TRANSFORMATION TERMS abc TO qdn FOR 9-PHASE SYSTEM WITH 3x9 MATRIX

Transcription

1 ISSN-L: -, ISSN: - Vol., No., Janary TRANSFORMATION TERMS TO qn FOR -PHASE SYSTEM WITH x MATRIX Aran Jaya Electronic Engineering Politeknik Institte of Srabaya, & Seplh Nopeber Institte of Technology INDONESIA aran@eepis-its.e Marihi Hery Prnoo Seplh Nopeber Institte of Technology INDONESIA hery@ce.its.ac.i Soebagio Seplh Nopeber Institte of Technology INDONESIA soebagio@ce.its.ac.i ABSTRACT Orer of the transforation atrix fro coorinates to qn for -phase inction otor was introce in a sqare atrix for with orer of x. The size of this large atrix orer can coplicate the process of changing fro -phase inction otor oel of the for into qn becoes less siple. Recing atrix orer obtains the efficiency of coptation cost. In this paper we propose the nit ector relationships both an qn coorinate transforation atrix obtaine qn ter to orer x in orer to recing significant paraeter. Throgh the silation can be shown that the response of the sae inpt an the reference rotary spee ifferent angles on the orer of x coorinate transforation can proce a ifferent response qn oltage an otpt oltage is the sae for each reference frae. Keywors: Nine-phase syste, coorinates, qn coorinates, the x transforation atrix INTRODUCTION All orers of the transforation atrix fro coorinates to qn that hae been introce hae sqare atrix for an the orer was se to ajst the nber of phases. Three-phase coorinate transforation introce by Park contains the orer x (Pal C. Krase, ), (Bial K. Bose, ). Transforation of other ariations has been eelope with the sae atrix size bt ifferent in the atrix eleents (R. Krishnan, ), (Ne Mohan, ). Orer of x atrix transforation has been introce as a tool to analyze the ynaic oel -phase inction otor (H. X, et al., ). The oel ynaic of -phase inction otor is se as an obserer in spee sensorless otor control (L. Parsa & H. A. Toliyat, ), (Libo Zheng et al.,). Orer of x- atrix transforation for the -phase has been introce an applie as a tool to analyze the -phase otor an the reslts obtaine is se as an obserer in the spee control (A. R. Mnoz & T. A. Lipo, ), (Renato O. C. Lyra, & T. A. Lipo, ). Orer x transforation atrix has also been introce to anticipate the eelopent of eqipent ries the increasingly consiere to yiel large power an sall iensions (E. Lei, et al.,). The transforation atrix for -phase syste has an orer of x. All fors of transforation are obtaine by sing ethos typical for then β α fnction which is place in a ector space then projecte to for projection of qn (R. Krishnan, ), (Ne Mohan, ), (E. Lei, et al.,). The transforation contains a constant that is / p or / p of which p states the nber of phases, while the orer is accoring to the nber of phases. Copyright SAVAP International

2 ISSN-L: -, ISSN: - Vol., No., Janary Althogh the etho of projection to for αβ has a certainty in eterining the for of inerse transforation becase of the transforation atrix of a sqare, the cost to be pai is expensie in the process of fining the inerse, an the siplification for the nine-phase inction otor oel of the for into qn can be ifficlt. Orer of the transforation atrix fro coorinates to qn on -phase inction otor was introce in the for of a sqare atrix with orer of x. The size of this large orer atrix can be ifficlt. Therefore, other ethos are offere to oercoe these ifficlties. In this etho, the fnction of -phase syetry is place in the coorinate syste of nine tally perpeniclar. Each of these fnctions fors a ector. The s of each ector proces a rotating ector space to the origin. Then, ector space occpies -axis an negatie eriation at q occpies in the coorinate axes qn. Unit ector in each coorinate axis at qn can be eterine. Frtherore, the relationship of nit ector is gaine between both an qn x atrix. The reference axis of rotation is n becase it generates space ector eqal to zero axis that is perpeniclar to the ector space. Unit ector in coorinates an qn can be eterine. Frtherore, the relationship of the two nit ectors is eterine to obtain the transforation atrix x of which refers the nber of phases. Variations in the reference anglar elocity are applie to proe the reslts obtaine. We clai this transforation atrix x in orer to increase efficiency on coptation. The prpose of the coorinate transforation is to siplify the fors an syste analysis. The prpose of this siplification can be achiee when the orer of the transforation atrix to for three rows that can escribe the aont of the qn axes coorinate an the nber of colns that escribes the nber of axes coorinates of nine phase. Ths, the for of -phase transforation atrices obtaine becoes siple. TO qn COORDINATES The f(t) fnction that can express the oltage, crrent, an the flx has a nber of -phase syetry. If the fnction f(t) expresses a negatie oltage seqence then: a V cosω t () b = V cos( ω t π () c = V cos( ω t π () a = V cos( ω t π / ) () b = V cos( ω t π () c = V cos( ω t + π () a = V cos( ω t + π / ) () b = V cos( ω t + π () c = V cos( ω t + π () Voltages are grope accoring to the orer of the last sbscription so that the ninth oltage is grope into three grops accoring oltage irection nit ector. The first grop is the oltage a, b, c, the secon grop a, b, c, an the thir a, b, an. Each grop is place on nine () coorinate axes, each axis is tally perpeniclar to each other. To siplify the isalization, then each grop is place on three-iensional coorinate syste separately as Figre a, b, an c. Eqation foring a rotating oltage ector to the origin is as follows: = a + b + c + a + b + c + a + b + () Sbstittion of the grop oltage in the eqation can create eqation. = V cosω t + V cos( ω t π + V cos( ω t π + V V cos( ω t π / ) cos( ω t + π / ) + V + V cos( ω t π cos( ω t + π + V + V cos( ω t + π cos( ω t + π + () Copyright SAVAP International

3 ISSN-L: -, ISSN: - Vol., No., Janary Vector agnite generate by the nine-phase oltage can be one by applying Pythagoras theore, naely: = a b c a b c a b Vector agnite obtaine is: = V () c c c c c b b Figre. Vector iagra of the coorinate syste of nine tally perpeniclar rawn separately Unit ector is obtaine by coparing ectors with greatness of the ectors sch as in eqation : = () (a) a a Sbstittion the grop oltage eqation into eqation reslts eqation. = (cosω t + cos( ω t π + cos( ω t π + cos( ω t π / ) + cos( ω t π + cos( ω t + π + b cos( ω t + π / ) + cos( ω t + π + cos( ω t + π ) Rotation ector is accoring to the tie fnctions an for ector space. As a starting point to facilitate analysis, the space ector is place in the qn coorinate syste that is, at first, asse to occpy the q-axis as shown in Figre. ω e ω e a b b (b) a b (c) a a () ω e q srface Figre. ector rotates in the plane The space ectors can be erie into the new ector space t that is place on the axis. The ins sign is taken as the consieration that the proct ltiplication is x q = n (negatie seqence). ector rotates in the space of q with center at the origin that is the reference rotation ector space an can be expresse in ters of eqialent n. = with n = +. Ths, the ector n is perpeniclar to the space of q; therefore, it wol coincie with ω as shown in Figre. Copyright SAVAP International

4 ISSN-L: -, ISSN: - Vol., No., Janary n axis n n q V q axis -V t axis Figre. The nit ector in the coorinate qn Unit ector in qn coorinates obtaine fro the thir, t, an n ectors is shown in Figre. In qn coorinates, nit ectors s, qs an ns are also epicte (Figre ). Unit ector q can be expresse by the following eqation : q = = () V Sbstittion fro eqation into eqation creates eqation. q = (cosω t + cos( ω t π + cos( ω t π + cos( ω t π / ) + cos( ω t π + cos( ω t + π + cos( ω t + π / ) + cos( ω t + π + cos( ω t + π Unit ector can be expresse in the following eqation : t t ω = = () t V Sbstittion erie fro eqation into eqation reslts eqation. = (sinω t + sin( ω t π + sin( ω t π + sin( ω t π / ) + sin( ω t π + sin( ω t + π + sin( ω t + π / ) + sin( ω t + π + sin( ω t + π Unit ector n can be expresse in the following eqation : n n = = ( ) () n ) ) () () The relationship between the nit ectors in qn coorinates an coorinates is obtaine by coparing the two nit ectors, then: Copyright SAVAP International

5 ISSN-L: -, ISSN: - Vol., No., Janary q n cosθ cos (θ π cos (θ π cos (θ π / ) = cos (θ π cos (θ + π cos (θ + π / ) cos (θ + π cos (θ + π sinθ sin (θ π sin (θ π sin (θ π / ) sin (θ π sin (θ + π sin (θ + π / ) sin (θ + π sin (θ + π Becase the nit ector atrix ltiplie qn with atrix transpose itself proces the ientity atrix, the inerse atrix of qn nit ectors will be obtaine fro naely: cosθ sinθ cos (θ π sin (θ π cos (θ π sin (θ π cos (θ π / ) sin (θ π / ) q = cos (θ π sin (θ π n cos (θ + π sin (θ + π cos (θ + π / ) sin (θ + π / ) cos (θ + π sin (θ + π cos (θ + π sin (θ + π () ector thatis expresse in qn coorinates can be seen fro: T () Copyright SAVAP International

6 ISSN-L: -, ISSN: - Vol., No., Janary = a a a b b b c c T so, = [ acosθ + acos (θ π + acos (θ π + cos (θ π / ) + cos (θ π + cos (θ + π + b c cos (θ + π / ) + b c sin (θ + π / ) + b c cos (θ + π + + [ asinθ + asin (θ π + asin (θ π + sin (θ π / ) + sin (θ π + sin (θ + π + + b c sin (θ + π + [ a + a + a + b + b + g + c + c + ] n b b cos (θ + π sin (θ + π When the oltage ector in qn coorinates is calle, it is clear that: = qn () an q qn = [ q n ] () n Therefore, it can be written that: ] ] q () q = [ acosθ + acos (θ π + acos (θ π + bcos (θ π / ) + bcos (θ π + bcos (θ + π + cos (θ + π / ) + cos (θ + π + cos (θ + π ] c c () = [ asinθ + asin (θ π + asin (θ π + bsin (θ π / ) + bsin (θ π + bsin (θ + π + sin (θ + π / ) + sin (θ + π + sin (θ + π ] c c () Copyright SAVAP International

7 ISSN-L: -, ISSN: - Vol., No., Janary = [ a + a + a + b + b + b + c + c ] () n + In atrix for, eqations, an can be written: q n = cosθ cos (θ π cos (θ π cos (θ π / ) cos (θ π cos (θ + π cos (θ + π / ) cos (θ + π cos (θ + π sinθ sin (θ π sin (θ π sin (θ π / ) sin (θ π sin (θ + π sin (θ + π / ) sin (θ + π sin (θ + π T In its copact for eqations, the oltage relationship between the two coorinates qn with in eqation can be expresse as: q [ ] T = T ( θ ) a a a b b b c c () n a a a b b b c c () RESULTS AND DISCUSSION To test the reslts of the analysis, silations are perfore with the chart as in Figre. Nine-phase oltage acts as inpt of the transforation of to qn ter. The otpt of the transforation becoes the inpt for the inerse transforation fro qn to. As a reference, the rotation is ω which can be replace when neee. Nine-phase oltage syetry has axi oltage / olts at Hz. Each phase is separate by an angle of o to the phase of each other. To eterine the response of otpt, soe changes are ae as the reference of anglar elocity ω in the three conitions ω of which they are eqal to synchronos spee ω e, eqal to the rotor spee ω r an eqal to zero (stationary). Inpt oltage to the treatent is shown in Figre ; when ω ses synchronos rotation ω e = ra/sec can generate a oltage response qn like Figre. In Figre, there are three otpts oltage i.e. q, an n foring c signal, an the ale of q an n is eqal to zero, while eqals to / of the peak inpt oltage. This qn oltage when se as an inpt inerse transforation response is obtaine as shown in Figre, which gies responses to the otpt atching the inpt. When inpt oltage is fixe as Figre an the ale of the reference is changeable, rotation becoes eqal to ω r = ra/sec an the qn oltage response an oltage are obtaine respectiely sch as in Figre an Figre. Both oltage responses an q ispte an angle of o with the q that is a negatie fnction of sinsoial an lags. Althogh qn as inpt inerse transforation atrix is a ifferent for, the response otpt oltage is eqal to inpt oltage. Copyright SAVAP International

8 ISSN-L: -, ISSN: - Vol., No., Janary Figre Chart silation sing atlab silink When reference is replace with a ron of ω = or stationary an the inpt the oltage eqals to Figre, reslting ifferent responses of the oltage qn of which its reference ω r is ifferent. There are changes in the fnction of q which is positie bt it still lags o sinsoi as shown in Figre. otpt oltage respon reains the sae like before sch as shown in Figre. a b c INPUT VOLTAGE a b a b c Tie (secons) Figre. inpt oltage is transfore to the for qn with ω = ω e Copyright SAVAP International

9 ISSN-L: -, ISSN: - Vol., No., Janary qn VOLTAGE n q Tie (secons) Figre. qn otpt oltage of oltages transforation with ω = ω e a b c OUT-PUT VOLTAGE a b a b c Tie (secons) Figre. otpt oltage of qn oltages transforation with ω = ω e Copyright SAVAP International

10 ISSN-L: -, ISSN: - Vol., No., Janary qn VOLTAGE - q n Tie (secons) Figre. qn otpt oltage of oltages transforation with ω = ω r a b c OUT-PUT VOLTAGE a b a b c Tie (secons) Figre. otpt oltage of qn oltages transforation with ω = ω r Copyright SAVAP International

11 ISSN-L: -, ISSN: - Vol., No., Janary qn VOLTAGE q - n Tie (secons) Figre. qn otpt oltage of oltages transforation with ω = a b c OUT-PUT VOLTAGE a b a b c Tie (secons) Figre otpt oltage of qn oltages transforation with ω = Copyright SAVAP International

12 ISSN-L: -, ISSN: - Vol., No., Janary CONCLUSION The transforation of -phase coorinates to the coorinates qn an it s silation hae been one in three kins of change of reference anglar elocity. When the synchronos rotation ω = ω e is se as a reference, anglar elocity can generate the otpt oltage ( qn ) inicating that a c signal with n an q is eqal to zero while the oltage is eqal to / ties the peak inpt the oltage. When the anglar elocity reference eqals to ω r the otpt oltages ( qn ) are obtaine, of which an q oltage responses are negatie fnction of sinsoi ispte with the angle of o an q is lagging. Meanwhile, if the reference anglar elocity is ω = or stationary, the oltage responses obtaine qn are ery ifferent fro the response qn oltages reference ω r. There is a change in the fnction of q that is a positie sinsoi bt it is still lags o. Althogh the oltage qn is ifferent fors, when it is transfore back to for throgh the inerse orer of x atrix, it can create the sae reslts with the for inpt the oltage. The reslts of the orer x atrix inerse of the three changes in the reference anglar elocity proce the otpt oltage which is the sae as the inpt oltage. It is concle that the transforation of phase syste to qn by x atrix with ifferent anglar elocity reference generates a response ( qn ) which is ifferent an the transforation qn to phase syste by inersing x atrix with ifferent anglar elocity reference generates otpt ( ) -phase syste. REFERENCES A. R. Mnoz & T. A. Lipo, Dal stator wining inction achine rie, IEEE Trans. In. on Applicat., ol., pp., Sept./Oct.. Bial K. Bose, Moern Power Electronics an AC ries, Prentice Hall PTR,. E. Lei, et al., Mltiphase inction otor ries a technology stats reiew, IET Electr. Power Appl., Vol., No., pp., Jly. H. X, et al., Fie-phase inction otor ries with DSP-base control syste, IEEE Trans. Power Electron., ol., no., pp., Jl.. Libo Zheng et al., Dal-Plane Vector Control of a Fie-Phase Inction Machine for an Iproe Flx Pattern, IEEE Trans. On In. Electronics., ol., pp., May. L. Parsa & H. A. Toliyat, Sensorless irect torqe control of fie-phase interior peranent-agnet otor ries, IEEE Trans. In. Appl., ol., no., pp., Jl./Ag.. Ne Mohan, Electric ries an integrate approach, McGraw-Hill,. Pal C. Krase, Analysis of Electric Machinery, MNPERE, Minneapolis,. Renato O. C. Lyra, & Thoas A. Lipo, Torqe Density Iproeent in a Six-Phase Inction Motor With Thir Haronic Crrent Injection, IEEE Trans. In. on Applicat., ol., pp., Sept./Oct.. R. Krishnan, Electric Motor Dries, Prentice Hall,. Copyright SAVAP International