New atheatcal pproach for Solvg the Equatos of Haroc Elato PW Roozbeh Nader Electrcal Egeerg Departet, Ira Uversty of Scece ad Techology Tehra, Tehra, Ira ad bdolreza Rahat Electrcal Egeerg Departet, Ira Uversty of Scece ad Techology Tehra, Tehra, Ira STRCT Selectve haroc elato s the optal cotrol techque for PW verters. The a challege ths techque s to solve the olear trascedetal equato syste obtaed fro the Fourer trasfor. The elato techque cojucto wth the Pade approato theory ad Resultat theory s oe of the effcet techques to solve ths equato syste. I ths paper, we propose a ew techque for elatg the varables aog odfed polyoal equatos of haroc elato techque based o a ew otato whch splfes the relatoshp betwee polyoal factors. Keywords: Selectve Haroc Elato, Pulse-Wdth odulato, Elato Theory, DC-C Iverter ad Polyoal Equatos.. INTRODUCTION Selectve haroc Elato (SHE s the optal cotrol techque for PW verters. SHE offers several advatages copared to covetoal atural PW techque cludg acceptable perforace wth low swtchg frequecy to fudaetal frequecy ratos, drect cotrol over output wavefor harocs, ad the ablty to leave trple harocs ucotrolled to tae advatage of crcut topology three-phase systes. These ey advatages ae SHE a vable alteratve to other ethods of odulato applcatos such as groud power uts, varable speed drves, or dual-frequecy ducto heatg []. I ths techque, a PW wavefor wth varable swtchg agles s cosdered to Fourer trasfor to tae t to the haroc doa. equato syste s the fored by cotrollg selectve harocs of ths PW wavefor. The resultg equato syste s olear ad trascedetal. Ths equato syste ca be solved by geeral calculatve techques such as Newto ad Iterato ethods or by specalzed techques such as Walsh fuctos, Geetc lgorths, ad Elato techques []. The assued syetry of the tal PW wavefor defes the for of the equato syste. The splest for s the quarter-wave syetry of the PW wavefor whch gves the followg equato syste: ( cos( θ = ( = Where s the uber of swtchg agles the quarter perod, θ are the swtchg agles, ad s costat. For sgle-phase verters s odd ad for three-phase verters s odd ad otrple: sgle-phase: =,,5,7,..., ubers three-phase: =, 5, 7,,,7,... For sgle phase syste the aswer s uque ad for three-phase syste several aswers et. The half-wave syetry assupto creases the soluto space for both sgle-phase ad three-phase systes ad therefore gves a good fleblty to the desger to choose the ost effcet aswer for decreasg total haroc dstorto (THD, acoustc ose, ad electroagetc terferece (EI. However, ths paper we focus o solvg the equato syste for quarter-wave syetry. The elato techque for solvg the equato syste of the SHE techque s based o covertg trgooetrc eleets of each equato to polyoal eleets. The objectve of the elato techque s to elate the varables aog these polyoal equatos ad covert t to a sgle-varable polyoal equato whose roots are the sae as the aswer of ths equato syste. Ths ca be acheved by Pade approato theory [] or Resultat theory [],[4]. Pade approato theory s used to obta the sgle polyoal equato for sgle-phase forulato ad the Resultat theory s used for the ore coplcated case of three-phase syste. I ths paper, we preset a ew atheatcal approach for elato based o the relatoshps betwee eve ad odd costats of polyoal equatos whch eables us to obta the sgle polyoal equato for sgle-phase systes. For the geeral case the proposed techque operates faster tha the Pade approato techque. For the future tred the eteso of ths techque ay be used wth the resultat theory
to splfy the equatos ad prove the degree of coplety that the resultat theory ca deal wth.. POLYNOIL EQUTION SYSTE ll the equatos for SHE-PW, Eq. (, are based o the ter cos(θ. Ths ter ca be coverted to su of powers of cos(θ usg the followg forulas: cos( θ = cos ( θ cos( θ = 4cos ( θ cos( θ 4 cos(4 θ = 8cos ( θ 8cos ( θ + 5 cos(5 θ = 6 cos ( θ cos ( θ + 5cos( θ 6 4 cos(6 θ= cos ( θ 48cos ( θ+ 8cos ( θ Iductvely, we have guessed the followg epresso for cos(θ: cos( θ = ( cos ( θ = ( O the other had, we ca epad cos (θ ters of the cose of ultples of θ accordg to the Fourer theory as: cos ( θ = ( cos( θ + cos ( θ = ( cos( θ + cos( θ 4 cos ( θ = ( cos(4 θ + 4 cos( θ + 5 4 cos ( θ = ( cos(5 θ + 5cos( θ + cos( θ 6 5 cos ( θ= cos(6 θ+ 6 cos(4 θ+ 5cos( θ+ ( The geeral forulato s: cos ( θ = cos (( θ = ( We have proved Eq. ( ad Eq. ( apped ad. Cosderg X = (- cos(θ ad usg Eq. ( the equato syste for sgle-phase verter, Eq. (, could be rewrtte as: X = c, j =,,..., (4 = The followg for ay ae a better sese of the polyoal syste: X + X + X + + X = c X + X + X + + X = c 5 5 5 5 X + X + X + + X = c 5 X + X + X + + X = c. NEW NOTTION The relatoshps betwee dfferet hgh order polyoals are very large. Therefore, we preset a ew otato whch eables us to epress the relatoshps: p p p Where s the set of varables. Ths otato represets the su of products of ay possble cobato of varables fro the for of the ter specfed. We call ths otato as SPC, the su of powers of the for (p +p + +p as the order of SPC, ad the uber of varables the for (uber of as the degree of SPC. The followg eaples ae t clear: = a,b,c,d ord. = deg. = { } = ab + ac + ad + bc + bd + cd ord. = 5 deg. = = abd + acd + bcd + abc + adc + bdc + acb + adb + cdb + bca + bda + cda ord. = deg. = 5 = s show the last eaple, f the for s ot achevable usg the set of varables (the degree of SPC s ore tha the uber of estg varables, the result s zero. We also assue the followg specfc case for the otato: = The equato systes we are dealg wth are the for of: = c (5 Therefore we eed to epress ay for of SPC ters of c, c,. Cosder the followg SPC eaple: To epress t ters of c, c, we preset the followg relato: c = 4 5 6 + + + Obvously, we have separated fro ad have ultpled ts SPC by the SPC of reag ter (. The result cossts of the SPCs of ay possble for whch ght be geerated by the ultplcato process. The estece of the last ter results two sets of slar ters. Ths s represeted by the coeffcet. We ca etract < > as: = c 4 5 6 ( It s clear that we have decoposed our SPC proble to several SPCs of oe ut lower degree. Usg Eq. (5, we repeat ths procedure several tes utl we reach to a epresso whch s oly cossts of c, c,. The followg recursve epresso reduces the proble of fdg the SPC of a specfed for to su of SPCs of other saller fors: cq tes tes tes tes tes tes q p p p p p p q = a tes tes tes tes p p p p+ q P + b = < (p p p The coeffcet a s the degree of q the correspodg SPC ter: f q = p a = + j j j + The coeffcet b s the degree of p q the correspodg SPC ter: f p + q = p b = + j j j If the ter we choose to separate fro the SPC for ( q has the greatest power aog all other ters (q p, p,, p, the all
b coeffcets are equal to. Thus, we ca wrte the followg recursve epresso for ay SPC for: tes tes tes p p p = tes tes tes p p p c p (6 tes tes tes tes p p p p+ p P = < (p p p < p Ths recursve epresso ca be easly pleeted by a fucto atheatcal softwares such as atlab, aple, or atheatca. 4. ELIINTING THE VRILES Now, suppose that we wat to obta a polyoal equato wth the roots R. We ca wrte t as: R = X,X,,X = { } (X X = Collectg X ths equato, we have: tes ( X = (7 = R Obvously, to specfy ths equato we eed to calculate the values for <>, <>,. For ths case, we ca splfy Eq. (6 as: tes tes tes = c tes tes tes q q+ = c R R R R q R R Usg these recursve equatos we have obtaed the drect epresso as: tes P p ( c = R Q W p! W : for each ter p = Q : ay possble for I the atr for ths ca be wrtte as the followg deterat: c c c c tes c c c = R c c (8! c However, the equato syste for sgle-phase SHE oly cotas odd powers of (c odd. Thus, we ust obta the relatoshp betwee eve ad odd. I ths case we ca use the epasos of the <> whch are: = + = + + 6 4 4 = + 4 + 6 + + 4! P P P = Q P!P! P! P + P + P = Q : ay possble for If we cosder the equatos fro [(+/] to, the ter whch oly cossts of ut powers of wll be elated. Usg Eq. (6, we ca reduce the degree of the reag ters to oe. y solvg the resultg equatos, the relatoshp betwee eve ad odd wll be obtaed. Thus we ca substtute c eve Eq. (8 ad obta all < > fors eeded to detere Eq. (7. s a eaple the followg s the fal equato for a syste wth fve varables: 5 4 PX PX + PX PX + PX P = 4 5 P = 9(c 75c c + 6c c 5c c + 5c c 89c + 5 5 7 7 5 7 5c c 5c c c 5 P = 9c (c 75c c + 6c c 5c c + 5c c 89c + 5 5 7 7 5 7 5c c 5c c c 5 P = 4c 5c c + 945c c c 7c c + + + + + + 575c c 5 4 7 5 5 9 6 9 7c c 5c c 7c c 575c c 5c c 9 5 7 4 7 945c c c 75c 97c c 5 5 P = c + 5c c c 5c c c 5c c c c + + + + + 4 5 7 7 5 4 89c c c 675c c 575c c 89c c c 4c c 5 7 9 5 5 8 7 4 6 8c c 575c c c + 6c c 75c c 54c c 5 9 7 567c c 5 P = c + 665c c 969c c 675c 8c c + + + + + 4 4 4 5 9 5 7 4 4 6 45c c 475c c c 44c c c 945c c c 7 5 7 8 5 88c c c 855c c c 798c c c 5c c 49c c 5 5 7 5 7 9 5 75c c 8c c 5c c c + 68c c + 44c c 5 7 9 5 9 P = c + 75c c 5c c c + 5c c 5c c + + + + + 7 5955c c c + 945c c c + 455c c c + 995c c c 4 5 8 5c c 555c c c 475c c c 5c c 9 7 7 6 5c c 5 9 5 5 5 5 6 5 97c c 5c c 57c 65c 555c c 5 9 5 9 4 855c c c 7875c c c 75c c c 95c c 5 7 7 5 7 5 5 5 7 5 9
5. CONCLUSION I ths paper we proposed a ew atheatcal approach for the elato of varables aog the polyoal equatos of the SHE PW ethod. Ths ew techque s based o the relatoshps betwee polyoal factors that are splfed through the proposed otato. lthough ths paper focuses o SHE equato syste for sglephase verters, the techque could be used to splfy the equato syste of three-phase verters ad cojucto wth the resultat techque ay prove the degree of coplety that the resultat theory ca deal wth. Ths could be a valuable atter for future research. 6. REFERENCES [] J. R. Wells,.. Nee, P. L. Chapa, ad P. T. Kre, Selectve Haroc Cotrol: Geeral Proble Forulato ad Selected Solutos, IEEE Trasactos o Power Electrocs, Vol., No. 6, Nov. 5, pp. 7-45. [] D. Czarows, D. V. Chudovsy, G. V. Chudovsy, ad I. W. Selesc, Solvg the Optal PW Proble for Sgle- Phase Iverters, IEEE Trasactos o Crcuts ad Systes I, Vol. 49, No. 4, pr., pp. 465-475. [] J. N. Chasso, L.. Tolbert, K. J. ckeze, ad Z. Du, Ufed pproach to Solvg the Haroc Elato Equatos ultlevel Coverters, IEEE Trasactos o Power Electrocs, Vol. 9, No., ar. 4, pp. 478-49. [4] J. N. Chasso, L.. Tolbert, K. J. ckeze, ad Z. Du, Elato of Harocs a ultlevel Coverter Usg the Theory of Syetrc Polyoals ad Resultats, IEEE Trasactos o Cotrol Systes Techology, Vol., No., ar. 5, pp. 6-. 7. pped pped : We have guessed the followg epresso for cos(θ, ductvely: cos( θ = ( cos ( θ = (9 ccordg to atheatcal ducto prcple, to verfy Eq. (9 we should obta cos((+θ for cos(θ. Cosder the followg trgooetrc detty: cos (( + θ = cos( θcos( θ s( θs( θ ( To calculate Eq. ( the ter s(θ s requred. Calculatg the frst-order dervatve of Eq. (9 wth respect to θ gves the followg epresso for s(θ: s( θ = s( θ ( ( cos( θ = ( Substtutg Eq. (9 ad Eq. ( Eq. (, we have: ( cos ( + θ = cos( θ ( cos ( θ = s ( θ ( ( cos( θ = Splfyg ths equato results : cos ( + θ = ( + ( cos ( θ = + + ( cos ( θ = C ( cos ( θ = ( To equalze the coses power betwee all ters, we should chage the paraeter to - ter C of Eq. (. Thus, we have: = + ew old C + ( + + ( cos ( θ = Regardless of beg ad ed pots of the suato ters, sug up ther geeral epressos/ters ad factorzg a eagful ter gves us: = + C ( + ( + ( + + ( ( (4 ( + + cos ( θ + The ters,, ad C Eq. (4 correspod to the ters,, ad C Eq. (. has a zero at = (+/, therefore we ca crease the fal pot of to [(+/] whle ths has o effect o the result of Eq. (. We ca also crease the fal pot of to [(+/] whle has two zeros at = / ad = (+/. C has a zero at =, thus we ca decrease the start pot of C Eq. ( to =. Therefore we ca eted Eq. (4 to all pots fro = to [(+/]: + + + + cos (( + θ = ( cos ( θ = + Clearly, the result cofrs Eq. (9. pped : We have obtaed the followg epresso for cos (θ: cos ( θ = cos (( θ = (5 We ca obta cos + (θ fro cos (θ as:
+ cos ( θ = cos( θ.cos ( θ = = ( cos ( = cos ( = cos ( θ cos( θ = ( + θ ( (6 + θ To equalze the cose ter betwee ad, we should chage the paraeter to - ter of Eq. (6. Thus we ca rewrte t as: = + ew old + cos ( = ( + θ (7 Substtutg (7 (6 ad factorzg a eagful ter gves us: + cos ( θ = + + cos (( + θ = + + + + cos (( + θ = + Note that the startg pot of suato s decreased to whle ths has o effects o the result. The two suatos ca be sply added fro to [(-/]. The reag pots are = (+/ for odd ad = /, /+ for eve: + = + cos ( θ = cos (( + θ + + + = odd = ( + + + + = cos( eve = = = + θ We ca erge the resultg two states to oe suato as: + + cos ( θ = cos (( + θ = + + = + + = + + cos (( + θ = + cos ( Obvously, the result cofrs (5. ( + θ