Parallel connection. The autotransformer. Regulation. Three- to two-phase conversion.

Transcription

1 4.1 Lecture 4 Trsformer Coectios rllel coectio. The utotrsformer. Regultio. Three- to two-phse coversio. Trsformers i rllel To be suitble for prllel opertio the trsformers must hve comptible polrity, rtio d phse displcemet. For exmple, trsformers of ideticl rted voltge coected Yd1 d Dy1 re comptible for prllel opertio, but trsformers Yd1 d Yd11 re ot, s their phse displcemets differ by 60. f ll of the bove coditios of comptibility re stisfied, the prllel coected trsformers my still fil to shre the lod i fir mer. The shrig of lod depeds o: (i) (ii) Trsformer impedces mll differeces i rtio Let: omil voltge rtio (primry/secodry) (4.1) ctul rtio of trsformer ctul rtio of trsformer (4.) (4.3) (4.4) ower Circuit Theory 011

2 4. equivlet circuit of two trsformers i prllel is show below: circ Figure 4.1 Two trsformers i prllel O o lod ( 0 ) we hve circultig curret betwee the trsformers (usig per-uit vlues): circ (4.5) where: per uit differece i rtio (4.6) Whe the we hve blced bridge circuit, d hece the circultig curret is idepedet of the lod curret. Eve if the trsformer impedces re ot equl, becuse the lod impedce is much lrger th the trsformer impedces, the couplig betwee the lod curret d the circultig curret is wek d c be igored. ower Circuit Theory 011

3 4.3 Therefore we c first clculte the brch currets by ssumig 0, d the dd the circultig curret to obti the totl curret i ech trsformer: (4.7) circ circ Exmple Two 50 M 11 / 66 k trsformers (T1 d T) re operted i prllel, supplyig lod of 80 MW d 60 Mvr t 66 k. The rectces r 6% d 8% for T1 d T respectively, resistces re cosidered egligible. oth trsformers hve o-lod tp chgers providig rtio djustmets i 3 discrete steps of 0.65% over the totl rge of 90% to 110% of the omil rtio. Clculte: ) er-uit mgitudes of trsformer currets whe both trsformers re operted t omil rtio. b) Tp djustmets to optimize lod shrig. olutio: Use 50 M, d lod voltge 1 j0 p.u. (referece). bse ) oth trsformers t omil rtio Lod power 80 j j1. p.u. Lod curret * 1.6 j p.u. j8 1.6 j j p.u. (overlod) j6 j8 ower Circuit Theory 011

4 4.4 j6 j6 j8 1.6 j j p.u. b) Rtio djustmet circ 1 j j0.06 j j0.686 j j0.514 j Note tht we c lter oly the rective compoets, the rel compoets re fixed. Optimum lod shrig occurs whe. uttig x , we hve: x x 0.38 olvig we obti x 0. 38, hece p.u. 3.33% % 0.65% 5.3 icremet differece, with. Usully the tps re provided i the H widig. Therefore icrese the 66 k tp settig of Trsformer by 3 icremets, d decrese Trsformer 1 by icremets. The 50.65% p.u. ower Circuit Theory 011

5 4.5 The phsor digrm is show below: circ before chgig tps - circ circ before chgig tps circ j0.06 j0.08 j0.3 p.u j j j j j j0.3 ower Circuit Theory 011

6 4.6 The utotrsformer Cosider the step-up utotrsformer with N primry turs d N secodry turs, with N N. There is svig of mterils by mkig the primry serve s prt of the secodry. N - - N N Figure 4. tep-up utotrsformer ctul costructio there re two widigs: commo widig with turs, d series widig with rrgemet re: N N N turs. The resos for this Differet coductor sizes Low lekge rectce Mechicl forces o short circuit Usig the idel trsformer model, the mmf blce is: N N N (4.8) or: N N (4.9) lso: ower Circuit Theory 011

7 4.7 N N (4.10) Therefore: 1 (4.11) Let the currets d voltges be the rted vlues for the utotrsformer. The the rted power is: (4.1) f the coils were rewired so tht the trsformer hs two seprte widigs with turs N d N N, the the rted power would be: e 1 1 for 1 (4.13) e is the equivlet two-widig rtig of the utotrsformer. ecuse e the utotrsformer c be mde physiclly smller th covetiol two-widig trsformer of the sme power rtig. ower Circuit Theory 011

8 4.8 Now let the equivlet two-widig trsformer hve lekge impedce ohms referred to the secodry widig, which is the series widig of the trsformer. N - - N N Figure 4.3 tep-up utotrsformer with lekge impedce The per-uit lekge impedce of the utotrsformer, bsed o its rtig is ow: uto p.u. (4.14) The per-uit lekge impedce of the equivlet two-widig trsformer, bsed o its rtig e is: w uto e p.u. (4.15) ower Circuit Theory 011

9 4.9 Therefore: uto 1 - p.u. for 1 w (4.16) Note tht the sme fctor 1 ppers i Eqs. (4.16) d (4.16). To obti step-dow utotrsformer 1, swp the primry d secodry termils. The: e for 1 (4.17) imilrly: uto 1 1- w p.u. for 1 (4.18) ower Circuit Theory 011

10 4.10 Regultio er-uit regultio of trsformer is defied s: NL NL p.u. (4.19) where: NL = secodry voltge o o lod = secodry voltge o specified lod (4.0) Whe the mgetisig curret d cpcitive effects re egligible, s is the cse i power trsformer, d we use per-uit vlues for ll qutities, the Eq. (4.19) gives: p.u. (4.1) equivlet circuit d phsor digrm i this cse is show below: R jx jx R Figure 4.4 Trsformer regultio ower Circuit Theory 011

11 4.11 Usully is the rted voltge, d the mgitude of the impedce voltge is smll, i.e. R jx d 1p.u. The d re erly i phse, d the followig pproximtio c be used: R R cos jx X si Figure 4.5 pproximte trsformer regultio Therefore: Rcos X si (4.) ower Circuit Theory 011

12 4.1 Three-hse to Two-hse Coversio Two-phse systems were the first polyphse systems. Two-phse geertors were used i the first hydroelectric power sttio t Nigr Flls i The dvtge of two-phse electricl power ws tht it llowed for simple, selfstrtig electric motors. Three-phse systems hve replced two-phse systems for commercil distributio of electricl eergy, but two-phse circuits re still foud i certi cotrol systems. Two sigle-phse trsformers re coected s show. This is kow s the cott coectio. Teser N =0 N (optiol) C Mi b b C C b Figure 4.6 cott coectio The cott coectio c be foud i few specil pplictios, such s cott coectios bck-to-bck for low-power three-phse trsformers. ower flow c be i either directio (3-phse to -phse or i reverse), but we will ssume the three-phse side to be the primry. ower Circuit Theory 011

13 4.13 The two sigle-phse trsformers hve equl power rtig: Mi trsformer turs rtio widig. N. Cetre tp, lbelled, o primry N Teser trsformer turs rtio 3 1. Optiol tp t primry turs. 3 ssume symmetricl three-phse voltge source is coected to the primry, d use idel trsformer modellig. For the momet, ssume the eutrl (tppig poit N) to be discoected. The secodry voltges re: b 3 C j 3 3 (4.3) (4.4) givig: b j (4.5) Now ssume tht the secodry is coected to blced two-phse lod (or two equl sigle-phse lods) with some rbitrry power fctor. The. b j ice the primry voltges re blced, the phsor digrm shows the tppig poit N to be t zero voltge, therefore o eutrl curret c exist whe the optiol eutrl coectio is mde. N 0 lwys, d C. lso, C b j j 3. ower Circuit Theory 011

14 4.14 Now we hve two simulteous equtios: C C j 3 (4.6) where: 3 (4.7) olvig Eq. (4.6) gives h d C h. This proves tht for blced twophse secodry currets we hve blced three-phse primry currets. From Eq. (4.7) d Eq. (4.3) we lso coclude tht the primry power fctor is the sme s the secodry power fctor. ower Circuit Theory 011

15 4.15 ummry To be suitble for prllel opertio the trsformers must hve comptible polrity, rtio d phse displcemet. rllel coected trsformers shre the lod i mer which depeds o the trsformer impedces d differeces i the turs rtio. utotrsformers provide higher rtig (compred to stdrd trsformer) for the sme cost. They therefore operte more efficietly sice the losses re the sme s for the ordiry coectio. The loss of electricl isoltio betwee the high- d low-voltge sides is usully the reso they re ot used extesively. Regultio of trsformer is mesure of the voltge vritio betwee full-lod d o-lod coditios. The cott coectio c be used to covert betwee three-phse system d two-phse system. Refereces Crmo, J.: ower Circuit Theory Notes, UT, Truupold, E.: ower Circuit Theory Notes, UT, ower Circuit Theory 011

16 4.16 Exercises 1. Two trsformers of power rtigs d hve ideticl voltge rtigs d widig coectios. Their per-uit impedces d re bsed o d respectively. rove tht the trsformers, operted i prllel, shre the lod ccordig to their cpbility if k trsformer hvig impedce of 1. j4.% is prlleled with similr 300 k trsformer hvig impedce of 1.5 j5.0%. oth impedces re referred to their respective k rtigs. The combied lod is 600 k t 0.75 p.f. lg. Fid the complex power delivered by ech trsformer. 3. The trsformer i Q6 hs tppig switch to chge the umber of H turs i icremets of 1.5 % of the pricipl tppig. Two of such trsformers, lbelled 1 d, re operted i prllel (icludig the tertiry widigs). Trsformer 1 is o the pricipl tppig, while trsformer is o the ext higher tp. Clculte the resultig circultig currets. 4. cott coected trsformers re used to supply 100 to two idepedetly switched sigle-phse lods. The three-phse supply voltge is 415, d ech sigle-phse lod is 5 k, uity power fctor. Determie the three-phse iput currets (mgitude d phse gle) for the followig coditios: () oth lods coected. (b) Lod o the mi trsformer. (c) Lod o the teser trsformer oly. ower Circuit Theory 011