]. The phase voltage Vas of the first groupof cascaded inverters with two insulated DC-link sources(Figure 1) is calculated in accordance with (1) [19]:

V01 = 13

(Va1 + Vb1 + Vc1 + Va2 + Vb2 + Vc2),

Vas = Va1 + Va2 −V01,

0

Time (s)

0.0250.005

0.01 0.015 0.02 0.03

Vxs

Vas

Va1

Va1b1

Va2

Vx1

Vx2

Vx1y1

Figure 4: Pole voltages Va1, Va2 and Vx1, Vx2, line-to-line voltagesVa1b1 and Vx1y1, and phase voltages Vas and Vxs of dual three-phasesystem with continuous synchronized PWM (F = 35 Hz, Vdc1 =Vdc2 = Vdc3 = Vdc4, m1 = m2 = m3 = m4 = 0.7, Fs = 1 kHz).

V02 = 13

(Vx1 + Vy1 + Vz1 + Vx2 + Vy2 + Vz2

),

Vxs = Vx1 + Vx2 −V02,

(1)

where Va1, Vb1, Vc1, Va2, Vb2, Vc2 and Vx1, Vy1, Vz1, Vx2,Vy2, Vz2 are the corresponding pole voltages of each group ofthree-phase inverters, and V01 and V02 are the correspondingzero sequence (triplen harmonic components) voltages.

At the same time, control of asymmetrical six-phaseinduction machines is based on the 30◦-phase shift ofcontrol and output signals of two above-mentioned groupsof four inverters [2, 5]. As an illustration of control of dualthree-phase system with synchronized PWM and with equalvoltages of all DC sources (Vdc1 = Vdc2 = Vdc3 = Vdc4),Figures 4–7 present results of MATLAB-based simulations,which include basic voltage waveforms (pole voltages Va1,


1

0.8

0.6

0.4

0.2

00 Order of voltage harmonics

20

40 60 80

100

Spectrum of Vas

Vk/V

dc2

Figure 5: Spectrum of the phase voltage Vas of the system withcontinuous synchronized PWM (F = 35 Hz, Fs1 = 1 kHz).

0

Time (s)

0.0250.005

0.01 0.015 0.02

Vxs

Vas

Va1

Va1b1

Va2

Vx1

Vx2

Vx1y1

Figure 6: Pole voltages Va1, Va2 and Vx1, Vx2, line-to-line voltagesVa1b1 and Vx1y1, and phase voltages Vas and Vxs of dual three-phasesystem with discontinuous synchronized PWM (F = 35 Hz, Vdc1 =Vdc2 = Vdc3 = Vdc4, m1 = m2 = m3 = m4 = 0.7, Fs = 1 kHz).

Va2, Vx1, Vx2, line-to-line voltages Va1b1, Vx1y1, and thephase voltages Vas and Vas (with the spectrum of the Vas

voltage)) of two groups of inverters, controlled by algorithmsof continuous (Figures 4 and 5) and discontinuous with the30◦-nonswitching intervals (Figures 6 and 7) synchronizedmodulation. The fundamental and switching frequencies ofeach inverter are equal correspondingly to F = 35 Hz andFs = 1 kHz, and modulation indices of all inverters are equalto m1 = m2 = m3 = m4 = 0.7 in this case. The phase outputvoltage of the system has nine levels (like output voltageof three-level neutral-clamped converter) in this controlmode. In particular, the presented voltage waveforms havesymmetry, and its spectra do not contain even harmonicsand subharmonics.

In the case of nonequal DC voltages, if Vdc1 /=Vdc2 andVdc3 /=Vdc4, but Vdc1 = Vdc3 and Vdc2 = Vdc4, in order toprovide equivalence of the output fundamental voltages (and

1

0.8

0.6

0.4

0.2


20

40 60 80

100

Spectrum of Vas

Vk/V

dc2

Figure 7: Spectrum of the phase voltage Vas of the system withdiscontinuous synchronized PWM (F = 35 Hz, Fs1 = 1 kHz).

0

Time (s)

0.0250.005

0.01 0.015 0.02

Vxs

Vas

Va1

Va1b1

Va2

Vx1

Vx2

Vx1y1

Figure 8: Pole voltages Va1, Va2 and Vx1, Vx2, line-to-line voltagesVa1b1 and Vx1y1, and phase voltages Vas and Vxs of dual three-phasesystem with continuous synchronized PWM (F = 35 Hz, Vdc1 =0.7Vdc2, Vdc3 = 0.7Vdc4, Vdc2 = Vdc4, m1 = m3 = 0.933, m2 = m4 =0.7, Fs = 1 kHz).

also power balancing) of two inverters of each inverter groupduring scalar V/F control of the system, it is necessary toprovide linear correlations between its modulation indicesand magnitudes of the DC voltages:

m1Vdc1 = m2Vdc2,

m3Vdc3 = m4Vdc4,(2)

In particular, for the case when Vdc1 = 0.7Vdc2, Vdc3 =0.7Vdc4 and m2 = m4 = 0.7, in accordance with (2)m1 = m3 = 0.93, and the last value of modulation indicescorresponds to overmodulation control mode of the twocorresponding inverters. To illustrate the above-mentionedprocess in dual three-phase drive system on the base of fourinverters supplied by four isolated DC sources with differentvoltages (Vdc1 = 0.7Vdc2, Vdc3 = 0.7Vdc4, Vdc2 = Vdc4),Figures 8–11 present results of simulation of this system.


0.8

0.6

0.4

0.2


20

40 60 80

100

Spectrum of Va1b1

Vk/V

dc2

(a)


20

40 60 80

100

Spectrum of Va2b20.8

0.6

0.4

0.2

0

Vk/V

dc2

(b)

1Spectrum of vas

0.8

0.6

0.4

0.2


20

40 60 80

100

Vk/V

dc2

(c)

Figure 9: Spectra of the Va1b1, Va2b2 and Vas voltages of the systemwith continuous synchronized PWM (F = 35 Hz, Fs = 1 kHz).

0

Time (s)

0.0250.005

0.01 0.015 0.02

Vxs

Vas

Va1

Va1b1

Va2

Vx1

Vx2

Vx1y1

Figure 10: Pole voltages Va1, Va2 and Vx1, Vx2, line-to-line voltagesVa1b1 and Vx1y1, and phase voltages Vas and Vxs of dual three-phasesystem with discontinuous synchronized PWM (F = 35 Hz, Vdc1 =0.7Vdc2, Vdc3 = 0.7Vdc4, Vdc2 = Vdc4, m1 = m3 = 0.933, m2 = m4 =0.7, Fs = 1 kHz).

In particular, Figures 8 and 9 show basic voltage waveformsand spectra of the Va1b1, Va2b2 and Vas voltages for the systemcontrolled by algorithms of continuous synchronized PWM.Figure 10 presents basic voltage waveforms during period ofthe fundamental frequency for the system with discontinu-ous synchronized PWM with the 30◦-nonswitching intervals,and Figure 11 shows spectra of the line and phase voltages ofthe first group of cascaded inverters. The fundamental andswitching frequencies of each inverter are equal to F = 35 Hzand Fs = 1 kHz. It is necessary to mention that spectra ofthe corresponding output voltages of inverters of the secondinverter group will have the same nature for the presentedcontrol mode.

For dual three-phase drive system on the base of fourinverters with nonequal voltages of DC links, in order toprovide the rated power ratio P1/P2 and P3/P4 betweenfour power sources of two groups of cascaded inverters(for scalar V/F control mode), it is necessary to providethe corresponding correlations between magnitudes of DCvoltages, modulation indices of four inverters, and the ratedpower ratio in accordance with (3) [14]:

m1Vdc1

m2Vdc2= P1

P2,

m3Vdc3

m4Vdc4= P3

P4.

(3)

For dual three-phase drive system it is practically impor-tant to provide equal power distribution between two groupsof inverters feeding asymmetrical six-phase induction motor:

P1 + P2 = P3 + P4. (4)

For this case, in accordance with (3), for balancing op-eration of dual three-phase system it is necessary to provide:

m1Vdc1P2 + m2Vdc2P1 = m3Vdc3P4 + m4Vdc4P3, (5)


0.8

0.6

0.4

0.2


20

40 60 80

100

Spectrum of Va1b1

Vk/V

dc2

(a)


20

40 60 80

100


0.6

0.4

0.2

0

Vk/V

dc2

(b)

1

0.8

0.6

0.4

0.2


20

40 60 80

100

Spectrum of Vas

Vk/V

dc2

(c)

Figure 11: Spectra of the Va1b1, Va2b2 and Vas voltages of the systemwith discontinuous synchronized PWM (F = 35 Hz, Fs = 1 kHz).

0

Time (s)

0.0250.005

0.01 0.015 0.02

Vxs

Vas

Va1

Va1b1

Va2

Vx1

Vx2

Vx1y1

Figure 12: Pole voltages Va1, Va2 and Vx1, Vx2, line-to-line voltagesVa1b1 and Vx1y1, and phase voltages Vas and Vxs of dual three-phasesystem with discontinuous synchronized PWM (F = 35 Hz, Vdc1 =0.8Vdc2, Vdc2 = 0.9Vdc4, Vdc3 = 0.75Vdc4, m1 = 0.97, m2 = 0.777,m3 = 0.933, m4 = 0.7, Fs = 1 kHz).

where corresponding power of each inverter can be describedas relative value of the total power of the system.

As an example of balanced operation of the systemwith four different levels of DC voltages, Figures 12 and 13present basic voltage waveforms and spectra of the Va1b1,Va2b2, Vas and Vxs voltages for the dual three-phase sys-tem with discontinuous synchronized PWM with the 30◦-nonswitching intervals. The fundamental and switchingfrequencies of each inverter are equal to F = 35 Hz andFs = 1 kHz. Relative magnitudes of DC voltages are equalfor this case, as parts of the maximum DC-voltage Vdc4:

Vdc1 = 0.72Vdc4; Vdc2 = 0.9Vdc4; Vdc3 = 0.75Vdc4.(6)

Modulation index of the fourth inverter, supplied by themaximum DC-voltage Vdc4, is equal to m4 = 0.7. Takingin consideration equal power distribution between thecorresponding sources (P1 = P2 = P3 = P4), in accordancewith (5), modulation indices of the corresponding invertersare equal to

m1 = 0.97; m2 = 0.777; m3 = 0.933. (7)

So, in this case control modes of two inverters, of the firstand the third, correspond to overmodulation control. Theproposed algorithm provides equal magnitudes of the phasefundamental voltages Vas and Vxs of dual three-phase system(see Figure 13), and, correspondingly, equal power distribu-tion between two sets of three-phase windings of six-phasemachine.

The motor phase voltages Vas and Vxs of six-phase driveson the base of four inverters with both continuous anddiscontinuous synchronized PWM have symmetry duringthe whole control range and for any operating conditions(see Figures 4–13), and their spectra do not contain even


0.8

0.6

0.4

0.2


20

40 60 80

100

Spectrum of Va1b1

Vk/V

dc4

(a)


20

40 60 80

100


0.6

0.4

0.2

0

Vk/V

dc4

(b)

1

0.8

0.6

0.4

0.2


20

40 60 80

100

Spectrum of Vas

Vk/V

dc4

(c)

1

0.8

0.6

0.4

0.2


20

40 60 80

100

Spectrum of Vxs

Vk/V

dc4

(d)

Figure 13: Spectra of the Va1b1, Va2b2Vas and Vxs voltages of the system with discontinuous synchronized PWM (F = 35 Hz, Fs = 1 kHz).

harmonics and subharmonics, which is especially importantfor high power/current systems.

In order to compare characteristics of asymmetrical dualthree-phase (six-phase) systems on the base of two inverters([2], standard topology of the system), and on the baseof four inverters (novel topology of six-phase system [6],analyzed in the paper), Figure 14 presents calculation resultsof Weighted Total Harmonic Distortion factor (WTHD)versus modulation index m for the motor phase voltage Vas

(averaged values of WTHD = (1/Vas1 )(∑1000

k=2 (Vask /k)2)0.5)for the six-phase drive with continuous (CPWM) and dis-continuous (DPWM) schemes of synchronized modulation.In particular, DC-voltage magnitudes are equal in this casefor all DC sources, so, modulation indices of all invertersare equal too. Control mode of the drive system correspondshere to standard scalar V/F control, and average switchingfrequency of each inverter is equal to Fs = 1 kHz.

The presented results show that integral spectral charac-teristics of the phase voltage of six-phase system on the base

of four inverters are much better than of the system on thebase of two inverters.

Figure 15 presents results of analysis of spectral compo-sition of the phase voltage Vas of dual three-phase systemon the base of four inverters with both continuous (CPWM)and discontinuous (DPWM) versions of synchronized PWMfor the case of nonequal magnitudes of DC voltages. Inparticular, for this case Vdc1 /=Vdc2 and Vdc3 /=Vdc4, butVdc1 = Vdc3 and Vdc2 = Vdc4. Two basic control modes havebeen analyzed:

(1) Vdc1 = 0.9Vdc2, Vdc3 = 0.9Vdc4, Kdc = 0.9 for thiscase;

(2) Vdc1 = 0.7Vdc2, Vdc3 = 0.7Vdc4, Kdc = 0.7 for thiscase.

The presented results show that a value of the WTHDfactor of the phase voltage of six-phase system on the baseof four inverters controlled by algorithms of discontinuous


WTHD of Vas2

1.5

1

0.5

0.2 0.4 0.6 0.8 1

Modulation index m

CPWM-2INVCPWM-4INV

DPWM-2INVDPWM-4INV

0

Wei

ghte

d to

tal h

arm

onic

dis

tort

ion

(%

)

Figure 14: Averaged WTHD factor of the phase voltage Vas versusmodulation index m for six-phase system on the base of twoinverters (CPWM-2INV, DPWM-2INV), and for the system on thebase of four inverters (CPWM-4INV, DPWM-4INV).

WTHD of Vas

0.2 0.4 0.6 0.8 1

Modulation index m

Wei

ghte

d to

tal h

arm

onic

dis

tort

ion

(%

)

CPWM: Kdc = 0.9CPWM: Kdc = 0.7

DPWM: Kdc = 0.9DPWM: Kdc = 0.7

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 15: Averaged WTHD factor of the phase voltage Vas versusmodulation index m = m2 = m4 for the six-phase system on thebase of four inverters with nonequal DC voltages.

synchronized PWM is not strongly sensitive to relativemagnitudes of DC voltages.

4. Conclusion

Space-vector-based algorithms of synchronized PWM, ap-plied for control of four voltage-source inverters feedingasymmetrical dual three-phase (six-phase) induction motor

with open-end windings, allow continuous synchronizationof the phase voltages in the system for any operating condi-tions. The presented results of MATLAB-based simulationsof the system show in details its behavior. In particular,algorithms of continuous and discontinuous synchronizedPWM provide voltage synchronization for any ratios (inte-gral or fractional) between the switching and fundamentalfrequencies, and for any ratios of voltage magnitudes of fourDC sources.

The phase voltages of six-phase drives on the basisof four inverters with synchronized PWM have symmetryduring the whole control range, including the zone ofovermodulation, and their spectra do not contain evenharmonics and subharmonics, which is especially importantfor high power/high current applications.

Simple linear correlations between modulation indices offour three-phase inverters, magnitudes of DC voltages, andrequired power sharing between inverters provide requiringpower balancing between DC sources and equivalence of thephase fundamental voltages of two groups of inverters.

Nomenclature

F: Fundamental frequencyFm: Maximum fundamental

frequencyi: Number of notches inside a

half of the clock intervalKs: Coefficient of synchronizationKov1: The first coefficient of

overmodulationKov2: The second coefficient of

overmodulationm1, m2, m3,, m4: Modulation indices of four

invertersP1, P2, P3, P4: Power of four DC sourcesVdc1, Vdc2, Vdc3,Vdc4: DC-sources voltagesVa1, Vb1, Vc1,Va2, Vb2, Vc2: Pole voltages of the first and

the second invertersVx1, Vy1, Vz1,Vx2, Vy2, Vz2: Pole voltages of the third and

fourth invertersVas, Vbs, Vcs: Phase voltages of the first

inverter groupVxs, Vys, Vzs: Phase voltages of the second

inverter groupVa1b1, Va2b2, Vx1y1, Vx2y2: Line-to-line voltages of

invertersVk: Amplitude of the k-harmonic

of voltageV0: Zero-sequence voltageWTHD: Weighted total harmonic

distortion factorβ1: Duration of the central active

switching signal inside theclock interval

β j : Duration of others activeswitching signals

γ j : Duration of the minor part ofthe active switching signals


λ j : Duration of notchesλ′: Duration of notches, generated at the clock

pointsβ′′: Duration of the β-signals, neighboring

with the λ′ notchesτ: Width of sub cycle (switching interval).

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