iecon05 paper

8/8/2019 Iecon05 Paper

1/6

VL

VC

id,conv id,load

iCiL1 iV1 iV2 iV3

V1 V2 V3

V4 V5 V6

iL2iL3

Figure 1. Three phase voltage source converter topology

DC Link Harmonics of Three Phase Voltage Source Converters Influenced by

the Pulse Width Modulation Strategy an Analysis

Michael Bierhoff Friedrich W. FuchsChair for Power Electronics and Electrical Drives Chair for Power Electronics and Electrical Drives

Christian-Albrechts-University of Kiel Christian-Albrechts-University of Kiel

Kaiserstr. 2, 24143 Kiel Kaiserstr. 2, 24143 KielGermany Germany

[email protected] [email protected]

Abstract Dc link current harmonics are the predominant

factor to be considered for dimensioning dc capacitors in

three phase PWM voltage source converters. In this article

an analysis of the dc link current harmonics applying

double Fourier series is derived. The analytical results for

the dc link current spectra of continuous and discontinuous

PWM are presented and compared with measurement

results taken from a converter test setup. A good match

between theoretically expected and actually obtained

experimental results can be stated. Moreover significant

differences between the investigated modulation strategies

regarding their dc link current spectra can be found.

I. INTRODUCTION Nowadays PWM voltage source converters constitute

the most important means to control adjustable speeddrives or generally to serve as an active rectifier with loweffects to the mains. An appropriate design of their powersection has to be done including the dc link capacitor. Thecapacitor sizing basically aims two issues, dc ripplevoltage and more important capacitor lifetime which candirectly be linked to its power losses. These subjects areaffected in particular by the capacitor currents whichcoincide with the dc current harmonics in steady stateoperation under certain presumptions.

There are already several articles on the calculation ofthe RMS value of the dc link ripple current [1][3] whicheventually all lead to the same results regardless of themodulation strategy. This value is only affected by theduty cycles of the active space vectors and thus it will beof the same quantity for all modulation waveforms as longas only one converter is considered to be the source ofharmonics on the dc bus. Another numerical approach is presented in [4] where the interference of two differentPWM converters connected to the same dc bus isexamined. There it turns out that the phase shift of the

respective switching periods would have an effect on thedc bus ripple current. However using differentmodulation strategies or switching frequencies as appliedonly to a single converter does not influence the dc linkripple current but the dispersion of the dc link currentspectra. Hence the dc link current spectrum has to beanalyzed for any applied modulation method to predictresulting capacitor ripple voltage for instance.

As the dc link capacitor constitutes an integrating filteron the dc link current a change of the modulation strategyand thus of the current spectra leads to a change of theripple voltage. This inherently can be seen in [2] wheredifferent modulation methods are compared analyticallybut only in time domain.

Regarding the aging of the capacitor, its heating andthus power losses have to be taken into account. Thesepower losses are a function of the capacitor current and its

series resistance. Usually the RMS value of the dc linkcurrent is used for the design but since the seriesresistance of a capacitor ESR() shows some dependencyon the frequency [5], [6] for detailed design it might benecessary to determine the capacitor current spectraexplicitly. The dc current spectrum already has beencalculated and measured for sine triangular modulation in[3] and determined by experiment for space vectormodulation as well [5].

This work contributes a comprehensive analyticalapproach to determine the dc bus current spectra explicitlyfor any kind of modulation strategy, thereby utilizing thedouble Fourier series introduced in [7]. This approach has been already applied successfully for the harmonicscalculation of the three phase currents of PWM voltagesource converters as reported by various sources [8], [9].

In section II of this article the principles of pulse widthmodulation are pointed out which will be important for theanalysis. Section III covers the derivation of the doubleFourier series according to Black and determination of thedc bus harmonics. Computed results compared withexperimental ones are presented in section IV that isfollowed by a conclusion.

II. PRINCIPLES OF PULSE WIDTH MODULATIONBasis for the analysis is a converter topology as shown

by fig. 1 operated with pulse width modulation. Fig. 2depicts the corresponding pulse scheme for switches V1 -V3 during an exemplary switching period. Involving theseswitches only into consideration is sufficient since the dclink current id,conv can be composed of the switchedcurrents iV1 - iV3. For this example the instantaneousvoltage command space vector is located in the first sectorof the vector space that is spanned by the six possibleswitching states of the basic space vectors. It is wellknown that the command space vector resembles the

sliding mean value of switched basic space vectors V1, V2and V0, V7 being turned on for the time period t1, t2, t0 andt7 respectively during the switching period Ts [9]. Thesespace vectors in turn correspond to specific switchingcombinations of the six semiconductor switches, see fig.2. During the time period t0 all upper switches (fig. 1) are


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t7/2

t1/2

t2/2

t0

TS

V1

V2

V3

V*

V2(V1,V2,V6)

V1

(V1,V4,V6)

t7/2

t1/2

t2/2

V0(V1,V2,V3)

V7(V4,V5,V6)

Figure 2. Switching scheme for upper bridge valves in the first sectorof the complex plane

2

3

0y = t

x = ct

y = x/p

iV(t)

2 30

0

Figure 3. The geometric wall model as applied on a cosine

modulation function in combination with a triangular carrier

turned on whereas during t7only the lower ones are turnedon. However during both time periods t0 and t7the ac loadis short circuited and the dc bus current flow isinterrupted. While the switches are turned on the switch

currents match the corresponding line currents. Henceduring time period t1 the instantaneous value of linecurrent iL1 determines the dc bus current and during timespan t2 the dc bus current equals the sum of currentsiL1 + iL2. For the presented analysis the line currents areassumed to be purely sinusoidal.

The presented calculus is derived by consideringcarrier based PWM. In [9] - [11] the direct relationbetween space vector and naturally sampled carrier basedPWM is explained. These relations yield the conclusionthat the carrier function that would be applied according tofig. 2 is a triangular function which can be deduced fromthe symmetrical arrangement of switching duty cycles.Moreover with this analogy the switching period Tswould be equal to the carrier function period of a comparablecarrier based PWM. The corresponding modulationwaveform is determined by the duty cycles of the twodifferent zero space vectors.

III. DC LINKCURRENT SPECTRA DESCRIBED BYDOUBLE FOURIERSERIES

A. The Geometric Wall ModelIn general the determination of the analytical spectra of

pulse width modulated values is a tedious issue. Even ifsolved, the appropriate equations yield infinite seriescomprising complex terms. One approach to determine the

dc link current spectra analytically is given by [3]. Thestandard approach utilizing the geometric wall model [7]resulting in double Fourier series is presented here. Thismethod provides a fairly simple way to obtain expressionsfor the Fourier coefficients of line side or dc side currentharmonics generated by any modulation strategy.

Initially the geometric wall model was applied for theanalysis of asynchronous sinusoidal modulation schemesto overcome the problems caused by non periodicbehaviour of the pulse waveform [8]. A plain view for theactually three dimensional considered model is given byfig. 3. The walls are represented by the boundaries of theshaded areas with an original height of the dc link voltageVdc [8], [9]. The important modification to apply this

method on the dc current is to adjust the height of thewalls to the instantaneous value of the line current iL(t).The walls are oriented perpendicular to the x-y plane. The

assumption of unipolar switching introduces an erroneousdc component when the line voltage harmonics areregarded [8]. But since the resulting pulse widths directlyagree with the switch duty cycles tn/TS, this model is a

perfect match of the pulse pattern as given by fig. 2.Moreover in this case it is valid for the entire range of onefundamental period. The intersections of the line y = x/pwith the modulation waveformsx = /2 ((2n+1) cos(t))indicate a pulse transition of iL(t). Here p is the carrierratio (p = c/), c is the carrier angular frequency and is the fundamental angular frequency. For illustration purposes the most simple case was chosen: a sinusoidalmodulation waveform along with natural sampling. Themodulation waveforms are symmetrically arranged aroundlines of multiples of to realise a model that supports atriangular carrier wave shape. As the widths of theresulting pulses are equal to the switching duty cycles ofone semiconductor switch and the instantaneous value ofthe corresponding phase current is assigned to their height,these pulses resemble the outline of one of the switchcurrents iV1, iV2, iV3 consequently. These currents solelydiffer from each other by the phase shift of the respectivemodulation function and phase current. Evidently the sumiV1+iV1+iV3 of all instantaneous valve currents finallydelivers the instantaneous dc bus current value id,conv.

It should be remarked that by applying this methodeven the influence of any other PWM inverter that isconnected to the dc bus can be taken into account bysimply adding its current portions to the foregoing sum.

B. Evolution of the Double Fourier SeriesAs the wall pattern is periodic in 2 for both axes, the pulse functionF(x,y) may be expressed as a double

Fourier series (1).

[ ]

=

++=1

00

00)sin()cos(

2),(

nnn

nyBnyAA

yxF

[ ]

=

++1

00)sin()cos(

mmm

mxBmxA (1)

[ ]

=

=

++++1 1

)sin()cos(m n

mnmnnymxBnymxA

+=+=

2

0

2

0

)(

2),(

2

1dxdyeyxFjBAC

nymxj

mnmnmn (2)

Where the complex Fourier coefficients can be determinedwith the double integral (2), see [7] - [9].


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M(t)

0 2

0

1

-1

t

Mod 2

Mod 3

Figure 4. The modulation waveforms Mod 2 (space vector with

equal zero space vector utilization) and Mod 3 (discontinuousmodulation waveform) at a modulation index of M = 1 each

C. Derivation of DC Link and Capacitor CurrentHarmonics

For all upcoming calculus natural sampling is assumedwhich is similar to regular sampling as long as asufficiently high carrier ratio p is assumed. Furthermore ahigh carrier ratio (p > 20) is presumed to satisfy the claim

that carrier side band interference can be neglected. Forthe example of natural sampled modulation with atriangular carrier like shown in fig. 3 the actual complexFourier coefficients mniV1 of switch current iV1 would beidentified by (3) which is derived from (2). However amodulation waveform combined with a saw tooth carrierwith a trailing edge requires a term like (4).

( )( )

( )

=

+

+

2

0

)(32

)(12

)(

21cos

2

dydxey

ii

yM

yM

nymxjL

V

mn(3)

( )( )( )

=

++

2

0

1

0

)(

21cos

2

dydxey

ii

yMnymxjL

V

mn(4)

Equations (5) through (7) render the modulationwaveforms for three most common PWM strategies asdiscussed here.

( )tMtM cos)( = (5)

( )


4/6

M

f/fc

C/ L

Figure 6. The calculated dc link current spectra for the spectra for

Mod 2 (space vector modulation with equal utilization of zero space

vectors) combined with a triangular carrier function, = 0

M

f/fc

C/ L

Figure 5. The calculated dc link current spectra for Mod 1 (cosine

modulation waveform) combined with a triangular carrier function,

= 0

M

f/fc

C/ L

Figure 7. The calculated dc link current spectra for Mod 3

(discontinuous modulation waveform) combined with a triangularcarrier function, = 0

capacitor current harmonics C are standardized with the peak value of the ac current fundamental

1L and the

frequencyfis scaled by multiples of the carrier frequencyfc. The derived formulas have been evaluated withMathcadTM. The power factor was set to cos() = 1 for allcalculations. A carrier ratio ofp = 60 was assumed and theside band sequences were confined to 40 elements each. Itshould be pointed out that all spectra of discontinuousmodulation waveforms look somewhat similar. But theone selected here ensures minimum switching losses forthe presumed power factor [9]. Plotting the spectra atdiscrete values of the modulation index (M = 0.1,0.2...1.1) results in a significant step form.

When comparing the harmonic performance of thedifferent modulation functions it is striking that for thediscontinuous one Mod 3 even when a triangular carrier isapplied the first carrier frequency band still is occupied bya significant bunch of spectral lines. This also applies forthe cosine modulation function in an alleviated manner asthe modulation index increases (unfortunately not visible

from this perspective, see also fig. 8, 11). In terms of dclink voltage harmonics reduction the modulation methodMod 2 seems to be favorable as already stated in [2].

B. Measurement ResultsMeasurements were taken from a voltage source

converter as depicted in fig. 1 with a rated power of22 kW and operated with a dc link voltage ofVdc = 500 V.A passive three phase ac load ofRL = 15 in series with afilter inductance ofLF = 4.5 mH was installed at the acterminals which causes a phase shift of = 5.3 at anapplied fundamental frequency off= 50 Hz. Thus it yieldsa power factor cos() of almost unity. The switching(carrier) frequencyfc was set to 3 kHz and the modulation

index was assigned to M= 1. Compliant to the analyticalconsiderations the dc bus was fed by a diode bridge todeliver a well smoothed dc load current id,load. The dc buscurrent was measured for steady state operation by acurrent probe. The current measurements were sampledand plotted with an ONO SOKKI CF-5210 FFT analyzer.The data evaluation including the Fourier analysis wasdone using Matlab

TM.

Once again the modulation waveforms shown in fig. 4were compared along with a triangle carrier wave shapeeach. Fig. 11 - 13 show the measured counterparts of thecalculated spectra in fig. 8 10. The measured andcalculated values correspond quite well. Remaining littledeviations originate from the experiment premises being

slightly different to the ideal assumptions. For instance theac line current does not possess an exact pure sinusoidaltrajectory and the phase shift is slightly different from = 0 as initially calculated which actually is a solelytheoretical operating point because a filtered line currentthat would converge to a pure sinusoid is only achievedwith a corresponding phase shift. Another source ofdeviation is that instead of natural sampling that wasassumed for the calculations the actual converter wasoperated with regular sampled PWM. The latter problemcan be overcome by little modification of the presentedcalculation methods [9]. Furthermore breakup errors ofthe Fourier analysis or deviations of the pulse rate from aninteger value can influence the measurement accuracy.


5/6

f/fc

C/

1L

Figure 10. The calculated dc link current spectra for Mod 3

(discontinuous modulation waveform according to fig. 4) combined

with a triangular carrier function, M = 1, = 0

f/fc

C/

1L

Figure 13. The measured dc link current spectra for Mod 3

(discontinuous modulation waveform according to fig. 4) combined

with a triangular carrier function, M = 1, = 5

f/fc

C/

1L

Figure 9. The calculated dc link current spectra for Mod 2 (space

vector modulation with equal utilization of zero space vectors)

combined with a triangular carrier function, M = 1, = 0

f/fc

C/

1L

Figure 12. The measured dc link current spectra for Mod 2 (space

vector modulation with equal utilization of zero space vectors)

combined with a triangular carrier function, M = 1, = 5

f/fc

C/

1L

Figure 8. The calculated dc link current spectra for Mod 1 (cosine


M = 1, = 0

f/fc

C/

1L

Figure 11. The measured dc link current spectra for Mod 1 (cosine


M = 1, = 5

V. CONCLUSIONA new method for the calculation of dc link current

harmonics for voltage source converters operated by anymodulation strategy is presented.

As means of analysis the geometric wall modelapproach for the determination of double Fourier seriesdescribing the dc link current harmonics is proposed andapplied to three selected cases. With this method it isfeasible to determine the dc bus current spectra for allkinds of modulation strategies with moderate effort.

Moreover it gives the opportunity to account for furtherconverters that are connected to the same dc bus as long as

the phase shift of the resulting harmonics is regardedcarefully.

Results of dc link current spectra calculated by meansof this method and their comparison are presented. Theyrender information on the dc bus current spectradispersion and thereby on the predominant influence ofthe modulation strategy. Different exemplary modulationfunctions are investigated and compared, continuous anddiscontinuous ones along with a triangle carrier each. Thetheoretical outcomes are proved by additionalmeasurements. The presented method contributes todetailed dc bus capacitor sizing as it facilitates the prediction of the dc bus current spectra for any kind ofmodulation method.


6/6

REFERENCES

[1] J.W. Kolar, H. Ertl, F.C. Zach, Calculation of the passive andactive component stress of three phase PWM converter systemswith high pulse rate,EPE 89, vol. 3, pp. 13031311, 1989.

[2] P.A. Dahono, Y. Sato, T. Kataoka, Analysis and minimization ofRipple components of input current and voltage of PWMinverters, IEEE Trans. on Ind. App., vol. 32, No.4, July/August

1996[3] F. Renken, Analytic calculation of the dc-link capacitor current

for pulsed three phase inverters,EPE-PEMC 04, Riga, CD ROMarticle, 2004

[4] M. Winkelkemper, S. Bernet, Design and optimization of the dc-link capacitor of PWM voltage source inverter with activefrontend for low-voltage drives, EPE 03, Toulouse, CD ROMarticle, 2003

[5] F.D. Kieferndorf, M. Frster, T.A. Lipo,Reduction of dc buscapacitor ripple current with PAM/PWM converter EPE 03,Toulouse, CD ROM article, 2003

[6] M.L. Gasperi, A method for predicting the expected life of buscapacitors,IEEE Industry Applications Society Annual Meeting,vol..2, pp. 1042-1047, 1997

[7] H.S. Black, Modulation Theory, Van Nostrand, 1953[8]

J.F. Moynihan, M.G. Egan, J.M.D. Murphy,Theoretical spectraof space-vector-modulated waveforms, IEE Proc.-Electr. PowerAppl., Vol. 145, No. 1, January 1998

[9] D.G. Holmes, T.A. Lipo, Pulse width modulation for powerconverters, IEEE press series on power engineering, Piscataway,2003

[10] K. Zhou, D. Wang, Relationship between space-vectormodulation and three-phase carrier based PWM: a comprehensiveanalysis, IEEE Trans. on Ind. Electr., vol. 49, No.1, February2002

[11] F. Jenni, D. West, Steuerverfahren fr SelbstgefhrteStromrichter, B. G. Teubner, Stuttgart, 1995

APPENDIX

As an example the following chapter presents the

dissolution of (3) for a cosine modulation waveform (5).As such (3) can also be written as (10) with (9).

( ) jj ee +=2

1)cos( (9)

( )( )

=

+

2

0

cos212

3

214

dyeee

mj

ii

yMmjynj

mjL

V

mn

( )( )

( )( )

( )( )

+

+

+

+

2

0

cos212

2

0

cos212

3

2

0

cos212

dyeee

dyeee

dyeee

yMmjynj

mj

yMmjynj

mj

yMmjynj

mj

(10)

With the identity given by (11), see also [7], (10) can beturned into (12) comprising Bessel functions of first kindand n

thorder.

( )

=

2

0

)cos(

2dee

jZJ jnjZ

n

n

(11)

=

+

MmJem

jii

n

mjn

L

V

mn

22

1

2

3

1

+

+

+

+

MmJe

MmJe

MmJe

n

mj

n

mj

n

mj

2

2

2

1

2

1

2

3

1

2

(12)

for n = -...+ and m = 1, 2,...

As can be seen above the case for m = 0 cannot beregarded by (12). Thus a distinction of cases yields (13) asdeduced from (3).

( ) +=

2

021

0 )cos()cos(2

dyyyMe

j

ii

yjnL

V

n(13)

It turns out that after employing lHospitals rule on(13) there are only values remaining for n = 0, 1, 2 as canbe seen by (14).

=

=

=

=

24

12

0)cos(2

1

0

nifej

iM

nifeji

nifj

iM

i

jL

jL

L

V

n

(14)

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