research article an improved phase disposition spwm strategy...

Research ArticleAn Improved Phase Disposition SPWM Strategy for CascadedMultilevel Inverter

Jinbang Xu1 Zhizhuo Wu1 Xiao Wu2 Fang Wu1 and Anwen Shen1

1 Huazhong University of Science and Technology Wuhan Hubei 430074 China2Guangdong University of Technology Guangzhou Guangdong 510006 China

Correspondence should be addressed to Anwen Shen sawyimailhusteducn

Received 21 August 2013 Revised 2 December 2013 Accepted 4 December 2013 Published 19 March 2014

Academic Editor Tao Li

Copyright copy 2014 Jinbang Xu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

TheCarrier Phase Shifted (CPS) strategy is conventional for cascadedmultilevel inverter because it can naturally ensure all cascadedcells to output balanced power However in point of spectra of the output line voltage CPS is suboptimal to Phase Disposition (PD)strategy because the latter can not naturally ensure power balance This paper presents an improved PD strategy inspiration fromthe disposition of CPS strategy triangle carriers Just reconstructing the triangle carriers of PD strategy it can not only reserve thewaveform quality of the line voltage to be optimal but also naturally ensure the output power of each cascaded cells to be balanced

1 Introductions

The cascaded multilevel inverters are widely implemented inthe high power AC drivers and power management systemswith a medium or high voltage level [1ndash5] The structure ofthe cascaded multilevel inverters is shown in Figure 1 It isconstructed with several full bridge inverters with the sametopologyThe advantages of the cascaded multilevel invertersinclude being simple to control being easy in output voltagelevel extension modular design and implementation and thefault tolerance control realization [6ndash10]

The Carriers Phase Shift (CPS) control is one of theconventional control schemes employed in the cascadedmultilevel inverters The control signals for each cascadedunit are generated by comparison between N (N is thenumber of cascaded units) triangle carrier and themodulatedsignal The carrier waveform is chosen with a phase shiftof 360N degree The fundamental component amplitude ofeach unit driving signal is the same So the output power ofeach unit is balanced [11ndash14]

Except for the CPS control method Phase Disposition(PD) Phase Opposite Disposition (POD) and AlternativePhase Opposite Disposition (APOD) are three other carrydeposition control strategies used in multilevel inverters [15]In [10] under the condition of same switching frequencythe output line voltage waveforms with both the CPS and

PD control methods are compared It is indicated that theexperiment result with the PD control method is better thanthat with the CPS controlHowever PD cannot achieve powerbalance spontaneously According to [16] each cascaded unitoutputs the PWM under the previous setting cycle and theoutput power will be balanced after N2 output periods Thisbalance strategy is easy but the power balance time increaseswith the cascade units increasing

To shorten the time of power balance this paper proposesan improved PD control strategy By improving the carrierwave for modulation the output power of each cascadedunit can be balanced in an output cycle and the switchingfrequency of each switch tube is equal to each other atthe same time The output line voltage waveform qualityis maintained to be optimized as that of the PD controlAn experimental prototype of a 5 kW three-phase cascadedmultilevel inverter is designed to verify the validity of thisimproved PD control

2 CPS and PD Control Strategy

21 Principle The principle of the CPS control strategybased on a two-unit cascaded multilevel inverter is shown inFigure 2(a) The triangular carrier of tri1simtri4 and thereference signal Vref are combined to generate the control

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 731574 9 pageshttpdxdoiorg1011552014731574

2 Mathematical Problems in Engineering

Uin

Uin

Sa1

Sa2

Sa3

Sa4

Sa5

Sa6

Sa7

Sa8

CHBb2

CHBb1 CHBc1

CHBc2

A B C

N

Figure 1 Cascaded five-level inverter

signal for the bridgesThe triangular carrier signal is arrangedwith 90 degree phase shift between the adjacent ones Thecontrol signal modulation is performed with the followingprinciple

The drive signal for Ha1 is generated by comparing Vrefwith tri1 and tri3 And the drive signal forHa2 is generated bycomparing Vref with tri2 and tri4 As is shown in Figure 2(a)the uHa1 and uHa2 are the output voltage of the full bridgesrespectively And the 119906

119860119873is the output voltage of phase A

In Figure 2(b) themodulation principle of the PD controlstrategy is illustrated Like that of CPS a combination offour triangular carrier signals is implemented with the samephase but different amplitude range The amplitude of thecarrier is distributed evenly in the whole carrier amplitudeThemodulation of PD strategy is comparing the Vref with tri1and tri4 to obtain the drive signal of Ha1 and with tri2 andtri3 to obtain the drive signal of Ha2 In Figure 2(b) the fullbridge output voltage of each inverter unit is shown as uHa1and uHa2 And the output voltage of phase A is shown as 119906

119860119873

It is obvious that the switching frequencies of the twounits are identical in the CPS strategy But with the PDstrategy the drive signal shows much difference betweenthe two units The switching frequencies of the two bridgesare different And at the same time when the PD strategyis employed the amplitudes of fundamental component inthe output voltage of the two units are different This meansthat the output power of the two units is not balanced Sowhen the PD strategy is directly implemented in the cascadedmultilevel inverter the output power and the switchingfrequency of the cascaded units are not the sameThis bringsa lot of practical design problems The rated power and thecooling design of the power units is hard to determine

22 VoltageHarmonics Comparison With the different trian-gular carrier waveform used in the twomodulation strategiesof PD and CPS the output voltage of phase A uAN producedis different The voltage harmonic is analyzed to reveal thedifference and the Flourier transformation is implementedto illustrate the spectral characteristics of the two voltagewaveforms

The modulation of the SPWM control is performed bycomparison between the reference signal and the triangularcarrier The Flourier expression of the phase voltage can bedivided into three parts The first part is the integer orderharmonics of the reference signal the second part is theinteger order harmonics of the triangular carrier signal thethird part is the sideband harmonics which are centralized tointeger order harmonics of the carrier signal

The extended Flourier expression of the voltagewaveformis shown in

119906

119860119873(119905) =

119860

00

2

+

infin

sum

119899=1

[119860

0119899cos (119899120596

119900119905) + 119861

0119899sin (119899120596

119900119905)]

+

infin

sum

119898=1

infin

sum

119899=minusinfin

[119860

119898119899cos (119899120596

119900119905 + 119898120596

119888119905)

+ 119861

119898119899sin (119899120596

119900119905 + 119898120596

119888119905)]

(1)

in which 120596

119900and 120596

119888are the frequency of the sinusoidal

reference signal and the triangular carrier m and n areinteger and119860

00is the dc component while119860

119898119899and 119861

119898119899are

the harmonic coefficient of the integer order harmonics andthe sideband harmonics

Then the output voltage of PD modulation is analyzedwith Double Fourier series expansion as follows

119906PD 119860119873 (119905) =119873119872119903119880in sin (120596119900119905)

+

infin

sum

119898=1

+infin

sum

119899=minusinfin

119861

1199011(119898 119899)

times sin [(2119898 minus 1) 120596

119888119905 + 2119899120596

119900119905]

+

infin

sum

119898=1

infin

sum

119899=minusinfin

119862

1199011(119898 119899)

times sin [2119898120596119888119905 + (2119899 minus 1) 120596

119900119905]

(2)

in which N is the cascaded unit number and 119880in is theinput voltage Assuming that 119880

119900119898is the amplitude of the

fundamental component in the output voltage of each unitthe voltage transfer ratio of each unit119872

119903can be calculated as

119872

119903=

119880

119900119898

119880in (3)

The Bp1(m n) and Cp1(m n) in (2) are the ampli-tude of the odd harmonics and the even harmonics

Mathematical Problems in Engineering 3

ref Tri1 Tri2 Tri3 Tri4

uAN

120596t

120596t

120596t

120596t

uH1198861

uH1198862

2Uc

Uc

minusUc

minus2Uc

minus2Uin

minusUin

minusUin

minusUin2Uin

Uin

Uin

Uin

0

0

0

0

(a) CPS strategy

2Uc

Uc

minusUc

minus2Uc

minus2Uin

minusUin

minusUin

minusUin2Uin

Uin

Uin

Uin

0

0

0

0

ref

Tri3 Tri4

Tri2Tri1

120596t

120596t

120596t

120596tuAN

uH1198861

uH1198862

(b) PD strategy

Figure 2 Modulation of CPS and PD control strategy

of the reference signal The value can be expressed asfollows

119861

1199011(119898 119899)

=

infin

sum

119896=0

4119880in(minus1)119896119869

(2119896+1)[(2119898 minus 1) 120587119873119872

119903]

120587

2(2119898 minus 1)

sdot [

1

2119899 minus 2119896 minus 1

times (minus sin [(2119899 minus 2119896 minus 1) 1205872

]

+ 2

119873minus1

sum

ℎ=0

cos (ℎ120587)

sdot sin [ (2119899 minus 2119896 minus1)

times arccos( ℎ

119873119872

119903

)])

+

1

2119899 + 2119896 + 1

times (minus sin [(2119899 + 2119896 + 1) 1205872

]

+ 2

119873minus1

sum

ℎ=0

cos (ℎ120587)

sdot sin [ (2119899 + 2119896 +1)

times arccos( ℎ

119873119872

119903

)])]

119862

1199011(119898 119899) =

119880in120587119898

119869

(2119899minus1)(2119898120587119873119872

119903)

(4)

In accordance with the phase voltage the line voltage119906PD 119860119861 can be expressed as follows

119906PD 119860119861 (119905) =radic3119873119872119903119880in sin(120596119900119905 +120587

6

)

+

infin

sum

119898=1

+infin

sum

119899=minusinfin

119861

1198971(119898 119899)

times sin [ (2119898 minus 1) 120596

119888119905

+ 2119899120596

119900119905 minus

2119899120587

3

]

+

infin

sum

119898=1

infin

sum

119899=minusinfin

119862

1198971(119898 119899)

times sin [2119898120596119888119905 + (2119899 minus 1) 120596

119900119905

minus

(2119899 minus 1) 120587

3

]

(5)

In which 1198611198971(119898 119899) and119862

1198971(119898 119899) are the amplitude of the

line voltage harmonics The value can be express in (6) and(7)

119861

1198971(119898 119899) = 2 sin(2119899120587

3

) sdot 119861

1199011(119898 119899) (6)

119862

1198971(119898 119899) = 2 sin [(2119899 minus 1) 120587

3

] sdot 119862

1199011(119898 119899) (7)


With the same principle the phase voltage of CPS controlstrategy 119906CPS 119860119873 can be expanded as follows

119906CPS 119860119873 (119905)

= 119873119872

119903119880in sin (120596119900119905)

+

infin

sum

119898=1

infin

sum

119899=minusinfin

119861

1199012(119898 119899) sin [21198981198731205961015840

119888119905 + (2119899 minus 1) 120596

119900119905]

(8)

in which 1198611199012(119898 119899) is the amplitude of the integer mul-

tiple harmonics with carrier frequency The value can beexpressed

119861

1199012(119898 119899) =

2119880in120587119898

119869

(2119899minus1)(119898120587119873119872

119903) (9)

And the output line voltage 119906CPS 119860119861 can be expressed asfollows

119906CPS 119860119861 (119905)

= 119873119872

119903119880in sin (120596119900119905)

+

infin

sum

119898=1

infin

sum

119899=minusinfin

119861

1198972(119898 119899)

times sin [21198981198731205961015840119888119905 + (2119899 minus 1) (120596

119900119905 minus

120587

3

)]

(10)

in which

119861

1198972(119898 119899) = 2 sin [(2119899 minus 1) 120587

3

] sdot 119861

1199012(119898 119899) (11)

It is shown through the comparison of (5) and (8)that when the following situation is satisfied the majorharmonics of the two control methods will appear with thesame frequency

120596

119888= 2119873120596

1015840

119888 (12)

When 119898 is 1 the harmonics of PD and CPS control withthe highest amplitude are around the carrier frequency 120596

119888

The high frequency filter design can be directly determinedby the amplitude of the harmonic at this frequency Sothe carrier frequency harmonics are compared And for

convenience the amplitudewith the carrier frequency of bothtwo control strategies is defined as follows

119867PD 119860119873 =radicsum

10

119899=1119861

2

1199011(1 119899)

119873119872

119903119880in

119867PD 119860119861 =radicsum

10

119899=1119861

2

1198971(1 119899)

radic3119873119872

119903119880in

119867CPS 119860119873 =radicsum

10

119899=1119861

2

1199012(1 119899)

119873119872

119903119880in

119867CPS 119860119861 =radicsum

10

119899=1119861

2

1198972(1 119899)

radic3119873119872

119903119880in

(13)

Taking two unit cascaded inverters as an example (119873 =

2) (13) is calculated to obtain the relationship between theamplitude of the carrier harmonic and the voltage transferratio119872

119903in the range of [0 1]The result is shown in Figure 3

Figure 3(a) shows that the phase voltage harmonicsproduced by the two control strategies are with the sameamplitude And in Figure 3(b) it is indicated that with boththe two control schemes the line voltage of is better thanthe phase voltage This can be verified by (6) and (11) Forthat when 119899 is an integral multiple of 3 in (6) the harmonicamplitude is 0 the harmonics with these frequencies arenaturally eliminated in the line voltage The same situationcan be observed in (11) assuming that (2119899 minus 1) is an integralmultiple of 3

And in Figure 3(b) the line voltage of the PD controlmethod contains fewer harmonic than that of the CPS con-trol For amultilevel cascaded inverter implementation in thethree-phase-three-wire system the quality of the line voltageis an important performance requirement And PD controlmethod shows obvious advantages with this consideration

3 Improved PD Control

As is analyzed above although the line voltage quality of PDcontrol is better than that of a CPS control the PD controlcannot be directly implemented Because the disadvantagesof the PD control method are unbalanced output power anddifferent switching frequencies of the cascaded unitsThe PDcontrol method has to be improved to be implemented in themulti-level cascaded inverter

The improvement of the PD control can be done bychanging the carrier signal waveform with observation of thedifference between the PD and CPS modulation The carriersignal of the CPS control in a carrier period is shown inFigure 4 for comparison

As is shown in Figures 4(a) and 4(b) the waveform ofthe carrier signals in both CPS and PD control is equallydivided into 8 parts which is shown in dash grid in Figures4(a) and 4(b)Through comparison of the carrier signal in thecorresponding parts of the period the following pattern canbe discovered


0 02 04 06 080

03

06

09

12

15

10

CPSPD

Mr

Phas

e vol

tage

har

mon

ic

(a) Amplitude of 120596119888 voltage harmonic in phase voltage

0 02 04 06 080

03

06

09

12

15

10

CPSPD

Mr

Line

vol

tage

har

mon

ic(b) Amplitude of 120596119888 voltage harmonic in line voltage

Figure 3 Amplitude of 120596119888voltage harmonic comparison of PD and CPS control

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(a) CPS carrier signal

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(b) PD carrier signal

Figure 4 Carrier signal comparison of CPS and PD control

(1) In the parts of120596119905 isin [0 1205874]cup[31205874 120587]cup[51205874 71205874]the slope of the CPS and PD carrier signal is the same

(2) In the part of120596119905 isin [1205874 31205874]cup[120587 51205874]cup[71205874 2120587]the slope of the CPS and PD carrier signal is theopposite

The shadow part of Figure 4(b) carrier waveform isseparately arranged as a set of carriers which is shown inFigure 5(a) Defined as Ca1 compared with tri1 in CPScontrol the carrier signal set is no longer a triangular carrierAnd the frequency of this carrier signal is the same as thatof the CPS triangular carrier The same separation can bedone in the other parts of the PD carrier signal with the same


2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(a) Ca1

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(b) Ca2

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(c) Ca3

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(d) Ca4

Figure 5 Carrier signal sets of the improved PD control method

principleThe results are shown in Figures 5(b) 5(c) and 5(d)which are defined as Ca2 Ca3 and Ca4 It is obvious thatthe phase shift of 90 degrees is the same pattern of that in theCPS control

Then the modulation of the improved PD control isdesigned to be similar to that of CPD control The referencesignal Vref is compared with Ca1 and Ca3 to generate thedrive signal for full bridge unitHa1The reference signal Vref iscompared with Ca2 and Ca4 to generate the drive signal forfull bridge unitHa2 The uHa1 and uHa2 are the output voltageof the full bridge unit and 119906

119860119873is the output phase voltage of

the cascaded inverter which has been shown in Figure 6It can be observed in Figure 6 that the switching frequen-

cies of the two bridge units are the same This is important

for heat dissipation design of the power unit At the sametime the fundamental component of the output voltage foreach unit can be measured to be the same The reasonis that the four sets of carrier signals are developed fromCPS control only with different sequence It means that theoutput power of each unit is balanced These two factorsare essential for practical implementation of this modulationmethod Because only with the balanced output power andequal switching frequency the practical consideration ofrated power and heat cooling design of the power unit canbe determined So this improved PD control method can beused in cascaded multilevel inverter In the following sectionthe experimental prototype and measured results will verifythe theory


2Uc

2Uc

Uc

Uc

minusUc

minus2Uc

minus2Uc

minus2Uin

minusUin

minusUin

minusUin2Uin

Uin

minusUc

Uin

Uin

0

0

0

0

0

Ca1

Ca2

Ca3

Ca4

ref

ref

uAN

120596t

120596t

120596t

120596t

120596t

uH1198861

uH1198862

Figure 6 Modulation of improved PD control

4 Experiment Results and Analysis

To verify the conclusion above a five-level three phasecascaded inverter is built with two cascaded units which isshown in Figure 7Themicrocontroller is the TMS320LF2812from Taxes Instruments The rated power is 5 kW and theswitching frequency is 5 kHz The input dc voltage for eachbridge unit is 180V The rated output phase voltage is 220Vwith the frequency of 50Hz The output filter inductor is400 120583H and the output filter capacitor is 15 120583F

The experiment result is shown is Figure 8 withmeasuredoutput voltage output current and the instantaneous powerwaveform In Figure 8(a) the 119906

119860119873represents the phase

voltage in the inverter side while the filtered output linevoltage is defined as 119906

119900 The load current 119894

119900is shown in

the brown color The inverter output line voltage 119906119860119861

is alsoshown in Figure 8(a) The spectrum analysis of the phasevoltage shows that the highest amplitude of phase voltageharmonic is at the switching frequency But in the line voltagethis harmonic is naturally eliminated It means that theimproved PD control method remains the advantage of PDcontrol The harmonic of output line voltage is optimized InFigure 8(b) the output instantaneous power waveform underrated power condition is measured and shown The drivingsignals of both units are shown in Figure 8(b) The outputinductor current is also shown in Figure 8(b) The averageoutput power of bridge units is measured using a poweranalyzer and shown inTable 1 It is clear that the improved PDcontrol method can balance the output power of each unit

Figure 9 shows the spectrum analysis results of the phasevoltage and line voltage with the improved PD method InFigure 9(a) major harmonics below 50 kHz are shown forthe phase voltage and line voltage separately The highestamplitude harmonics around the 4 times of the switchingfrequency (20 kHz) are magnified in Figure 9(b) for a clearerview The analysis result for phase voltage is shown in the

Figure 7 Three phase cascaded multilevel inverter experimentalprototype

Table 1 Output power measure in different situations

Output power Output power of unit 1 Output power of unit 215 kW 750w 751 w2 kW 999w 1001 w

up side and line voltage in the low side It can be observedthat in the situation of phase voltage the highest amplitudevoltage harmonic appearswith the frequency of 20 kHzwhilein the situation of line voltage the voltage harmonic at 20 kHzis naturally eliminated with inconsiderable amplitude Theanalysis result shows clearly that the improved PD controlmethod retains the PD controlrsquos advantage of optimized linevoltage harmonic

5 Conclusions

In this paper a Fourier series expansion method is usedto analyze the output phase voltage and line voltage of thePD and CPS control The spectrum character of the twocontrol methods is analyzed to come to the conclusion thatin the range of the whole modulation ratio the line voltagequality of PD control is better than that of CPS control Thecarrier signals of CPS and PD control are compared and animproved PD control method is discussed in this paper Thenew method can be practically implemented in the cascadedmultilevel invert With the improved PD control method theoutput power of each full bridge unit is balanced and theswitching frequencies of the units are kept the same At thesame time the improved PD method retains the advantageof PD control the optimized line voltage harmonic To verifythe theory a 5 kW three phase cascaded multilevel inverterprototype is built in the laboratory The experiment resultsshow the feasibility of the improved PD control method andthe advantages of this novel control method


Time (5msdiv)

uAB (500Vdiv)

uAN (500Vdiv)

uo (500Vdiv)

C2

C4

C3

io (10Adiv)

(a) Output voltage and current waveform

uH1198861 (250Vdiv)

uH1198862 (250Vdiv)

Time (5msdiv)

pH1198862 (1250Wdiv)

pH1198861 (1250Wdiv)

C1

F1

F2

C3

C2

iL119886 (250Vdiv)

(b) PhaseA output voltage and instantaneous output power of eachunit

Figure 8 Experiment waveform for improved PD control method

F3

F4

uAN (100Vdiv)

uAB (100Vdiv)

Frq (5kHzdiv)

(a) Major harmonic spectrum

F3

F4

uAB (20Vdiv)

Frq (200 kHzdiv)

uAN (20Vdiv)

(b) Harmonic spectrum around 4 times of the switching frequency

Figure 9 Spectrum analysis result of output phase voltage and line voltage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M Malinowski K Gopakumar J Rodriguez and M A PerezldquoA survey on cascaded multilevel invertersrdquo IEEE Transactionson Industrial Electronics vol 57 no 7 pp 2197ndash2206 2010

[2] J Ebrahimi E Babaei and G B Gharehpetian ldquoA new mul-tilevel converter topology with reduced number of power elec-tronic componentsrdquo IEEETransactions on Industrial Electronicsvol 59 no 2 pp 655ndash667 2012

[3] Z Du B Ozpineci and L M Tolbert ldquoModulation extensioncontrol of hybrid cascaded H-bridge multilevel converterswith 7-level fundamental frequency switching schemerdquo inProceedings of the 38th IEEEAnnual Power Electronics SpecialistsConference (PESC rsquo07) pp 2361ndash2366 June 2007

[4] C Buccella C Cecati and H Latafat ldquoDigital control ofpower convertersmdasha surveyrdquo IEEE Transactions on IndustrialInformatics vol 3 no 8 pp 437ndash4447 2012

[5] R Rathore H Holtz and T Boller ldquoGeneralized optimalpulsewidthmodulation ofmultilevel inverters for low switchingfrequency control of medium voltage high power industrial ac

drivesrdquo IEEE Transactions on Industrial Electronics vol 10 no60 pp 4215ndash4224 2013

[6] L M Tolbert F Z Peng and T G Habetier ldquoA multilevelconverter-based universal power conditionerrdquo IEEE Transac-tions on Industry Applications vol 36 no 2 pp 596ndash603 2000

[7] Z Wu Y Zou and K Ding ldquoAnalysis of output voltagespectra in a hybrid diode-clamp cascade 13-level inverterrdquoin Proceedings of the 36th IEEE Power Electronics SpecialistsConference (PESC rsquo05) pp 873ndash8879 2005

[8] J Wang and F Z Peng ldquoUnified power flow controller usingthe cascade multilevel inverterrdquo IEEE Transactions on PowerElectronics vol 19 no 4 pp 1077ndash1084 2004

[9] H Akagi S Inoue and T Yoshii ldquoControl and performance ofa transformerless cascade PWM STATCOM with star configu-rationrdquo IEEE Transactions on Industry Applications vol 43 no4 pp 1041ndash1049 2007

[10] S Yin H Luo and S X Ding ldquoReal-time implementation offault-tolerant control systems with performance optimizationrdquoIEEE Transactions on Industrial Electronics vol 61 no 5 pp2402ndash2411 2014

[11] X Wang Y Zang B Xu and J-Q Lin ldquoResearch on switch-based control method for cascaded H-bridge inverter failuresrdquoProceeding of the CSEE vol 27 no 7 pp 76ndash81 2007


[12] P W Hammond ldquoEnhancing the reliability of modularmedium-voltage drivesrdquo IEEE Transactions on Industrial Elec-tronics vol 49 no 5 pp 948ndash954 2002

[13] Y-F Sun and X-B Ruan ldquoPower balance control schemes forcascaded multilevel invertersrdquo Proceeding of the CSEE vol 26no 4 pp 126ndash133 2006


[15] Z-Y Yang J-F Zhao X-J Ni J-M Lu and B Chen ldquoElim-inating common-mode voltage of cascaded medium-voltagevariable frequency driver with phase-difference STS-SVMrdquoProceeding of the CSEE vol 28 no 15 pp 32ndash38 2008

[16] P Lezana and G Ortiz ldquoExtended operation of cascade mul-ticell converters under fault conditionrdquo IEEE Transactions onIndustrial Electronics vol 56 no 7 pp 2697ndash2703 2009

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Uin

Uin

Sa1

Sa2

Sa3

Sa4

Sa5

Sa6

Sa7

Sa8

CHBb2

CHBb1 CHBc1

CHBc2

A B C

N

Figure 1 Cascaded five-level inverter

signal for the bridgesThe triangular carrier signal is arrangedwith 90 degree phase shift between the adjacent ones Thecontrol signal modulation is performed with the followingprinciple

The drive signal for Ha1 is generated by comparing Vrefwith tri1 and tri3 And the drive signal forHa2 is generated bycomparing Vref with tri2 and tri4 As is shown in Figure 2(a)the uHa1 and uHa2 are the output voltage of the full bridgesrespectively And the 119906

119860119873is the output voltage of phase A

In Figure 2(b) themodulation principle of the PD controlstrategy is illustrated Like that of CPS a combination offour triangular carrier signals is implemented with the samephase but different amplitude range The amplitude of thecarrier is distributed evenly in the whole carrier amplitudeThemodulation of PD strategy is comparing the Vref with tri1and tri4 to obtain the drive signal of Ha1 and with tri2 andtri3 to obtain the drive signal of Ha2 In Figure 2(b) the fullbridge output voltage of each inverter unit is shown as uHa1and uHa2 And the output voltage of phase A is shown as 119906

119860119873

It is obvious that the switching frequencies of the twounits are identical in the CPS strategy But with the PDstrategy the drive signal shows much difference betweenthe two units The switching frequencies of the two bridgesare different And at the same time when the PD strategyis employed the amplitudes of fundamental component inthe output voltage of the two units are different This meansthat the output power of the two units is not balanced Sowhen the PD strategy is directly implemented in the cascadedmultilevel inverter the output power and the switchingfrequency of the cascaded units are not the sameThis bringsa lot of practical design problems The rated power and thecooling design of the power units is hard to determine

22 VoltageHarmonics Comparison With the different trian-gular carrier waveform used in the twomodulation strategiesof PD and CPS the output voltage of phase A uAN producedis different The voltage harmonic is analyzed to reveal thedifference and the Flourier transformation is implementedto illustrate the spectral characteristics of the two voltagewaveforms

The modulation of the SPWM control is performed bycomparison between the reference signal and the triangularcarrier The Flourier expression of the phase voltage can bedivided into three parts The first part is the integer orderharmonics of the reference signal the second part is theinteger order harmonics of the triangular carrier signal thethird part is the sideband harmonics which are centralized tointeger order harmonics of the carrier signal

The extended Flourier expression of the voltagewaveformis shown in

119906

119860119873(119905) =

119860

00

2

+

infin

sum

119899=1

[119860

0119899cos (119899120596

119900119905) + 119861

0119899sin (119899120596

119900119905)]

+

infin

sum

119898=1

infin

sum

119899=minusinfin

[119860

119898119899cos (119899120596

119900119905 + 119898120596

119888119905)

+ 119861

119898119899sin (119899120596

119900119905 + 119898120596

119888119905)]

(1)

in which 120596

119900and 120596

119888are the frequency of the sinusoidal

reference signal and the triangular carrier m and n areinteger and119860

00is the dc component while119860

119898119899and 119861

119898119899are

the harmonic coefficient of the integer order harmonics andthe sideband harmonics

Then the output voltage of PD modulation is analyzedwith Double Fourier series expansion as follows

119906PD 119860119873 (119905) =119873119872119903119880in sin (120596119900119905)

+

infin

sum

119898=1

+infin

sum

119899=minusinfin

119861

1199011(119898 119899)

times sin [(2119898 minus 1) 120596

119888119905 + 2119899120596

119900119905]

+

infin

sum

119898=1

infin

sum

119899=minusinfin

119862

1199011(119898 119899)

times sin [2119898120596119888119905 + (2119899 minus 1) 120596

119900119905]

(2)

in which N is the cascaded unit number and 119880in is theinput voltage Assuming that 119880

119900119898is the amplitude of the

fundamental component in the output voltage of each unitthe voltage transfer ratio of each unit119872

119903can be calculated as

119872

119903=

119880

119900119898

119880in (3)

The Bp1(m n) and Cp1(m n) in (2) are the ampli-tude of the odd harmonics and the even harmonics



uAN

120596t

120596t

120596t

120596t

uH1198861

uH1198862

2Uc

Uc

minusUc

minus2Uc

minus2Uin

minusUin

minusUin

minusUin2Uin

Uin

Uin

Uin

0

0

0

0

(a) CPS strategy

2Uc

Uc

minusUc

minus2Uc

minus2Uin

minusUin

minusUin

minusUin2Uin

Uin

Uin

Uin

0

0

0

0

ref

Tri3 Tri4

Tri2Tri1

120596t

120596t

120596t

120596tuAN

uH1198861

uH1198862

(b) PD strategy



119861

1199011(119898 119899)

=

infin

sum

119896=0

4119880in(minus1)119896119869

(2119896+1)[(2119898 minus 1) 120587119873119872

119903]

120587

2(2119898 minus 1)

sdot [

1

2119899 minus 2119896 minus 1


]

+ 2

119873minus1

sum

ℎ=0

cos (ℎ120587)


times arccos( ℎ

119873119872

119903

)])

+

1

2119899 + 2119896 + 1

times (minus sin [(2119899 + 2119896 + 1) 1205872

]

+ 2

119873minus1

sum

ℎ=0

cos (ℎ120587)

sdot sin [ (2119899 + 2119896 +1)

times arccos( ℎ

119873119872

119903

)])]

119862

1199011(119898 119899) =

119880in120587119898

119869

(2119899minus1)(2119898120587119873119872

119903)

(4)


119906PD 119860119861 (119905) =radic3119873119872119903119880in sin(120596119900119905 +120587

6

)

+

infin

sum

119898=1

+infin

sum

119899=minusinfin

119861

1198971(119898 119899)

times sin [ (2119898 minus 1) 120596

119888119905

+ 2119899120596

119900119905 minus

2119899120587

3

]

+

infin

sum

119898=1

infin

sum

119899=minusinfin

119862

1198971(119898 119899)

times sin [2119898120596119888119905 + (2119899 minus 1) 120596

119900119905

minus

(2119899 minus 1) 120587

3

]

(5)

In which 1198611198971(119898 119899) and119862



119861

1198971(119898 119899) = 2 sin(2119899120587

3

) sdot 119861

1199011(119898 119899) (6)

119862

1198971(119898 119899) = 2 sin [(2119899 minus 1) 120587

3

] sdot 119862

1199011(119898 119899) (7)



119906CPS 119860119873 (119905)

= 119873119872

119903119880in sin (120596119900119905)

+

infin

sum

119898=1

infin

sum

119899=minusinfin

119861

1199012(119898 119899) sin [21198981198731205961015840

119888119905 + (2119899 minus 1) 120596

119900119905]

(8)



119861

1199012(119898 119899) =

2119880in120587119898

119869

(2119899minus1)(119898120587119873119872

119903) (9)


119906CPS 119860119861 (119905)

= 119873119872

119903119880in sin (120596119900119905)

+

infin

sum

119898=1

infin

sum

119899=minusinfin

119861

1198972(119898 119899)

times sin [21198981198731205961015840119888119905 + (2119899 minus 1) (120596

119900119905 minus

120587

3

)]

(10)

in which

119861

1198972(119898 119899) = 2 sin [(2119899 minus 1) 120587

3

] sdot 119861

1199012(119898 119899) (11)


120596

119888= 2119873120596

1015840

119888 (12)


119888



119867PD 119860119873 =radicsum

10

119899=1119861

2

1199011(1 119899)

119873119872

119903119880in

119867PD 119860119861 =radicsum

10

119899=1119861

2

1198971(1 119899)

radic3119873119872

119903119880in

119867CPS 119860119873 =radicsum

10

119899=1119861

2

1199012(1 119899)

119873119872

119903119880in

119867CPS 119860119861 =radicsum

10

119899=1119861

2

1198972(1 119899)

radic3119873119872

119903119880in

(13)











0 02 04 06 080

03

06

09

12

15

10

CPSPD

Mr

Phas

e vol

tage

har

mon

ic


0 02 04 06 080

03

06

09

12

15

10

CPSPD

Mr

Line

vol

tage

har

mon



2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872


2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872







2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(a) Ca1

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(b) Ca2

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(c) Ca3

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(d) Ca4









2Uc

2Uc

Uc

Uc

minusUc

minus2Uc

minus2Uc

minus2Uin

minusUin

minusUin

minusUin2Uin

Uin

minusUc

Uin

Uin

0

0

0

0

0

Ca1

Ca2

Ca3

Ca4

ref

ref

uAN

120596t

120596t

120596t

120596t

120596t

uH1198861

uH1198862








119900is shown in








5 Conclusions



Time (5msdiv)

uAB (500Vdiv)

uAN (500Vdiv)

uo (500Vdiv)

C2

C4

C3

io (10Adiv)


uH1198861 (250Vdiv)

uH1198862 (250Vdiv)

Time (5msdiv)

pH1198862 (1250Wdiv)

pH1198861 (1250Wdiv)

C1

F1

F2

C3

C2

iL119886 (250Vdiv)



F3

F4

uAN (100Vdiv)

uAB (100Vdiv)

Frq (5kHzdiv)


F3

F4

uAB (20Vdiv)

Frq (200 kHzdiv)

uAN (20Vdiv)





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











uAN

120596t

120596t

120596t

120596t

uH1198861

uH1198862

2Uc

Uc

minusUc

minus2Uc

minus2Uin

minusUin

minusUin

minusUin2Uin

Uin

Uin

Uin

0

0

0

0

(a) CPS strategy

2Uc

Uc

minusUc

minus2Uc

minus2Uin

minusUin

minusUin

minusUin2Uin

Uin

Uin

Uin

0

0

0

0

ref

Tri3 Tri4

Tri2Tri1

120596t

120596t

120596t

120596tuAN

uH1198861

uH1198862

(b) PD strategy



119861

1199011(119898 119899)

=

infin

sum

119896=0

4119880in(minus1)119896119869

(2119896+1)[(2119898 minus 1) 120587119873119872

119903]

120587

2(2119898 minus 1)

sdot [

1

2119899 minus 2119896 minus 1


]

+ 2

119873minus1

sum

ℎ=0

cos (ℎ120587)


times arccos( ℎ

119873119872

119903

)])

+

1

2119899 + 2119896 + 1

times (minus sin [(2119899 + 2119896 + 1) 1205872

]

+ 2

119873minus1

sum

ℎ=0

cos (ℎ120587)

sdot sin [ (2119899 + 2119896 +1)

times arccos( ℎ

119873119872

119903

)])]

119862

1199011(119898 119899) =

119880in120587119898

119869

(2119899minus1)(2119898120587119873119872

119903)

(4)


119906PD 119860119861 (119905) =radic3119873119872119903119880in sin(120596119900119905 +120587

6

)

+

infin

sum

119898=1

+infin

sum

119899=minusinfin

119861

1198971(119898 119899)

times sin [ (2119898 minus 1) 120596

119888119905

+ 2119899120596

119900119905 minus

2119899120587

3

]

+

infin

sum

119898=1

infin

sum

119899=minusinfin

119862

1198971(119898 119899)

times sin [2119898120596119888119905 + (2119899 minus 1) 120596

119900119905

minus

(2119899 minus 1) 120587

3

]

(5)

In which 1198611198971(119898 119899) and119862



119861

1198971(119898 119899) = 2 sin(2119899120587

3

) sdot 119861

1199011(119898 119899) (6)

119862

1198971(119898 119899) = 2 sin [(2119899 minus 1) 120587

3

] sdot 119862

1199011(119898 119899) (7)



119906CPS 119860119873 (119905)

= 119873119872

119903119880in sin (120596119900119905)

+

infin

sum

119898=1

infin

sum

119899=minusinfin

119861

1199012(119898 119899) sin [21198981198731205961015840

119888119905 + (2119899 minus 1) 120596

119900119905]

(8)



119861

1199012(119898 119899) =

2119880in120587119898

119869

(2119899minus1)(119898120587119873119872

119903) (9)


119906CPS 119860119861 (119905)

= 119873119872

119903119880in sin (120596119900119905)

+

infin

sum

119898=1

infin

sum

119899=minusinfin

119861

1198972(119898 119899)

times sin [21198981198731205961015840119888119905 + (2119899 minus 1) (120596

119900119905 minus

120587

3

)]

(10)

in which

119861

1198972(119898 119899) = 2 sin [(2119899 minus 1) 120587

3

] sdot 119861

1199012(119898 119899) (11)


120596

119888= 2119873120596

1015840

119888 (12)


119888



119867PD 119860119873 =radicsum

10

119899=1119861

2

1199011(1 119899)

119873119872

119903119880in

119867PD 119860119861 =radicsum

10

119899=1119861

2

1198971(1 119899)

radic3119873119872

119903119880in

119867CPS 119860119873 =radicsum

10

119899=1119861

2

1199012(1 119899)

119873119872

119903119880in

119867CPS 119860119861 =radicsum

10

119899=1119861

2

1198972(1 119899)

radic3119873119872

119903119880in

(13)











0 02 04 06 080

03

06

09

12

15

10

CPSPD

Mr

Phas

e vol

tage

har

mon

ic


0 02 04 06 080

03

06

09

12

15

10

CPSPD

Mr

Line

vol

tage

har

mon



2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872


2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872







2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(a) Ca1

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(b) Ca2

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(c) Ca3

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(d) Ca4









2Uc

2Uc

Uc

Uc

minusUc

minus2Uc

minus2Uc

minus2Uin

minusUin

minusUin

minusUin2Uin

Uin

minusUc

Uin

Uin

0

0

0

0

0

Ca1

Ca2

Ca3

Ca4

ref

ref

uAN

120596t

120596t

120596t

120596t

120596t

uH1198861

uH1198862








119900is shown in








5 Conclusions



Time (5msdiv)

uAB (500Vdiv)

uAN (500Vdiv)

uo (500Vdiv)

C2

C4

C3

io (10Adiv)


uH1198861 (250Vdiv)

uH1198862 (250Vdiv)

Time (5msdiv)

pH1198862 (1250Wdiv)

pH1198861 (1250Wdiv)

C1

F1

F2

C3

C2

iL119886 (250Vdiv)



F3

F4

uAN (100Vdiv)

uAB (100Vdiv)

Frq (5kHzdiv)


F3

F4

uAB (20Vdiv)

Frq (200 kHzdiv)

uAN (20Vdiv)





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906CPS 119860119873 (119905)

= 119873119872

119903119880in sin (120596119900119905)

+

infin

sum

119898=1

infin

sum

119899=minusinfin

119861

1199012(119898 119899) sin [21198981198731205961015840

119888119905 + (2119899 minus 1) 120596

119900119905]

(8)



119861

1199012(119898 119899) =

2119880in120587119898

119869

(2119899minus1)(119898120587119873119872

119903) (9)


119906CPS 119860119861 (119905)

= 119873119872

119903119880in sin (120596119900119905)

+

infin

sum

119898=1

infin

sum

119899=minusinfin

119861

1198972(119898 119899)

times sin [21198981198731205961015840119888119905 + (2119899 minus 1) (120596

119900119905 minus

120587

3

)]

(10)

in which

119861

1198972(119898 119899) = 2 sin [(2119899 minus 1) 120587

3

] sdot 119861

1199012(119898 119899) (11)


120596

119888= 2119873120596

1015840

119888 (12)


119888



119867PD 119860119873 =radicsum

10

119899=1119861

2

1199011(1 119899)

119873119872

119903119880in

119867PD 119860119861 =radicsum

10

119899=1119861

2

1198971(1 119899)

radic3119873119872

119903119880in

119867CPS 119860119873 =radicsum

10

119899=1119861

2

1199012(1 119899)

119873119872

119903119880in

119867CPS 119860119861 =radicsum

10

119899=1119861

2

1198972(1 119899)

radic3119873119872

119903119880in

(13)











0 02 04 06 080

03

06

09

12

15

10

CPSPD

Mr

Phas

e vol

tage

har

mon

ic


0 02 04 06 080

03

06

09

12

15

10

CPSPD

Mr

Line

vol

tage

har

mon



2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872


2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872







2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(a) Ca1

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(b) Ca2

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(c) Ca3

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(d) Ca4









2Uc

2Uc

Uc

Uc

minusUc

minus2Uc

minus2Uc

minus2Uin

minusUin

minusUin

minusUin2Uin

Uin

minusUc

Uin

Uin

0

0

0

0

0

Ca1

Ca2

Ca3

Ca4

ref

ref

uAN

120596t

120596t

120596t

120596t

120596t

uH1198861

uH1198862








119900is shown in








5 Conclusions



Time (5msdiv)

uAB (500Vdiv)

uAN (500Vdiv)

uo (500Vdiv)

C2

C4

C3

io (10Adiv)


uH1198861 (250Vdiv)

uH1198862 (250Vdiv)

Time (5msdiv)

pH1198862 (1250Wdiv)

pH1198861 (1250Wdiv)

C1

F1

F2

C3

C2

iL119886 (250Vdiv)



F3

F4

uAN (100Vdiv)

uAB (100Vdiv)

Frq (5kHzdiv)


F3

F4

uAB (20Vdiv)

Frq (200 kHzdiv)

uAN (20Vdiv)





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 080

03

06

09

12

15

10

CPSPD

Mr

Phas

e vol

tage

har

mon

ic


0 02 04 06 080

03

06

09

12

15

10

CPSPD

Mr

Line

vol

tage

har

mon



2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872


2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872







2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(a) Ca1

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(b) Ca2

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(c) Ca3

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(d) Ca4









2Uc

2Uc

Uc

Uc

minusUc

minus2Uc

minus2Uc

minus2Uin

minusUin

minusUin

minusUin2Uin

Uin

minusUc

Uin

Uin

0

0

0

0

0

Ca1

Ca2

Ca3

Ca4

ref

ref

uAN

120596t

120596t

120596t

120596t

120596t

uH1198861

uH1198862








119900is shown in








5 Conclusions



Time (5msdiv)

uAB (500Vdiv)

uAN (500Vdiv)

uo (500Vdiv)

C2

C4

C3

io (10Adiv)


uH1198861 (250Vdiv)

uH1198862 (250Vdiv)

Time (5msdiv)

pH1198862 (1250Wdiv)

pH1198861 (1250Wdiv)

C1

F1

F2

C3

C2

iL119886 (250Vdiv)



F3

F4

uAN (100Vdiv)

uAB (100Vdiv)

Frq (5kHzdiv)


F3

F4

uAB (20Vdiv)

Frq (200 kHzdiv)

uAN (20Vdiv)





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(a) Ca1

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(b) Ca2

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(c) Ca3

2Uc

Uc

minusUc

minus2Uc

1205872 2120587120587

120596t0

31205872

(d) Ca4









2Uc

2Uc

Uc

Uc

minusUc

minus2Uc

minus2Uc

minus2Uin

minusUin

minusUin

minusUin2Uin

Uin

minusUc

Uin

Uin

0

0

0

0

0

Ca1

Ca2

Ca3

Ca4

ref

ref

uAN

120596t

120596t

120596t

120596t

120596t

uH1198861

uH1198862








119900is shown in








5 Conclusions



Time (5msdiv)

uAB (500Vdiv)

uAN (500Vdiv)

uo (500Vdiv)

C2

C4

C3

io (10Adiv)


uH1198861 (250Vdiv)

uH1198862 (250Vdiv)

Time (5msdiv)

pH1198862 (1250Wdiv)

pH1198861 (1250Wdiv)

C1

F1

F2

C3

C2

iL119886 (250Vdiv)



F3

F4

uAN (100Vdiv)

uAB (100Vdiv)

Frq (5kHzdiv)


F3

F4

uAB (20Vdiv)

Frq (200 kHzdiv)

uAN (20Vdiv)





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2Uc

2Uc

Uc

Uc

minusUc

minus2Uc

minus2Uc

minus2Uin

minusUin

minusUin

minusUin2Uin

Uin

minusUc

Uin

Uin

0

0

0

0

0

Ca1

Ca2

Ca3

Ca4

ref

ref

uAN

120596t

120596t

120596t

120596t

120596t

uH1198861

uH1198862








119900is shown in








5 Conclusions



Time (5msdiv)

uAB (500Vdiv)

uAN (500Vdiv)

uo (500Vdiv)

C2

C4

C3

io (10Adiv)


uH1198861 (250Vdiv)

uH1198862 (250Vdiv)

Time (5msdiv)

pH1198862 (1250Wdiv)

pH1198861 (1250Wdiv)

C1

F1

F2

C3

C2

iL119886 (250Vdiv)



F3

F4

uAN (100Vdiv)

uAB (100Vdiv)

Frq (5kHzdiv)


F3

F4

uAB (20Vdiv)

Frq (200 kHzdiv)

uAN (20Vdiv)





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Time (5msdiv)

uAB (500Vdiv)

uAN (500Vdiv)

uo (500Vdiv)

C2

C4

C3

io (10Adiv)


uH1198861 (250Vdiv)

uH1198862 (250Vdiv)

Time (5msdiv)

pH1198862 (1250Wdiv)

pH1198861 (1250Wdiv)

C1

F1

F2

C3

C2

iL119886 (250Vdiv)



F3

F4

uAN (100Vdiv)

uAB (100Vdiv)

Frq (5kHzdiv)


F3

F4

uAB (20Vdiv)

Frq (200 kHzdiv)

uAN (20Vdiv)





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article an improved phase disposition spwm strategy...

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