Chapter 04.00E Physical Problem for Electrical Engineering Simultaneous Linear Equations
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1 hpter 04.00E Phyicl Prblem fr Electricl Egieerig Simulteu Lie Equti Prblem Sttemet Three-phe ytem e the rm fr mt idutril pplicti. pwer i the frm f vltge d curret it delivered frm the pwer cmpy uig three-phe ditributi ytem d my lger ld e three-phe ld i the frm f mtr, cmprer, r imil. Surce d ld c be cfigured i either wye (where urce r ld e cected frm lie t eutrl/grud) r delt (where urce r ld e cected frm lie t lie) cfigurti d mixig betwee the type i cmm. Figure 1 hw the geerl wirig f wye-wye three-phe ytem mdelig ll f the impedce typiclly fud i uch ytem. Durig the typicl lyi udertke i mt rcuit textbk, it i umed tht the ytem i etirely blced. Thi me tht ll the urce, lie, d ld impedce e equivlet, tht i, b c b c N bn N Uder thi umpti, the rcuit i the typiclly reduced t igle-phe equivlet rcuit mdel d the reultt rcuit i lved with igle lp equti. Wht hppe, hwever, whe the ytem i ublced? Typiclly becue the three ld impedce, d N N N e t equl, which reult i differet curret thrugh ech ld, i fte meured i term f the percetge differece betwee the ld curret E.1
2 04.00E.2 hpter 04.00E Figure 1 Three-Phe Wye-Wye Sytem with Pitive Phe Sequece retig imblce i three-phe ytem i t ll tht difficult. ider mll buie pertig i ilted leg f the pwer grid tht lclized pect f ld e t blced by ther eighbrig ld. Let ume tht the primy ld fr thi ytem i 45 kv et f three-phe mtr t 0.8 pwer fctr lggig d, further, tht the electri tht did the wirig fr the lightig mitkely cected tw bk f light t the phe, e t the phe d e t the phe etig imblce i the ytem. Ech f thee lightig ld i 1500 W. The ld fr thi ytem i hw i Figure 2.
3 Phyicl Prblem fr Electricl Egieerig: Simulteu Lie Equti 04.00E.3 N Figure 2 Mdel f the Sytem Ld The impedce f ech f the ld c be determied by exmiig the pwer cumed i ech phe f the ytem : / j j / kv : / j j / : / kv V vertig thee t impedce uig the frmul S with V 120V yield: / R j j N N / R j j N / R j j Fr the ret f thi lyi we will ume tht ech phe f the ytem h equivlet urce d lie impedce f R j j d tht the grud retur wire h impedce f R j j Thi yield the equivlet rcuit f Figure 3. 2 kv
4 04.00E.4 hpter 04.00E Figure 3 Equivlet ircuit Mdel fr the Wrkig Prblem The rcuit c be lyzed uig three lp equti uig the curret, b, d c hw i Figure 3. Fr lp thi yield the cmplex equti: Lp : V 0 R j R j b c R j 0 with lp d yieldig imil reult. umig tht ur imulteu equti lver i t cpble f hdlig cmplex umber we c tur the lp equti it tw epte -cmplex equti ddreig bth the rel d imgiy pt. Uig j d cllectig term yield: Rel : R R R R R 120 (1) mgiy : R R R R R 0 (2) pplyig the me lyi t the d lp yield the remig equti fr the ytem. Rel : R R R R R 60 (3) mgiy : R R R R R (4) Rel :
5 Phyicl Prblem fr Electricl Egieerig: Simulteu Lie Equti 04.00E.5 R R R 60 R R (5) mgiy : R R R R R (6) Thi yield ytem f ix lie equti d ix ukw,,,,, d tht c be lved by y cvetil me. Thi i hw i mtrix frm i Figure Figure 4 mplete Sytem f Equti (7) Oce the curret e kw it i imple prcedure t determie the vltge the three mtr termil V N, VN, dvn uig Ohm Lw. VN j R j VN j R j VN j R j T better evlute the imblce, the percetge differece i curret thrugh the ctul ld elemet i mre fte dered. The bet wy t thik f thi three peple pullig d puhig tgether. f they d t pull d puh i blce, thig c becme utble. the ce f three-phe mtr, thi c reult i igifict wbble with repdig we i the beig d ther pt. T determie the curret thrugh ech ld i g cmputed uig Ohm Lw. VN VN VN ld, ld ld j where 3 w cmputed elier. D t frget tht the lightig ld e epte. QUESTONS 1. Wht wuld be the rmificti f lvig the prblem directly uig the three cmplex lie equti? uld we d it uig pprch like Gu-Jrd Elimiti? Wht but me f the ther umericl methd ued t lve imulteu lie equti? 2. Thi prblem i ly iteretig if the grud retur leg N i -zer. Otherwie, we hve three lp equti tht e cmpletely idepedet f ech ther d c be lved directly. Why i tht the ce?
6 04.00E.6 hpter 04.00E 3. much mre iteretig d prcticl prblem ccur whe the mtr ld i Delt cfigurti. Sice it de t hve the grud retur lie i the middle it reult i dditil lp equti. Sketch the equivlet rcuit fr ytem with Wye urce d mix f Delt d Wye ld. Write the et f equti tht reult frm thi ytem. Slve them. SMULTNEOUS LNER EQUTONS Tpic Simulteu Lie Equti Summy Three phe ld i ytem Mjr Electricl Egieerig uthr Hery Welch Dte Jue 30, 2005 Web Site
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