pwm vetorial

8/8/2019 PWM Vetorial

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A Hybrid PW M Strategy Combining Modified Space Vector andTriangle Comparison MethodsVladim ir BlaskoRockwell Automation - Allen BradleyMequon, WI 53092, USA

Abstract - Classical space vector PWM w ith equal durationof application of zero state vectors VO an d V7 was modified.The time of application of vector V7 (and VO) was madechangeable from 0 to 100% over the time To for their com-bined application. The ratio of the duration of application ofvector VO vs. V7 can be kept constant or changed on a sampleby sample basis with a significant impact on the characteris-tics of the PW M. Correlation between modified space vectorand triangle comparison methods was established. It wasproved in the paper that modified space vector PWM (with aclassical space vector PWM as a special case) can be imple-mented as triangle comparison method with added zero se-quence. A new algorithm suitable for implementation ofmodified space vector method on digital or analog hardwarefor triangle comparison PWM was proposed. Because thealgorithm combines theory of space vector PWM with easeof implementation of a triangle comparison PW M it wasnamed Hybrid PWM (HPWM).

I. INTRODUCTIONThe Triangle Comparison Pulse With Modulator(TCPW M) compares a high frequency triangular carrier with

three reference signals and creates gating pulses for theswitches in the power circuit. It can be easily implem ented inanalog or digital domain. Almost all the motor controllersavailable on the market today have hardware for the digitalimplementation of symmetrical (updated once per cycle) orasymmetrical (updated twice per cycle) PWM or both. Toextend linearity a zero sequence signal is added to all of thethree phase reference voltages [l]. The zero sequence moveslocally averaged voltage of motor neutral n (averaged overone period of the carrier) with respect to the center point ofthe DC link voltage n, Fig. l(a). However neutral of the mo-tor is not connected and locally averaged motor phase volt-ages remain undistorted and follow reference voltages.

* The author is with Rockwell Automation - Allen BradleyStandard Drives Development, 6400 W. Enterprise Drive,Mequon, WI 53092 USA, vblasko@meqlanl .remnet.ab.com

By adding a third harmonic zero sequence [11, linearity of thePW M is extended for about 15.5%.1 1

c , x {O,l, ... }( A B C ) , A,B,c E 1 (conducting),O non onducting)}

(a>-v,(010)

\+ /,/ ,R eSector 6

(001) (1 01)(b)

Fig. 1. (a) three phase inverter with switches A , 2 ,B , B , C,c and the definition of switching state vectors (b) in thecomplex dq plane

0-7803-3500-7/96/$5.000 996 IEEE 1872

Authorized licensed use limited to: Universidad Federal de Pernambuco. Downloaded on May 10,2010 at 16:36:11 UTC from IEEE Xplore. Restrictions apply.
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The space vector method [ 2 ] (SVPWM) operates in acomplex plane divided in the six sectors separated by theswitching state vectors, Fig. l.( b). The switching state vec-tors are defined by com bination of conductinghonconductingswitches in the power circuit of inverter, Fig. I(a). The com -plex reference vector FY: is used to locate two adjacentswitching state vectors and to compute time (T I and T2) orwhich they are active. For the rest of sampling time T, zeroswitching state vectors V , or Vo (connecting all of the threephase winding to positive or negative rail of the DC bus) areactive. SVPWM locally averages, over sampling period T,,active and zero state vectors to be equal to the reference vec-tor.In this paper, classical SVPWM (with sequence VO, V I, V2,V7, V7, V2, VI, VO over two sampling intervals in the frstsector and with equal duration of zero state vectors VU andV7, [2]) was modified. The ratio of duration of application ofzero vectors VO and V7 over the interval To for their com-bined application was made variable. The duration of appli-cation of vector VO can be changed from 0% to 100% at theexpense of duration of application of vector V7 and viceversa. The correlation between modified space vector methodand triangle comparison method was established. A newHybrid PWM (HPW M) algorithm combining good features ofSVPW M and TCPW M was developed

11. SPACE VECTOR Pwh.4The reference voltage vector T4; = Vq*- Vd * can be ap-

proximated in first sector using local averaging concept bythe following:

(1 )t; = v * - j v * = v + T p

r , T 2d Y dTime intervals T I an d T2 determine durations of applicationsof vectors VI and V2 respectively. Zero state vectors V7 orVO are applied for the rest of sampling interval:

In all the six sectors time intervals T, and T2 can be calcu-lated from [3]:A * 7r * 7 r7; = ~-[v ,cos(-)+v,s in(->]2 3 3 ( 3 )

where:m - sector number, m=1, ..6; i fm=l thanm- 1=6,

- normalized q and d components ofreference vector = v ~ : ~( v d C / 2 )

For sectors 1 to 5, T I and T2 determine duration of applica-tions of vectors with lower and higher indices respectively.In sector 6 , T I and T2 determine duration of application of V6an d VI respectively.The relationship between reference voltages in three phasesystem a,b,c and two phase d,q system is defined by:

* I I * *vy = - (2v , - b - c)2* 43Vd = -(vf -vi)3

Equations ( 2 ) to (6) define space vector PWM . They givecorrelation between normalized three phase reference volt-ages v:,~,~ which are inputs into PWM) and time intervals To ,T I and T2 during which zero state and active space vectorsare applied.111. CORRELATION BETWEEN MODIFIEDSPACE VECTOR AND TRIANGLE COM-

PARISON P w MIn the classical SVPWM zero state vectors VO and V7 ar eapplied each OST,. Figure 2.(a) shows sequence of applica-tions of switching vectors in sector one in modifiedSVPWM. Zero vectors V7 and VO were applied for the timeintervals ko To and (1-ko) To respectively, where 0 4 ,, 4 .

Durations T,, Tb, and T, of gating pulses for switches A, B,and C are:T, = k , Toi-, +T,& = k , , T + &T , = k,,&

For the triangle comparison PW M method, Fig. 2(b), thenormalized triangle signal vf (part of v, with positive slope)can be described by:

where V, an d V , are instantaneous and peak values of tri-angle wave form respectively. It is convenient for the analy-sis to normalize V, o the same base value as three phase ref-erence voltages i. e. Vrp= v&. Thus Vt pulsates between +/-1 and modulation index m (defined as the ratio of peak valueof reference voltage and peak value of triangle) becomes

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equivalent to the amplitude of normalized input referencevoltage.

Fig. 2 (a) timing of switching of space vectors and (b)timing of gating pulses based on triangle com parisonmethodBy introducing (7) to (9) into (10) i.e. the equation of tri-

angle wave form in Fig. 2(b); reference voltages vu ,vb an dvc ,which produce the same gating pulses and give the sameresults as the modified space vector method, are obtained:

** *** I

** LV b = -(k ,T, + T * ) - 1 ,v, =-(k,T,)-l ,

r,T,

* * 2

with time intervalsT, T , ,2 2r, =-(vu - v b ) = - V u b ,

T = Z ( *b - v , ) = - v b c .r , *2 2 2Time intervals T I and T2 were comp uted from (3) to ( 6 ) fo rthe first sector. Note that T I and T2are proportional to line to

line reference voltages. The same results can be obtaineddirectly by the inspection of the Fig. 2(a). From Fig. 2(a) itis evident that during T I and T2 bus voltage V d , is appliedbetween lines ab and bc respectively. Time intervals T I andT2 can be computed from the required equilibrium of volt -seconds over sampling interval Ts :7; V, = Vu; and

V, = Vbzc . Such an approach eliminates the need for useof space vector method (at least equations (3) and (4)) in theanalysis. It also confirms that space vector method deals withline to line voltages and as a consequence it is able to providefull dc bus voltage Vdc as locally averaged peak value of lineto line voltage. On the other side, the triangle comparisonmethod without added zero sequence, deals with phase volt-ages. It can provide only VdJ2 as the peak value of locallyaveraged phase voltage. Evidently for the same Vdc , SVPWMprovides without distortion higher output voltage for the fac-tor of 2/& = 1.155 than the TCPW M.

Introducing (2), (14) and (15) into (1 I ) to (13) the new setof reference voltages is obtained:** * *vu = v, +v,, (16)

V b = V b +v, (17)v, = v, +v, (18)vi s =-[(1-2k,)+k0v: +(l- k,)v ;] (19)

** * 1* I * *

For the "classical" SPWM, with VO and V7 applied for theTo12 each and for the balanced system (v: + v i +v,* = 0),zero sequence follows fiom (19) as a special case with ko=0.5:

(20)I *v:, = -0.5(vU + v c )= 0 . 5 ~ ;The above procedure was repeated for all the other sectors.The results are summarized in the Table I, Fig. 3 and in Eqs.

(21) to (23):

* *where ofv,,, ,v , , ~nd v;, are minimum, middle and maxi-mum values of input reference voltages v : , ~ , ,llustrated inFig. 3.

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TABLE I. DEPENDENCEF T I ,Tz AN D MINIMUM, AXIMUMAN D MEDIUM VOLTAGES ON SECTOR NUM BE RSND INPUTPHASE AND LINETO LINE REFERENCE VOLTAGES

cation for the classical SVP WM implementation on TCPWMhardware. The algorithm ffor the determination of the zerosequence voltage with factor ku as a variable is shown in Fig.4 ( 4 .

I I Sector-1~~

0.02t [SI0 * 0.005 * 0.01 * 0.015v mm l d - - - - -v m --Fig. 3. Derivation of v:in , i ld nd v i a x from the inp ut refer-

ence voltages v:,~,,

The new set of normalized reference voltages v::b,c wa sgenerated in (21) by adding zero sequence voltage vzs to theold set of reference voltages v:,~,, The zero sequence con-tains a dc component (1-2 ku ) and the combination of theoriginal - initial phase reference voltages. Work ing with me-dium, maximum and minimum voltages simplifies the algo-rithm and eliminates need for explicit sector identification.Equations (21) - (23) are used as a basis for the block dia-gram representation in Fig. 4.The algorithm is convenient formicroprocessor implementation. All the operations requiredincluding sorting to determine maximum and minimum inputreference are simple and fast. The algorithm is also conven-ient for the analog implementation. The principle of opera-tion of analog sorting block is shown in Fig. 4(b). Two sim-ple three phase rectifier bridges with compensated voltagedrops on diodes can be used to sort minimum and m aximumvalues (envelopes) of input voltages. Fig. 4(c) shows blockdiagram of the algorithm (23) for the zero sequence identifi-

USorting(a )

Fig. 4 (a) Block diagram of HP WM, (b) analog implemen-tation of sorting block (c) zero sequence generation forko=0.5, and (d) generation of zero sequence with ko as avariable

111.FACTOR k,,AND CHARACTERISTICSOF HPWM

SVPWM and TCPW M have tw o or three input variablesvi>dor v : , ~ , ~ .PWM has one additional input, factor k,,which can be constant or a variable changed on a samplinginterval basis. Factor ku effects voltage and current waveforms of the load and thus evidently characteristics of PWM.Equation of the voltage on one phase of three phase sym-metrical load (subscripts for phase assignment are omitted)

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will be used to investigate effects of ko on PWM characteris-tics:

d idtU = L - - + i R + e ,

u = U + u _ , i = I + i _ , e = E ,where:R LU,Z,u,i-

- resistance and inductance of the load,-fundamental components of voltage, current and- ripple components of vo ltage and current.electromotive force,

The voltage U, current i and counter electromotive force ein (25) are split into slowly changing fimdamental compo-nents (quasi - constant) and ripple components with a signifi-cant change over carrier period. introduction of (25) into (24)splits (24) into two equations, one for fundamental and an-other for ripple voltage. Voltage U from the equation withfundamental quantities correspond to the voltage V* which isinput into PWM. More interesting is the equation with theripple components:

(26)d i_U _= L - +i _ R ,dtwhich after integration gives the trajectory of the ripple cur-renti _ ( z ) = - / u - d t + - / l _ d t + i _ ( O ) , 0 2 z I n T , ,where n is a PWM update factor n=1,2. For PWM updatedevery T, or twice per carrier n=l . For PWM updated every2T, or once per carrier n=2. Depending on n , factor ko can bechanged at least once or twice per ca rrier.The results of simulation showing the influence of ko on perunit wave forms of current and voltage ripple, are shown inFig. 5. A three phase, predominantly inductive load with atime constant of L/R=lOT, was assumed.

(27)1 ' R ' .LO LO

A . Factor ko Constant or Changed O nce PerCarrierFactor ko is kept constant over carrier period 2T, at values

k , = 0.1 and 0.7, Fig. 5(a-b). Pulsating voltage U- and currenti- on the load are even and o dd functions respectively aroundthe center of the carrier regardless of value of ko. Their aver-age values are zero and the both of he first terms in (27) arezero. It follows from (27) that i-(2TJ=i-(O)=O i.e. ripple cur-rent i- crosses zero at the centers of sym metry of voltage U-Current sampled at these instances corresponds to the fun-damental component and is ripple free. However change o f ko, every even multiple of T,, effects the w ave form of the ripple

voltage U-. It effects the distance between fixed high and lowvoltage levels and thus effects the amplitude of the ripplecurrent, Fig. 5(a-b).Increase of ko from 0 to 1 widens gating pulses A, B, and C,Fig. 2. In the first sector gating pulse fo r the phase c narrowsto zero if ko=O. On the other hand for ko=l gating pulse forphase a stretches across the whole carrier cycle. Evidently forko = constant = 0 or 1 the devices in one leg of the bridge stopswitching. Currentdvoltages of the three phase load are con-trolled by switching the other two legs in the bridge. Switch-ing losses are reduced approximately by U3.

1 r-. I

10 50

-0.5-11

0 50

-0 5-1

2t/T,.5(b )10.5

2( c ) UT,0 51

0.50

-0.5- 1

Fig. 5. Wave forms of a carrier vt , current i- and voltage rippleU- on the RL load with (a) ko= 0.1, (b) ko=0.7, (c) with kochanged from 0.1 to 0.7 in the m iddle of the carrier cycle and(d) steady state (after 58TJ with ko periodically changed from

0.1 to 0.7 every T,B. Factor ko Changed Twice Per Carrier

By computing v : : ~ , ~ith ko different at different samplingintervals T A , he center of gating pulses A, B and C in Fig.2(a) is shifted to the left or to the right from the center of the

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triangle. The effects o f change o f ko=kol=O. to ko=koz=0.7atthe cen ter of the triangle of carrier are illustrated in F ig. 5(c).Change of ko , every T, or even number of T,, effects the am -plitude of the current ripple and also adds a bias to it effect-ing the fundamental component.

If ko is periodically changed between the same values ev eryT,, the second term in (27) causes the bias from ripple currentto decay towards zero with the time constant of the load( L / R ) , Fig. 5(d). However the zero crossing of the current(in steady state) is shifted from the p eaks of the triangle to thecenter of (shifted) voltage pulses. The current and voltagewave forms in Fig. 5(d) are identical to the those obtainedwith ko=0.5(kol+koz)=0.4=constant ve r 2T, without phaseshift.

IV . THEORETICALND EXPERIMENTALRESULTS

The analysis of the HPWM was done at different modula-tion index es m and for different facto rs ko. Fig. 6 shows zerosequence and reference voltages v , , v, and v:, for (a) ko=O,(b) ko =l, (c) ko=0.5 and (d) ko=0.75 with modulation indexm=1.154. Reference voltages vi * for ko=O and ko =l are mir-ror images of each other. For the one third of (hnd ame ntal)period they are equal to the + or - peak value of v, causingcontinuou s conduction and reducing switching losses by -113.Voltage reference v , for all the factors k, f 0.5 are nonsymmetrical around time axis.Figures 6(e-f) illustrates operation of HPWM with facto r kochanging during operation. Factor ko pulsates betw een 0 an d1 with frequency three time s higher than the frequ ency of thereference v: . The modulation indexes m is 0.9 and 1.154.Wave form vi* is symmetric around time axis. Switching ofthe bridge is reduced by 113.The reference wave forms shown in Fig. 6 (with exceptionof wave form in Fig. 6(d)) have been generated by the d iffer-ent PWM schemes, reported in [2], [4 - 61. From Fig. 6. it isevident that a single HPWM algorithm can generate any ofthem.The results of simulation in Fig. 7. illustrate the operationof HPWM with random change of ko around 0.5 everyTS=2/f,;fc= carrier frequency = 2000Hz. The random changeof ko adds random component to v:* and makes spectrum C,of current i in Fig. 7(c) uniformly distributed. It effec ts theacoustic noise produce d by the drive. The spectrum of thecurrent obtained under the same conditions but withko=0.5=constant is mostly below 60dB for frequency rangebetween 200 and 1500 Hz .Figure 8 . gives experimental verification o f HPWM . It pre-sents wave forms of v , , , , , and i, with ko changed be-

* **

I*

. .. I

l0.5

0-0.5

-11

0.50

-0.5-1 0.015 . 0.020.005 0.01(b ) ko= 1 1 LSl

0 0.005 0.01 0.015 0.02(c) ko= 0.51

0.50

-0.5-1

0 0.005 0.01 0.015 0.02(d) k,= 0.75 [I1

0.50

-0.5-1

0 0.005 0.01 0.015 11 0.02(e ) k,=f((t), m=0.91

0.50

-0.5-1 0 0.005 0.01 0.015 t [I 0.02(f) k,=f(t), m=1.154

Fig. 6. Wave forms of m odified reference voltage v ,reference voltage v: and zero sequ ence voltage v : ~ t

different ko

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tween 0 an d 1 with three times fundamental frequency as inFig. 6(e). Th e wave forms were taken on a lOOkW threephase regenerative voltage source converter with unity powerfactor [6], at input voltage of 48OV, DC bu s voltage of 75 0Vand the line reactor of 1m H.

10.5

0-0.5-10.035 0.04 0.045 (a) 0.05 t [SI 0.055

I I

-2 I I I I0.05 t [SI 0.055(b)0.035 0.04 0.045

10 10 1o4

V . CONCLUSIONSpace vector PWM was modified to enable change of the

duration of the application of zero vectors VO an d V7 . Thecorrelation between modified space vector an d triangle com-parison PWM was established . A variable k,, proportional tothe time of application of vector V7 , was introduced. Bychanging ko from 0 to 1, the duration of application of V7can be changed fi-om 0 to 100% of the time for the applica-tion of both zero vectors VO and V7. I t made PWM moreuniversal and flexible. The transitions from different PW Mschemes reported in literature can be easily achieved.Factor ko can be changed within every sampling intervalthus enabling movement of the center of the PWM patternaround center of the triangle effecting the spectrum of thecurrent and audible noise of the drive. Change of ko everytwo or even numbers of sampling intervals T, effects the cur-rent ripple without changing the hnd am enta l component.The developed algorithm is suitable for microprocessorand analog implementation. It provides extended linearity(with modulation index up to 1.154), reduced switchinglosses (during ko = 0 or 1) and enables change of the spec-trum of the current by changing k,.

REFERENCES[l ] J. A. Houldsworth and D. A, Grant, The Use of Har-monic Distortion to Increase O utput Voltage of a Three -Phase PWM Inverter, IEEE Trans. Ind. Appl., Vol. IA-20, No. 5, SeptemberiOctober, 1984, pp. 1124 - 1228.. - . - - - . ,.. 1 . 1 - . 1Fig. 7. Wave forms of: (a) normalized discrete reference voltage

v, , riangle v, , factor k,, (b) current i switching hnction A--_ _ - - _**

and (c) spectrum C, of current la ; ko changed randomly around0.5 with twice the carrier frequencyv j l p d d r v ]

L L ] H. w . van der Broeck, H. Ln . x u a e m y a na cr. > t a m e ,Analysis and Realization of a Pulse Width ModulatorBased on Voltage Space Vectors, IEEE IAS AnnualMeeting, Denver, USA, 1986, pp. 244-251 .[3 ] J. S. Kim and S. K. SUI,A Novel Voltage ModulationTechnique of the Sp ace Vector PWM , IPEC - Yokohama

[4] H. W. van der Broeck, Analysis of The Harmonics inVoltage Fed Inverter Drives Caused by PWM Schemeswith Discontinuous Switching Operation, EPE AnnualMeeting , Firenze, Italy, 1 991, pp. 3-261 to 3-266.[5] D. R. Alexander and S. M. Williams, An Optimal PWMAlgorithm Implementation in a High Performance125kVA Inverter, APE C Annual Meeting, San Diego,[6] Vikram Kaura, Vladimir Blasko, Operation of a VoltageSource Converter at Increased Utility Voltage, Con$ Rec.of PESC-9j Ann. Mtg, Atlanta, USA, 1995, pp.523 - 527.

95, pp.742-747.

USA, 1993, pp. 771 - 777.

Fig. 8. Experimental results - wave forms of v: , c vr, andcurrent i with ko changed between 0 an d 1 as in Fig. 6(e)

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pwm vetorial

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