VECTOR ANALYSIS APPLICATION IN ROTATING MAGNETIC FIELDS
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1 VECTOR ANALYSIS APPLICATION IN ROTATING MAGNETIC FIELDS runo Osorno Dprtnt of Eltril And Coputr Enginring Cliforni Stt Univrsity Northridg Nordhoff St Northridg CA Astrt Rotting gnti filds in thr-phs ltril hins hs n on of th hrdst topis to onvy to our studnts in th r of nrgy onvrsion. Ths filds r trnsford into phsors (vtors tht rott in sp round th sttor of n ltril hin. Th thtil proof of suh rottion will shown nd MATLA siultion dsriing ths vtors will givn. Clssi txtooks pproh this onpt thtilly nd usully th studnts r lft to igin th sp vtor rottion. This is no longr th s, now w n siult nd SEE in our dsktop oputrs, using Mtl TM, rotting phsors. Polyphs Anlysis Thr phs indution hins r th work-hors of industry, nd ths hins hv rotting gnti fild. W will giv rif physil dsription; onsidr squn -- nd sytri distriution of th phss y 12 ltril dgrs h in sp nd round th ir gp. Th si thr-phs hin will hv thr oils tht w onsidr to hv th following trinls: Ths oils r xitd y ti dpndnt sinusoidl ltrnting urrnt produing sinusoidl gnto-otiv for (f wv t th ntr of th gnti xis of prtiulr phs. Thrfor th thr-sp sinusoidl f wvs r displd 12 ltril dgrs in sp. Figur 1 shows how w dtrin th gnti xis of oil. Figur two indits thr gnti-xis of thr oils pld in sp round th sttor. A thr-phs syst rquirs thr oils to rt thr gnti filds tht will intrt ong h othr to otin rsulting gnti oponnt. Furthror, this rsultnt oponnt will rott in sp round th ir gp of th ltri hin. Prodings of th 22 Arin Soity for Enginring Edution Annul Confrn & Exposition Copyright 22, Arin Soity for Enginring Edution Pg
2 X Currnt out Mgnti flux round th ondutor X' Mgnti Axis Currnt in Mgnti flux round th ondutor Figur 1. Mgnti xis of singl oil Figur 1 shows oil with urrnt oing out ondutor X with dot nd ondutor X with urrnt gtting into th ondutor. Using th right hnd rul [3], w dtrin th dirtion of th gnti flux. Th gnti xis is, thrfor, rtd prpndiulr to th oil. Figur 2 shows thr oils pld in sp round th sttor of priitiv thr-phs hin. Noti tht th gnti xis of h oil is prpndiulr to thir orrsponding pln. gnti xis oil -' ' ' gnti xis oil -' gnti xis oil -' ' Figur 2. Mgnti xis of thr oils displd sytrilly For lnd onditions th instntnous urrnts, for h oil, r: i = I sin ωt (1 Prodings of th 22 Arin Soity for Enginring Edution Annul Confrn & Exposition Copyright 22, Arin Soity for Enginring Edution Pg
3 i = I sin( ω t 12 (2 i = I sin( ω t 24 (3 In vtor for, qutions 1 thru 3 o: j I = I (4 I I j12 = I (5 24 j = I (6 Whr I is th xiu vlu of urrnt. Fro ltrognti thory [4] Φ inditing tht th gnti flux dnsity is proportionl to th gnti flux. If I Φ thn I nd onsquntly qutions 4, 5 nd 6 o: = sin( ωt = sin( ωt 12 = sin( ωt 24 (7 (8 (9 Th gnti flux dnsitis of qutions 7, 8 nd 9 r lotd in sp s inditd in figur 2. Thn w hv th following vtors to solv in ordr to dtrin th rsultnt gnti flux dnsity. = sin( ωt j = sin( ωt 12 = sin( ωt 24 j12 j24 (11 (12 (1 Finlly th rsultnt gnti flux dnsity is otind y dding th vtors fro qutions 1, 11 nd 12. Thn : R = + + (13 As xpl, lt ωt qul to zro, thn r os: r = sin( j j12 sin( sin( 24 3 j9 r = 2 Figur 3 indits th vtor ddition nd th rsultnt vtor t zro dgrs. j24 Prodings of th 22 Arin Soity for Enginring Edution Annul Confrn & Exposition Copyright 22, Arin Soity for Enginring Edution Pg
4 Rsultnt gnti xis R gnti xis oil -' ' gnti xis oil -' ' ' Figur3. Rsultnt gnti flux t ωt= At ωt =9 o th rsultnt gnti flux is qul to rsultnt gnti flux. o j 1.5. Figur 4 shows th vtor digr for this Mgnti xis oil -' ' ' gnti xis oil -' ' Rsultnt gnti xis R gnti xis oil -' Prodings of th 22 Arin Soity for Enginring Edution Annul Confrn & Exposition Copyright 22, Arin Soity for Enginring Edution Pg
5 Figur4. Rsultnt gnti flux for ωt = 3 o Siultion This progr dvlopd using Mtl TM tht prfors th lultions for r for 36 dgrs. Th nition prt of it nd th grphis indit vry lrly th vtor rottion of ltri gnti filds in n ltri thr-phs hin. % % CALIFORNIA STATE UNIVERSITY NORTHRIDGE % ROTATING MAGNETIC FIELD % JANUARY 22 % % CALCULATE NET MAGNETIC FIELD PRODUCED Y A THREE-PHASE STATOR % MAX = 1 % FREQUENCY = 6 HZ % GENERATE THREE COMPONENTS MAGNETIC FIELD hz = 6 t = :1/6:1/6 og = hz*2* j = sqrt(-1 x = sin(og*t*(os( + j*sin( y = sin(og * t - 2*pi/3*(os(2*pi/3+ j*sin(2*pi/3 z = sin(og*t + 2*pi/3*(os(-2*pi/3+ j*sin(-2*pi/3 r = x + y + z irl = 1.5*(os(og*t+ j*sin(og*t % plot th gnitud nd dirtion of th rsultnt gnti fild r 1.5 * x*ngl ii = 1:lngth(t k = 1 %suplot(3,1,1 %plot([ rl(x(ii],[ j*(x(ii],'k','linwidth',2 %suplot(3,1,2 %plot(irl, 'k' Prodings of th 22 Arin Soity for Enginring Edution Annul Confrn & Exposition Copyright 22, Arin Soity for Enginring Edution Pg
6 %suplot(3,1,3 %plot (r,t iii = 59 for iii=1:11 k(1=x(iii k(2=y(iii k(3=z(iii k(4=r(iii %suplot(1,2,1 opss(k pus(.1 nd %suplot(1,2,2 %polr(ngl(r,s(r %opss(r Output Th grphil output is shown low. This grph is n nitd grph tht n only shown if w run th progr. In th nti w show only th lst lultd point. Osrv tht th xiu vlu of th rsultnt phsor is 1.5 s it is inditd y th irl. Prodings of th 22 Arin Soity for Enginring Edution Annul Confrn & Exposition Copyright 22, Arin Soity for Enginring Edution Pg
7 Plot of x, y, z nd r Rotting Mgnti Fild Conlusions MATLA is vry powrful siultion pkg tht w r trying to inlud in our nrgy onvrsion urriulu. As it n sn in th output of th siultion th nuril prt is quit lr, yt, th grphil prt phsizs th rsult nd rings ho th onpt of hving onstnt vtor gnitud of1.5 for th rsultnt gnti flux rotting in sp round th sttor. Th fft tht this siultion hs hd in our studnts is grt, du to th ft tht thy gt vry otivtd to ontinu studying our ltri hins nd looking forwrd to prfor siultions tht would th thir howork or onpts xplind in lss. For furthr rsrh w will look into th rotting phsors for synhronous hins oprting t diffrnt powr ftors. Prodings of th 22 Arin Soity for Enginring Edution Annul Confrn & Exposition Copyright 22, Arin Soity for Enginring Edution Pg
8 iliogrphy [1] runo Osorno Eltril Mhins Ltur Nots, Cliforni Stt Univrsity Northridg, [2] Turn Gonn Eltril Mhins, Powr Intrntionl Prss, First Ed [3] A.E. Fitzgrld, t.l., Eltri Mhinry. MGrw-Hill, 5 th Ed [4] Syd A. Nsr, Eltri Mhins nd Powr Systs, MGrw-Hill, 1 st Ed [5] Guru nd Hiziroglu, Eltri Mhinry nd Trnsforrs, Sundrs Collg Pulishing, 2 nd Ed RUNO OSORNO, is profssor of ltril nd oputr nginring t Cliforni Stt Univrsity Northridg. H is th ld fulty r in th Powr Systs nd Powr Eltronis progr. Profssor Osorno hs writtn ovr 2 thnil pprs. His urrnt intrst in rsrh is Fuzzy Logi pplitions in powr ltronis nd ltri otor drivs. Prodings of th 22 Arin Soity for Enginring Edution Annul Confrn & Exposition Copyright 22, Arin Soity for Enginring Edution Pg
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