PDH ourse E450 (4 PDH) Understnding Three-Phse Trnsformers Rlph Fehr, Ph.D., P.E. 2014 PDH Online PDH enter 5272 Medow Esttes Drive Firfx, V 22030-6658 Phone & Fx: 703-988-0088 www.pdhonline.org www.pdhenter.om n pproved ontinuing Edution Provider
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org Understnding Three-Phse Trnsformers Rlph Fehr, Ph.D., P.E. Trnsformers re indispensile omponents of power system. By llowing voltge nd urrent levels to e djusted, trnsformers solve mny prtil prolems tht would otherwise e very diffiult to overome. For exmple, if power is generted t resonle genertion voltge, sy 18 kv, without the ility to hnge voltge levels, ll iruits supplied y tht genertor would hve to operte t 18 kv. Generting power t sustntilly higher voltge is not prtil due to insultion design issues involving rotting mhines. For given mount of power, n 18 kv iruit must rry ertin numer of mperes of urrent. f the genertor is rted t 500 MV, the three-phse urrent would e 500,000 kv 18 kv 3 16,038. Suh high urrent would produe exessive voltge drop nd losses. Both of these issues ould e overome y reduing the urrent to more mngele level. When this is done, the voltge will inrese y the sme rtio tht the urrent ws deresed. The term step-up trnsformer is used to desrie suh trnsformer, s this desries wht the trnsformer does to the voltge. The reltionships involving voltge nd urrent will e explored in the next setion. The del Trnsformer Trnsformers re est understood y first onsidering nd omprehending the idel trnsformer. n idel trnsformer is simply two oils of wire tht re eletrilly isolted from eh other nd re mgnetilly oupled. Being idel, the mgneti oupling etween the oils is perfet. n other words, ll the mgneti flux produed y one winding links the other winding, so there is no stry flux. The self-indutne nd the resistne of eh winding re lso disregrded. Only the mutul indutne etween the windings is onsidered. Per Frdy s lw, when time-vrying mgneti flux uts ondutor formed into oil with N turns, voltge is indued ross the oil. dφ V N dt Lenz s lw determines the polrity of tht voltge. The negtive sign in Frdy s lw indites tht the indued voltge is developed with polrity tht would produe urrent (if the oil eing onsidered is prt of losed iruit) tht would produe mgneti flux tht would oppose the originl flux. Fig. 1 shows, y the right-hnd rule, tht the urrent produes flux tht opposes the originl flux. This stipultion is neessry for the onservtion of energy. 2014 Rlph Fehr Pge 2 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org Originl Flux Produed Flux + V _ FGURE 1 Lenz s Lw The time-vrying flux is produed y time-vrying urrent flowing in one of the oils. By onvention, the oil ross whih voltge is pplied is lled the primry winding, nd the oil ross whih voltge is indued is lled the seondry winding. Sine it my not e known whih winding will hve voltge pplied ross it, the use of primry nd seondry n e risky. Trnsformer mnufturers lwys designte the windings high voltge nd low voltge to void this issue. n n idel trnsformer, the rtio of the numer of turns on the high-voltge winding (N H ) to the numer of turns on the low-voltge winding (N L ) defines the turns rtio (n) for the trnsformer. The turns rtio lso desries the rtio of the voltges ross the two windings, so is lso referred to s the voltge rtio. f eh winding in the idel trnsformer is prt of losed iruit, the rtio of the urrents tht flow through eh winding is inversely proportionl to the turns rtio. These rtios re shown in Fig. 2. H L VH V L n N N H L N H VH N H L N L VL N L H FGURE 2 The del Trnsformer NH N L Sine the idel trnsformer is lossless, the power hndled y the high-voltge winding must equl the power hndled y the low-voltge winding. V H H = V L L 2014 Rlph Fehr Pge 3 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org lso, n impedne put ross the low-voltge windings ppers to e muh higher impedne when viewed from the high-voltge iruit. This n e seen y writing the highvoltge iruit quntities in terms of the low-voltge iruit quntities: V H = n V L 1 n H L Dividing the voltge eqution y the urrent eqution gives Z H = n 2 Z L. Trnsformer iruit Model The hrteristis tht were disregrded in the idel trnsformer model n e ounted for in lumped-prmeter iruit model to relistilly represent n tul trnsformer. Figure 3 shows iruit model representing n tul trnsformer. H R H X H R L XL L VH G B VL N H ideln L FGURE 3 Trnsformer iruit Model The resistnes R H nd R L represent the resistnes of the high-voltge nd the low-voltge windings, respetively. The indutive retnes X H nd X L represent the self-indutnes of the high-voltge nd the low-voltge windings, respetively, multiplied y the rdin frequeny. X H = ωl H = 2πf L H X L = ωl L = 2πf L L G prlleled with B mke up the ore rnh of the model. G is the shunt ondutne representing hysteresis nd eddy urrents. B is the shunt suseptne whih represents the mgnetiztion urrent for the ore. The totl urrent flowing through oth G nd B voltge dependent nd is the no-lod urrent of the trnsformer, nd n e mesured using the open iruit test, whih mesures H with the low-voltge terminls open-iruited. 2014 Rlph Fehr Pge 4 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org The losses in trnsformer re very low ompred to the mount of power the trnsformer n hndle; in ft, trnsformers re mong the most effiient mhines tht n e uilt, with effiienies of lrge trnsformers often exeeding 99%. But even t loss rte of 1%, the losses in trnsformer should not e overlooked. Think out the losses in 750 MV genertor step-up trnsformer with n effiieny of 99% -- tht s 7.5 MW of losses (enough to supply out 1500 homes with eletriity!) The idel trnsformer provides the turns rtio for the model. lthough firly strightforwrd, the turns rtio omplites lultions involving systems ontining trnsformers. n the next setion, lultion method will e introdued tht mkes eh trnsformer turns rtio in the system pper to e 1:1, therey effetively eliminting the trnsformers nd mking the entire eletril system to pper s single iruit. Tht lultion method is the per-unit system. The Per-Unit System of lultion The omplitions used y the trnsformer turns rtio re tully quite serious nd need to e nlyzed. onsider the following one-line digrm, showing trnsformer with turns rtio n supplied from us operting t voltge V H nd drwing urrent H. The trnsformer, then, delivers urrent L to the low voltge us, whih opertes t voltge V L. lod impedne Z L is onneted to the low voltge us. V H H V L L n FGURE 4 One-Line Digrm Z L The first step in nlyzing the system shown in Fig. 4 is to onvert the one-line digrm to iruit. To do this, n equivlent iruit must reple the trnsformer one-line symol. The iruit shown in Fig. 3 would work, ut for mny types of lultions, simpler model would suffie. For exmple, the ore rnh of the trnsformer equivlent iruit n e ignored when doing voltge drop lultions, s doing so hs little effet on the nswer. nd for short iruit lultions, the winding resistnes n lso e ignored, sine the very indutive fult urrent produes little voltge drop ross the resistnes. nd sine the X/R rtio of trnsformer tends to e lrge, X>>R, giving further justifition to neglet the resistnes. So for fult lultions, the trnsformer equivlent iruit ould e shown s single indutive retne (the X on one side of the idel trnsformer ould e refleted to the other side of the 2014 Rlph Fehr Pge 5 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org idel trnsformer, then the two retnes ould e omined in series to yield single retne. This iruit is shown in Fig. 5. V H H V L X xfmr-equiv. L FGURE 5 iruit representtion of Figure 4 V H V L Z L Exmining Fig. 5, serious prolem is relized: the urrent entering the indutne representing the trnsformer is different from the urrent leving the indutne ( H L ). This seems like violtion of Kirhhoff s urrent Lw (KL), ut KL pplies to n eletri iruit. The trnsformer is tully two eletri iruits linked y mgneti iruit. While tht tehnility explins the pprent KL violtion, the ft tht H L is still prolem. Rell the three equtions desriing how mgnitudes hnge when moving from one eletri iruit of trnsformer to the other: V H = n V L, 1 H L, nd Z H = n 2 Z L. n There is one speifi vlue of turns rtio tht mkes high-side urrent nd low-side urrent equl, nmely n=1. f n=1, then the iruit in Fig. 5 n e nlyzed. n ft, if n=1, then V H = V L, H = L, nd Z H = Z L. But other thn uilding trnsformer with n equl numer of turns on oth windings (s with n isoltion trnsformer), how n n=1? Perhps some mthemtil trikery n e used to mke n pper to hve vlue of 1. This trikery is the per-unit system. The per-unit system uses dimensionless quntities insted of eletril units (volts, mps, ohms, wtts, et.). The per-unit system, when properly pplied, hnges ll trnsformer turns rtios to 1. This wy, s one moves from the high-voltge iruit of the trnsformer to the low-voltge iruit, the voltge, urrent, nd impedne re ll unffeted, mking the two eletri iruits pper s one. The per-unit system relies on the estlishment of four se quntities. The required se quntities re: se power, se voltge, se urrent, nd se impedne. The se quntities re seleted y the person doing the prolem. The numeri vlues of these se quntities re ritrry ny numer will work exept zero. This is euse the per-unit system is liner mthemtil trnsformtion. The prolem to e solved is trnsformed into the per-unit system, solved, nd then trnsformed k to eletril quntities. Sine the trnsformtion k to eletril quntities is the inverse of the trnsformtion into the per-unit 2014 Rlph Fehr Pge 6 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org system, the se quntities whih define the trnsformtions n ssume ny non-zero numeri vlue. Of the four se quntities, two re mthemtilly independent. The other two re then defined y the first two. For exmple, if voltge nd urrent re ssumed to e independent, power n e thought of s the produt of voltge nd urrent, nd impedne s the quotient of voltge nd urrent. The typil wy to pply the per-unit system is to ritrrily ssign the power nd voltge ses, then using the mthemtil reltionships etween the eletril quntities to determine the urrent nd impedne ses. Typilly, the se power (kv or MV se) is seleted ritrrily, often s 10 or 100 MV. The power se is onstnt through the entire system. The se voltge (kv se) is frequently ssigned s the nominl operting voltge t given point in the system. t every voltge trnsformtion, the se voltge is djusted y the trnsformer turns rtio. Therefore, mny different se voltges my exist throughout the system. The proper seletion of voltge ses throughout the system effetively mkes the trnsformer turns rtios equl to one, thus removing the omplitions introdued y the trnsformer turns rtios. fter the power nd voltge ses re hosen, the other two se quntities n e lulted from the estlished ses y using the formuls Bse urrent Bse kv 3 3 Bse kv LL nd Bse kvl L Bse mpedne, Bse MV 3 where the susripts 3Φ nd L L respetively denote three-phse power nd line-to-line voltge. tul eletril quntities re onverted to dimensionless per-unit quntities using the formul tul Quntity Per Unit Quntity. Bse Quntity Often, per-unit quntity must e onverted from prtiulr se to new se. Towrd tht end, we use the reltionship kv Bse Old kv Bse New Per Unit Quntity New Per Unit Quntity Old. kv Bse New kv Bse Old 2 2 2014 Rlph Fehr Pge 7 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org onsider the power system shown elow: Utility G Xn Rted Voltges: 1 Xn T1 T2 Utility Soure: 12.47 kv Genertor: 460 V F GURE 6 Exmple Power System M1 2 T3 Trnsformers T1: 13.2 kv / 4.16 kv T2: 460 V / 4 kv T3: 4.16 kv / 480 V Motors M1: 4000 V M2: 460 V M2 f the se voltge t Bus 1 is ritrrily hosen to e 4.16 kv, wht must e the se voltges t the utility onnetion point, the genertor terminls, nd Bus 2 so tht the turns rtio of eh trnsformer ppers to e 1? Strting with the given se voltge t Bus 1 nd pplying the turns rtio of trnsformer T1, 13.2 kv Vse (Utility) 4.16 kv 13.2 kv. 4.16 kv Strting with the given se voltge t Bus 1 nd pplying the turns rtio of trnsformer T2, 460 V Vse (Genertor ) 4.16 kv 478.4 V. 4000 V Strting with the given se voltge t Bus 1 nd pplying the turns rtio of trnsformer T3, 480 V Vse (Bus 2) 4.16 kv 480 V. 4160 V So the se voltge t one rndom us is ritrry, ut every other us hs its se voltge defined y the ritrrily hosen voltge se multiplied y the produt of the turns rtios of eh trnsformer one must pss through to get from the ritrrily-ssigned us to the us eing onsidered. This proess of ssigning se voltges mkes eh turns rtio pper to e 1. 2014 Rlph Fehr Pge 8 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org Trnsformer mpedne trnsformer s impedne is n importnt prmeter tht helps determine its suitility for given pplition. Trnsformer impedne is expressed in perent sed on the self-ooled kv rting. The impedne is numerilly equl to the perentge of rted voltge tht would hve to e pplied to the primry terminls to use rted urrent to flow from the short-iruited seondry term inls. This onept is illustrted in Fig. 7, where x equls the nmeplte vlue %Z of the trnsformer. x x FGURE 7 Definition of Trnsformer mpedne Note tht the perent impedne, whih is simply the per unit impedne times 100, is numerilly the sme whether referred to the high-voltge iruit or the low-voltge iruit. f the impedne were expressed in ohms, the ohmi vlue would differ y ftor of n 2 when viewed through the trnsformer. lso, the ohmi impedne would vry tremendously, over orders of mgnitude, when ompring very lrge trnsformers to very smll trnsformers, while the perent impedne vries over rther smll rnge, sy 2% to 14%, regrdless of the size of the trnsformer. Trnsformer impedne vries ording to design prmeters, prtiulrly kv rting nd si lightning impulse insultion level (BL). n generl, impedne inreses with kv rting, nd lso inreses with BL. Most susttion lss trnsformers hve n impedne in the 5.5% to 7.5% rnge, ut speifi designs n use the impedne to lie outside this rnge. Sine trnsformer impedne is ritil for lulting voltge drop nd short iruit mgnitudes, tul nmeplte dt should e used whenever possile. ommon Three-Phse onnetions lthough mny three-phse trnsformer onnetions, two prtiulr onnetions, the delt nd the wye, mke up most of the three-phse onnetions ommonly enountered. t is importnt 2014 Rlph Fehr Pge 9 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org delt-wye trnsformtion n e omplished in two different wys. Three identil singlephse trnsformers n e externlly onneted to form delt-wye nk. Or, three sets of windings onneted in delt nd three sets of windings onneted in wye n e wound on single three-phse ore. While the two designs re very muh different in their mgneti iruits, they re eletrilly identil. While mny delt-wye trnsformers my e onstruted s s ingle three-phse unit, it my e lerer to visulize nd nlyze the devie s three single-phse trnsformers. to understnd the dvntges nd disdvntges of eh onnetion type so the proper trnsformer onnetions n e determined for given pplition. The Delt onnetion The delt onnetion is usully three-wire, ungrounded iruit. Delt iruits work well to supply lned lods, suh s three-phse motors. Requiring only three ondutors, deltonneted iruits re eonomil to onstrut. s the lod urrents supplied y the delt iruit eome unlned, the voltges lso eome unlned. Unlned voltges n led to opertionl prolems, prtiulrly for three-phse motors. lso, eing n ungrounded fult, ground fult protetion is omplited nd ground fults require more time to ler thn they would on omprle grounded system. Delt-onneted iruits lok zero-sequene urrents nd triplen hrmonis y providing pth in whih they n irulte, whih n e dvntgeous on systems ontining lrge mgnitudes of hrmoni urrents. nd delt- only one three-phse voltge the line-to-line onneted iruits provide voltge. The Wye onnetion The wye onnetion is usully four-wire, grounded iruit. Wye iruits re desirle to supply single-phse lods, s these lods frequently eome unlned. s the lod urrents supplied y the wye iruit eome unlned, the voltges tend to remin well lned, s the urrent imlne is returned to the soure y the fourth, or neutrl, ondutor. Wyeonneted windings hve no effet on zero-sequene urrents or triplen hrmonis. The ground referene provides fst detetion nd lering of ground fults, nd wye-onneted systems provide oth line-to-line nd line-to-neutrl voltges. Single-phse lods re usully onneted line-to-neutrl to keep the phse voltges lned. The Delt-Wye Trnsformer Mny of the dvntges of the delt onnetion nd the wye onnetion n e relized when onstruting three phse trnsformer hving one set of windings onfigured in delt nd the other set of windings onfigured in wye. This onfigurtion, the delt-wye trnsformer, is the most ommon three-phse trnsformer onfigurtion. 2014 Rlph Fehr Pge 10 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org FGURE 8 Three Single-Phse Trnsformers in Delt-Wye Bnk FGURE 9 ore nd oil ssemly of Three-Phse Delt-Wye Trnsformer Figure 10 shows delt-wye trnsformer in four different wys: s onnetion digrm, s shemti digrm, s phsor digrm, nd mthemtilly. Some reltionships my e seen more esily in one of the four formts thn the others. By hving ll four in front of you, you will more esily lern how the delt-wye trnsformer works. H1 H1 H2 Sutrtive Polrity X1 X2 X0 X1 n n B H1 H3 H2 X1 X0 X2 X3 n n B H2 H1 X1 X2 B B H3 H2 H1 X2 X1 X3 = B = = - - - = x = x = x 3 30 3 30 3 30 H2 X2 B FGURE 10 Delt-Wye Trnsformer 2014 Rlph Fehr Pge 11 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org Strting with the onnetion digrm on the left, oth single-phse nd three-phse implementtions n e seen. Looking t the lrge dshed retngle nd ignoring the three smller dshed retngles, the three-phse implementtion is shown. The lrge dshed retngle represents the tnk of the three-phse unit. Everything inside the retngle is internl to the trnsformer. The internl onnetions re mde t the ftory nd nnot e hnges. Seven ushings provide ess to instll the trnsformer. On the left, ushings H1, H2, nd H3 re the high-voltge ushings nd provide onnetion points for the high-voltge phses. On the right, ushings X1, X2, nd X3 provide onnetion points for the low-voltge phses, nd ushing X0 provides ess to the neutrl. lterntely, the lrge dshed retngle n e ignored, nd the three smller dshed retngles represent the tnks of three single-phse trnsformers tht n lso provide the delt-wye trnsformtion. Eh single-phse trnsformer hs four ushings (H1, H2, X1, nd X2). The onnetions etween these ushings re mde up with insulted le jumpers interonneting the three trnsformers. These jumpers re lerly seen in Fig. 8. dentifying the delt- nd wye-onneted windings on the onnetion digrm is diffiult, lthough y strting t the H1 end of the red winding, moving through the red winding, then following the jumper from red H2 to lue H1, through the lue winding, then following the jumper from lue H2 to green H1, through the green winding, then following the jumper from green H2 to red H1, the high-voltge delt n e tred. Similrly, the non-polrity (X2) ends of the three low-voltge windings re tied together nd onneted to the X0 ushing, while the polrity (X1) ends of the windings re onneted to ushings X1 through X3, thus forming wye onnetion. The delt nd wye re muh more ovious on the shemti digrm in the top right of Fig. 10. To develop the phsor digrm nd the mthemtil reltionships, we must nlyze the ehvior of the delt-wye trnsformer. We n perform the nlysis using voltge-sed or urrent-sed method. Voltge, eing sutrtion of two potentil phsors (potentil differene) is very strt nd diffiult, if not impossile, to visulize. urrent on the other hnd is simply flow of eletril hrges, nd n esily e visulized. So we will develop method of urrent-sed nlysis nd pply it to the delt-wye trnsformer. This generl method n e pplied to ny type of trnsformer onnetion, regrdless of its omplexity. f voltge nlysis is needed, the urrent-sed method n still e used, nd sine Ohm s lw sttes tht voltge nd urrent re linerly dependent, the vriles n e repled with V vriles in the lst step of the nlysis, thus giving voltge nlysis of the trnsformer. To egin the urrent-sed nlysis, we strt with the eletri iruit where the line urrent equls the phse urrent. This is the wye iruit sine the line urrents (leled,, nd ) re the sme s the urrents through the windings (phse urrents), s n e seen in Fig. 11. This is not the se with the delt iruit. 2014 Rlph Fehr Pge 12 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org H1 X1 X0 n H3 X2 n B H2 X3 FGURE 11 Delt-Wye Trnsformer Shemti Digrm fter ll the urrents in the wye iruit re leled, the urrents in the delt-onneted windings n e leled. The urrent flowing in high-voltge winding is smller thn the urrent flowing in the orresponding low-voltge winding y ftor of 1/n, where n is the turns rtio. rrying ftors of 1/n through the nlysis would e umersome. f per-unit urrents re used, the urrent flowing through the high-voltge winding equls the urrent flowing through the orresponding low voltge winding, sine the per-unit system mkes the turns rtio pper to e 1. The diretion of the high-voltge winding urrents is determined y the dot onvention. f n instntneous urrent diretion is out of the dotted terminl in the low-voltge iruit, the high-voltge winding will hve urrent flowing into its dotted terminl. So, using per-unit urrents, the urrents through the high-voltge windings n e leled, s shown in Fig. 12. H1 X1 X0 n B H3 H2 X2 X3 n FGURE 12 High-Voltge Winding urrents Leled Finlly, the high-voltge iruit line urrents n e determined y writing node equtions t the orners of the delt. Upper-se susripts re used to distinguish high-voltge line urrents from low-voltge line urrents, nd the high-voltge line urrents re drwn flowing into the trnsformer, s shown in Fig. 13. 2014 Rlph Fehr Pge 13 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org B H1 H3 H2 X1 X0 X2 X3 n n B FGURE 13 High-Voltge iruit Line urrents leled t the top orner of the delt, Similrly, t the other two nodes, =. B = nd =. f lned set of positive sequene (mening the phsors re sequened --) urrents re ssumed, the reltionship etween the high-voltge line urrents nd low-voltge line urrents n e found. Let = 1/0, = 1/240, nd = 1/120. Sustituting these vlues into the node equtions developed ove, = = 1/0 1/240 = 3 /30 B = = 1/240 1/120 = 3 /270 = = 1/120 1/0 = 3 /30. onstruting phsor digrm showing the three low-voltge iruit line urrents nd the three high-voltge line urrents, s in Fig. 14, revel the mthemtil reltionships etween the urrents. FGURE 14 Phsor Digrm of Delt-Wye Trnsformer B t n e seen tht eh ounterprt y ftor of high-voltge iruit phsor is longer thn its low-voltge iruit 3 nd leds it y 30. These reltionships re shown in Fig. 15. 2014 Rlph Fehr Pge 14 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org = B = = - - - = x = x = x 3 30 3 30 3 30 FGURE 15 Mthemtil Reltionships etween High-Voltge iruit Line urrents nd Low-Voltge iruit Line urrents n this exmple, the high-voltge phsors led the low-voltge phsors y 30. This onstitutes n NS Stndrd delt-wye trnsformer. Lgging high-voltge phsors would onstitute n NS Non-Stndrd delt-wye trnsformer. By using oth stndrd nd non-stndrd trnsformers, nd y onneting the phses to different ushings (insted of to H1, B to H2, nd to H3 s in this exmple), delt-wye trnsformer n produe phse shifts of 30, 90, 150, 210, 270, nd 330. Plese note tht if voltge reltionships re needed insted of urrent reltionships, the vriles simply need to e hnged to V vriles. The per-unit urrent-sed nlysis of the delt-wye trnsformer tht ws just ompleted n e used to nlyze ny type of trnsformer onnetion. Let s pply it to more omplited onnetion. Figure 16 shows three-winding trnsformer used to supply six-phse system from three-phse soure. V H H L1 L2 V L1 V L2 FGURE 16 Three-Winding Trnsformer One-Line Digrm V H is three-phse system whih supplies the delt-onneted high-voltge windings. The two wye-onneted low-voltge windings differ in phse y 180 reltive to eh other, so the two low-voltge uses V L1 nd V L2 tully omprise six-phse system. High phse order system suh s this re exellent for supplying eletroni lod, suh s vrile-frequeny drives, s they produe less hrmoni ontent thn three-phse system would. The shemti digrm for this trnsformer is shown in Fig. 17. Plese note tht lthough the neutrl is drwn s prt of us V L1, it is tully shred y oth V L1 nd V L2. 2014 Rlph Fehr Pge 15 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org V L1 B n V H f d f V L2 e d FGURE 17 Three-Winding Trnsformer Shemti Digrm When trnsferring the trnsformer winding urrents from the low-voltge iruits to the highvoltge iruit, reful ttention must e pid to the winding polrities. Looking t the lue winding, urrent leves the dotted terminl, so n equl per-unit urrent must enter the dotted terminl on the high-voltge side. But in iruit V L2, urrent d enters the dotted terminl of the lue winding, so urrent d must leve the lue winding in the high-voltge iruit. This mens nd d oppose eh other in the high-voltge, nd tht they re 180 out of phse in the low-voltge iruits. So in the high-voltge iruit, net urrent of d will e shown flowing into the dotted terminl. The other windings re treted similrly, s shown in Fig. 18. e V L1 B - e - f n V H - d f d f V L2 e d FGURE 18 Shemti Digrm with ll urrents Leled e 2014 Rlph Fehr Pge 16 of 17
www.pdhenter.om PDHonline ourse E450 www.pdhonline.org Now, lned set of positively-sequened phsors n e ssumed for one of the low-voltge iruits. Strting with V L1, we n ssume = 1/0, = 1/240, nd = 1/120. Sine the polrities of the V L2 windings re opposite from the V L1 windings. We know the orresponding urrents re 180 out of phse from their V L1 ounterprts. This mens d = 1/180, e = 1/60, nd f = 1/300. Using these vlues, the high-side line urrents n e found y writing node equtions t eh orner of the delt. = ( d ) ( e ) = + e d = 1/0 + 1/60 1/240 1/180 = 2 3 /30 B = ( e ) ( f ) = + f e = 1/240 + 1/300 1/120 1/60 = 2 3 /270 = ( f ) ( d ) = + d f = 1/120 + 1/180 1/0 1/300 = 2 3 /150 Now, phsor digrm n e onstruted showing ll nine line urrents. e d f B FGURE 19 Three-Winding Trnsformer Phsor Digrm 2014 Rlph Fehr Pge 17 of 17