Chapter 8 Three-Phase Power System and Three-Phase Transformers

Transcription

1 Chpter 8 Three-Phse Power System nd Three-Phse Trnsformers Single-phse systems re dequte for residentil pplitions up to n pprent power S of 5 kva (per residene), nd 25 kva single-phse pole trnsformers supply power to out 5 10 residenes. In rurl res irrigtion pplitions require single-phse motor rtings of up to 50 kva. Aove this rting three-phse power equipment must e employed for ommeril nd industril pplitions. 8.1 Three-Phse AC Systems Genertion, trnsmission nd hevy power utiliztion is sed on three-phse systems, introdued more thn 100 yers go [1] following the invention of the two-phse (or poly-phse) indution motor y Tesl [2]. Although Edison proposed DC trnsmission t tht time, it ws only 50 yers lter when the first DC trnsmission lines were ommissioned. Due to stility prolems ssoited with trnsmission of AC eletril power over distnes of more thn 1,000 miles, the utiliztion of high-voltge DC trnsmission tie lines eme populr nd osteffetive (see Chp. 3) Genertion of Three-Phse Voltges Consider the elementry 3-phse, 2-pole genertor of Fig. 8.1; on the rmture (memer, where voltges re indued) there re three oils (phse elts) 0, 0, nd 0, whose xes re displed 120 in spe from eh other. Note tht this text uses in this setion lokwise (w) nottion for the phse (elt) sequene of three-phse system. When the field winding residing on the rotor (moving memer) of Fig. 8.1 is exited nd rotted, voltges will e indued in the three phse elts in ordne with Frdy s lw. If the field winding struture is so designed tht the flux f is distriuted sinusoidlly round the rotor periphery (y), the flux linkges of phse elt will vry sinusoidlly with time E.F. Fuhs nd M.A.S. Msoum, Power Conversion of Renewle Energy Systems, DOI / _8, # Springer SieneBusiness Medi, LLC

2 340 8 Three-Phse Power System nd Three-Phse Trnsformers phse mgneti xis N wt = 0 N-turn oil phse mgneti xis N N-turn oil wt = 90 f e f e phse N N mgneti xis f N -turn field oil phse mgneti xis N N -turn field oil phse mgneti xis f N mgneti xis of rotor field (oiniding with phse mgneti xis) Fig. 8.1 Three-phse, two-pole genertor with two rotor positions; () ot ¼ 0,() ot ¼ 90 Fig. 8.2 Three-phse voltge system e 0 (t), e 0 (t), nd e 0 (t) in time domin e (t) e (t) e (t) wt l 0 ¼ NfðtÞsin ot (8.1) nd ording to Frdy s lw the indued voltge in phse elt 0 is e 0ðtÞ ¼dl 0=dt ¼ onfos ot þ Nðdf/dtÞsin ot ¼ speed voltge þ trnsformer voltge: (8.2) The osinusoidl speed voltge (flux f is sttionry with respet to rotting rotor, nd does not hnge in time with respet to rotor) e 0ðtÞ ¼Nfo os ot ¼ E mx os ot (8.3) will e indued in phse elt 0 (Fig. 8.2). Beuse phse elt 0 is displed y 120 mehnil degrees (for 2-pole onfigurtion) from phse elt 0 one onludes tht whtever hppens to phse elt t time t ¼ 0 orresponding to time ngle of ot ¼ 0 will hppen to phse elt 0 fter time of t ¼ 120 /o or when time ngle of ot ¼ 120 hs elpsed; the sme hppens to the voltge

3 8.1 Three-Phse AC Systems 341 indued in phse elt 0. As shown elow, these three voltge wves e 0 (t), e 0 (t) nd e 0 (t) will e displed 120 eletril degrees in time s result of the phse elts eing displed y 120 mehnil degrees in spe. Note, for two-pole (p ¼ 2) mhine the mehnil degrees re identil with the eletril degrees. This is not so for mhine with pole numers lrger thn two (e.g., p ¼ 4, 6, 8...), s will e shown in Chp. 10. The sinusoidl time-domin voltges e 0 (t), e 0 (t), nd e 0 (t) n e represented in the phsor domin (see Fig. 8.3) s p e 0ðtÞ ¼ ffiffi p 2 Ee j0 ¼ ffiffiffi 2 Eff 0 ; (8.4) p e 0ðtÞ ¼ ffiffiffi p 2 Ee j120 ¼ ffiffi 2 Eff 120 ; (8.4) p e 0ðtÞ ¼ ffiffi 2 Ee j240 ¼ p ffiffiffi 2 Eff 240 ¼ p ffiffiffi 2 Eff þ 120 ; (8.4) where E ¼ Nfo/ 2 is the rms vlue of the indued phse voltge [see (8.3)]. A three-phse AC system employs voltge soures (see Fig. 8.4) whih onsist of three voltges equl in mplitude nd displed y eletril phse ngles of e (t) CW Fig. 8.3 Phsor digrm of indued three-phse voltge system e (t) 120 e (t) Fig. 8.4 ( ) Three-phse voltge soures nd lods; () Three seprte systems, () three-phse system in D-onnetion, where 0 ¼, 0 ¼, 0 ¼, nd () D three-phse system with D nd Y-onneted lods, where N ¼ 0 ¼ 0 ¼ 0 e (t) e - - e lod (t) lod _ e lod (t) lod e lod e e e _ lods e lod N lods

4 342 8 Three-Phse Power System nd Three-Phse Trnsformers Fig. 8.5 Y nd D onfigurtions for rmture windings representing voltge soures 360 /3 ¼ 120 from one nother. A three-phse system requires the lest mount of opper/luminum for the trnsmission lines to trnsmit given mount of power from point A to point B. The three individul voltges of three-phse soure my eh e onneted to its own independent lod iruit. We would hve then three seprte single-phse systems (Fig. 8.4). Alterntively, symmetril eletri onnetions n e mde etween the three voltges nd the ssoited lod iruitry to form three-phse system (Figs. 8.4, ). Note tht the word phse now hs two distint menings: l It my refer to portion of three-phse system or iruit or l In the fmilir stedy-stte iruit theory, it my e used in referene to the ngulr displement or phse shift etween voltge nd urrent phsors. The three phses of the rmture winding (representing voltge soures) my e interonneted in two possile wys, s shown in Fig. 8.5: in Y-onnetion with or without neutrl (N) resulting in 4-wire or 3-wire three-phse system, respetively, or D onnetion resulting in 3-wire three-phse system. The three-phse voltges re equl in mplitude nd re phse-displed in time y 120 eletril degrees, generl hrteristi of lned three-phse system. Furthermore, the lod impedne in ny one phse (see Fig. 8.4) is equl to tht in either of the other two phses, so tht the resulting phse urrents re equl in mplitude nd phse displed from eh other y 120 eletril degrees. Likewise, equl rel power nd equl retive power flow in eh phse exist. It is importnt to note tht only lned systems re treted in this setion nd none of the methods developed or onlusions rehed pply to unlned systems s they my our in rel-life power pplitions Three-Phse Voltges, Currents nd Power When the three phses of n rmture winding re Y-onneted, the phsor digrm of voltges is tht shown in Fig The polrity of the voltges is defined y the indies, for exmple e N ðtþ ¼ e N (t), where is the positive terminl of phse, nd N is the neutrl N, tht is, the negtive terminl: e N ðtþ; e N (t); e N ðtþ re the line-to-neutrl voltges or phse voltges, nd e (t); e (t); e (t) re lled line voltges or line-to-line voltges. From Fig. 8.6 one otins pffiffi 3 je (t) j ¼ 2 je N (t) jsin 60 ¼ 2 je N (t) j 2 ¼ p ffiffi 3j en (t) j; (8.5)

5 i A (t) A -e N (t) e (t) e (t) e N (t) i (t) N i (t) e (t) e N (t) en (t) i (t) e (t) i B (t) i C (t) B C -e N (t) e N (t) i (t) e (t) e N (t) f - i(t) e N (t) i(t) e (t) -e N (t) e N (t) -e N (t) e N (t) o o o 30 e(t) = 2 en(t) sin60 o 3 = 2 e (t) 3 N = en(t) 2 isoseles tringle i(t) e (t) i(t) i(t) e (t) e (t) ia(t) ib(t) ic(t) A B C e (t) ia(t) i(t) - i(t) - i(t) e (t) f i i(t) A(t) - i(t) i(t) e (t) o ia (t) = 2 i(t) sin 60 = 2 i(t) = 3 i (t) 3 2 o 30 o 60 i(t) o 30 isoseles tringle Fig. 8.6 () Phsor digrm of phse (e.g., e N ðtþ) voltges, line-to-line ðe:g:;e ðtþþ voltges, nd phse (line) ðe:g:;i (t) ¼ i A ðtþþ urrents for Y-onnetion () Phsor digrm of phse (line-to-line) ðe:g:;e ðtþþ voltges nd phse (e.g.,i (tþ) nd line (e.g.,i A (t)) urrents for D-onnetion

6 344 8 Three-Phse Power System nd Three-Phse Trnsformers tht is, for Y-onnetion the line-to-line voltge mgnitude e.g., je (t) j ¼ je (t) j ¼ je (t) j ¼ V L L,is 3 times the phse voltge mgnitude e.g., je N (t) j ¼ je N (t) j ¼ je N (t) j ¼ V ph, nd the line urrent mgnitude i A (t) ¼ i A (t) ¼ i A (t) ¼ IL is identil with the phse urrent mgnitude i (t) ¼ i (t) ¼ i (t) ¼ Iph. When the three phses re D-onneted, one n show tht the lineto-line voltge mgnitude V L-L is identil with the phse voltge mgnitude V ph nd the line-urrent mgnitude I L is 3 times the phse urrent mgnitude I ph. When the three phses of the windings re D-onneted, the phsor digrm of voltges is tht shown in Fig Constny of Power Flow in Three-Phse System For oth the Y nd D- onneted systems it will e shown tht the totl of the instntneous power p(t) for ll three phses of lned three-phse system is onstnt nd does not vry with time, s the voltges nd urrents do. For lned three-phse system one n write for the phse voltges [3] e N ðtþ ¼ e N ðtþ ¼ e N ðtþ ¼ p ffiffiffi 2 Eph os ot; p ffiffiffi 2 Eph osðot 120 Þ; p ffiffi 2 Eph osðot 240 Þ; (8.6) (8.6) (8.6) nd the phse urrents leding the phse voltges y n ngle of f i ðtþ ¼ i ðtþ ¼ i ðtþ ¼ p ffiffiffi 2 Iph osðot þ fþ; p ffiffiffi 2 Iph osðot 120 þ fþ; p ffiffi 2 Iph osðot 240 þ fþ: (8.7) (8.7) (8.7) The totl instntneous power of ll three phses is p(t) ¼ p ðtþþp ðtþþp ðtþ; (8.8) with osos ¼ (1/2){os(-)os()} one otins p ðtþ ¼e N (t)i ðtþ ¼E ph I ph fosð2ot þ fþþosfg; (8.9) p ðtþ ¼e N (t)i ðtþ ¼E ph I ph fosð2ot þ f 240 Þþosfg; (8.9)

7 8.1 Three-Phse AC Systems 345 p ðtþ ¼e N (t)i ðtþ ¼E ph I ph fosð2ot þ f 480 Þþosfg; (8.9) or p(t) ¼ 3E ph I ph os f þ E ph I ph fosð2ot þ fþþosð2ot þ f 240 Þ þ osð2ot þ f 480 Þg: (8.10) It will e shown elow tht the seond term of (8.10) dds up to zero! Proof: fosð2ot þ fþþosð2ot þ f 240 Þþosð2ot þ f 480 Þg ¼ 0; (8.10) with os( ) ¼ osos sinsin nd os( g) ¼ osososg sinsinosg sinossing ossinsing Eqution 8.10 eomes in n expnded mnner: os2otosf sin2otsinf os2ot osfos120 sin2ot sinf os120 sin2ot osf sin120 os2ot sinf sin120 os2ot osf os120 sin2ot sinfos120 sin2ot osf sin120 os2ot sinf sin120 ¼ 0, or with os120 ¼ (1/2): os2ot osf sin2ot sinf (1/2)os2ot osf (1/2)sin2ot sinf (1/2) os2otosf (1/2) sin2otsinf ¼ 0. This shows tht the time-dependent powers of lned three-phse system dd up to zero nd the power flow within suh system is P ¼ p(t) ¼ 3E ph I ph os f ¼ independent of time! (8.11) The totl instntneous power for lned three-phse system is onstnt nd is equl to 3 times the verge power per phse. This is of prtiulr dvntge in the opertion of three-phse (or poly-phse) motors, for exmple, euse it mens tht the shft-power output is onstnt nd tht no torque pulstions rise due to the time-dependent sinusoidl vritions of AC urrents nd voltges. In the following [4] the time-dependent power reltions of single, two, nd three-phse systems will e visulized. Figures 8.7, illustrte the voltge, urrent nd power t unity-power ftor, nd lgging power ftor (onsumer nottion), e.g., osf ¼ orresponding to F ¼ 45, of single-phse system, respetively. Figure 8.8 shows the voltge, urrent nd power t unity-power ftor of two-phse system. Figures 8.9, depit the voltge, urrent nd power t unity-power ftor, nd lgging power ftor (onsumer nottion), e.g., osf ¼ 0.80 orresponding to F ¼ 36.87, of three-phse system, respetively. The influene of 5% imlne is studied in Fig ndthtoflossofphsein Fig

8 346 8 Three-Phse Power System nd Three-Phse Trnsformers p(t) [degree] i(t) v(t) p(t) v(t) i(t) [degree] Fig. 8.7 () Voltge, urrent nd power s funtion of time for single-phse system t unitypower ftor of osf ¼ 1.0 (Courtesy of John M. Cowdrey, P.O. Box 847, Lyons, CO 80540, [4]) () Voltge, urrent nd power s funtion of time for single-phse system t lgging-power ftor (onsumer nottion) of osf ¼ (Courtesy of John M. Cowdrey, P.O. Box 847, Lyons, CO 80540, [4]) 8.2 Three-Phse Trnsformers [5 7] Equivlene of Y nd D-Conneted Ciruits For D (Fig. 8.12) nd Y (Fig. 8.12)-onneted lod iruits the line-to-line voltge (V L-L ), line urrent (I L ), power ftor (osf), totl rel power (P tot ), totl retive power (Q tot ) nd totl pprent power or voltmpere power (S tot ) re

9 8.2 Three-Phse Trnsformers 347 Fig. 8.8 Powers s funtion of time for two-phse system t unity-power ftor of osf ¼ 1.0 (Courtesy of John M. Cowdrey, P.O. Box 847, Lyons, CO 80540, [4]) preisely equl for oth ses. In other words, onditions viewed from the terminls A, B, nd C of Figs. 8.12, re identil nd one nnot distinguish etween the two iruits from their terminl quntities. It will lso e seen tht the impedne (resistne nd retne) per phse of the lned Y onnetion is extly onethird of the orresponding vlues per phse of the lned D onnetion. Consequently, lned D onnetion n e repled y lned Y onnetion, providing tht the iruit impednes per phse oey the reltion Z Y ¼ ð1=3þz D : (8.12) A generl omputtionl sheme for lned iruits n e sed entirely upon Y-onneted iruits (or entirely D-onneted iruits) Three-Phse Trnsformer Ciruits Three single-phse trnsformers n e onneted to form three-phse nk in ny of the four wys shown in Figs d. Applition Exmple 8.1: Three-Phse (Idel) Trnsformer Connetions Figure 8.13 represents the Y-D onnetion, tht is, the primry is onneted in Y nd the seondry in D. Strting out with the primry line-to-line voltge mgnitude V p L L one otins the primry line-to-neutrl (phse) voltge mgnitude

10 348 8 Three-Phse Power System nd Three-Phse Trnsformers p totl (t) p (t) p (t) p (t) [degree] p (t) totl p (t) p (t) p (t) [degree] Fig. 8.9 () Powers s funtion of time for three-phse system t unity-power ftor of osf ¼ 1.0 (Courtesy of John M. Cowdrey, P.O. Box 847, Lyons, CO 80540, [4]) () Powers s funtion of time for three-phse system t lgging-power ftor (onsumer nottion) of osf ¼ 0.80 (Courtesy of John M. Cowdrey, P.O. Box 847, Lyons, CO 80540, [4]) V p ph ¼ Vp L L = p ffiffi 3. The primry nd seondry of phse re wound on the sme iron-ore lim nd, therefore, the primry ( V p ph ) nd seondry ( V s ph ) re in phse for the given dot mrkings of Fig If the turns rtio etween primry (N p ) nd seondry (N s ) phse windings is ¼ (N p /N s ) then V s ph ¼ Vp ph ¼ Vp L L = pffiffi 3 :

11 8.2 Three-Phse Trnsformers 349 p totl (t) p (t) p (t) p (t) [degree] Fig Powers t n imlne of 5% s pplied to three-phse system (Courtesy of John M. Cowdrey, P.O. Box 847, Lyons, CO 80540, [4]) p totl (t) p (t) p (t) [degree] Fig Powers t loss of phse s pplied to three-phse system (Courtesy of John M. Cowdrey, P.O. Box 847, Lyons, CO 80540, [4])

12 350 8 Three-Phse Power System nd Three-Phse Trnsformers I A A V AB I B B V BC I C C V CA S tot ;P tot ; Q tot ; osf Z D Z D Z D S tot ;P tot ;Q tot ; osf I A A V AB B I B Z V Y Z Y CA V BC Z Y I C C Fig (, ) Equivlene of D nd Y-onneted iruits Fig () Y-D onnetion of three-phse trnsformer () D-Y onnetion of three-phse trnsformer () D-D onnetion of three-phse trnsformer (d) Y-Y onnetion of three-phse trnsformer

13 8.2 Three-Phse Trnsformers 351 The seondry line-to-line voltge mgnitude for D-onnetion is identil with the phse voltge mgnitude V s L L ¼ Vs ph ¼ Vp L L = p ffiffiffi 3 : Strting out with the primry line urrent I p Lwhih is identil with the primry phse urrent for Y onnetion I p ph ¼ I p L one otins for the seondry phse urrent I s ph ¼ Ip ph ¼ Ip L. The seondry line urrent is then Is L ¼ p ffiffi 3 I s ph ¼ pffiffi 3 I p ph ¼ p ffiffi 3 I p L. The primry (input) pprent power is S p ¼ 3V p ph Ip ph ¼ 3 Vp L L pffiffiffi I p ph 3 ¼ p ffiffi 3 V p L L Ip L nd the seondry (output) pprent power is S s ¼ 3V s ph Is ph ¼ 3Vs L L Is ph ¼ 3 Vp L L = p ffiffi 3 I p L ¼ p ffiffi 3 V p L L Ip L ¼ Sp : This mens the input pprent power is identil to the output pprent power euse trnsformer losses hve een negleted due to the use of the idel trnsformer reltions. The reder is enourged to perform the orresponding nlysis for Figs d. Applition Exmple 8.2: Applition of Short Ciruit nd Open Ciruit Tests to 3-Phse D-Y Trnsformer nd Power Flow in 3-Phse Feeder t Blned Lod A D-Y onneted nk of three identil single phse trnsformers, with one trnsformer hving the rtings S ¼ 100 kva, V p ¼ 2,400 V, V s ¼ 120 V t 60 Hz, is supplied with power through feeder whose impedne is Z P F ¼ Rp F þ jxp F ¼ ð0:30 þ j0:80þ O per phse. The line-to-line voltge mgnitude t the sending end of the feeder is held onstnt y the utility t V send L -L ¼2,400 V. The results of short-iruit test on one of the trnsformers with its low (seondry)-voltge terminls short-iruited re V p_s ¼ 57.5 V, f ¼ 60 Hz, I p_s ¼ 41.6 A, nd P p_s ¼ 875 W. () Determine the seondry line-to-line voltge when the nk delivers rted urrent to lned three-phse unity (osf ¼ 1.0) power ftor lod. () Compute the urrents in the trnsformer primry nd seondry windings nd in the feeder wires if solid three-phse short-iruit ours t the seondry line terminls. Solution First find single-phse equivlent iruit prmeters euse test results re ville for one trnsformer (see Fig. 8.14).

14 352 8 Three-Phse Power System nd Three-Phse Trnsformers Fig Single-phse trnsformer under shortiruit ondition V p I p short-iruit Fig Single-phse equivlent iruit referred to primry without short-iruit I p V p = I s R eq jx eq V s The equivlent iruit of trnsformer of Fig referred to primry is shown in Fig The equivlent iruit prmeters re: nd R p eqd ¼ X p eqd ¼ P s ði p s I p s 875 ¼ Þ2 ð41:6þ 2 ¼ 0:5056 O; Zp eqd ¼ Vp s ¼ 57:5 ¼ 1:382 O; 41:6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz p eqd Þ2 ðr p eqd Þ2 ¼ ð1:382þ 2 ð0:5056þ 2 ¼ 1:286O: The single-phse trnsformer tested is one phse of D-onneted primry of the three-phse trnsformer, therefore, the equivlent-iruit series impedne is tht of D onnetion referred to the primry Z p eqd ¼ Rp eqd þ jxp eqd ¼ð0:5056 þ j1:286þo. The D-onneted primry of the three-phse trnsformer (see Fig. 8.16) is now trnsformed to n equivlent Y-onneted primry (see Fig. 8.17) This ove-mentioned trnsformtion will e performed y the reltion (wellknown D-Y trnsformtion for lned system) Z p eqy ¼ð1=3ÞðRp eqd þ jxp eqd Þ¼ ð0:16853 þ j0:42867þo. Now we n redue the three-phse trnsformer prolem (Fig. 8.17) to single-phse trnsformer prolem (see Fig. 8.18, with n idel trnsformer, nd Fig without n idel trnsformer) y nlyzing one phse only, nd we n proeed s we hve lerned it in Chp. 7. After suh preliminry onsidertions we re redy to nswer the questions posed. () Determintion of seondry line-to-line voltge for the lod ondition power ftor of osf ¼ 1.0, nd P trnsformer ¼ S trnsformer osf ¼ S trnsformer ¼ 100 kva ¼ 100 kw:

15 8.2 Three-Phse Trnsformers 353 Fig Three-phse trnsformer with D-onneted primry Fig Three-phse trnsformer with equivlent Y-onneted primry Fig Single-phse equivlent iruit for feeder, trnsformer (with idel trnsformer) pffiffiffi nd lod, where the sending-end voltge is mintined onstnt t V sendphy ¼ 2400= 3 ¼ 1385:68V Fig Single-phse equivlent iruit for feeder, trnsformer (without idel trnsformer) nd lod

16 354 8 Three-Phse Power System nd Three-Phse Trnsformers Fig Simplified single-phse equivlent iruit of feeder, trnsformer nd lod Fig Phsor digrm for unity power ftor, not to sle the lod urrent referred to primry side is I 0 s rted ph ¼ Srted trnsformer ¼ 100kW V 0 s rted 1385:68ff 0 0 ¼ 72:167ff 00 A; ph nd the remining nlysis n e performed sed on the simplified equivlent iruit of Fig. 8.20: The prmeters of Fig re V send ph y ¼ 1385:68V, R p tot ¼ R p F þ Rp eqy ¼ ð0:1685 þ 0:3Þ ¼0:4685O, X p tot ¼ X p F Xp eqy ¼ ð0:4286 þ 0:8Þ ¼1:2286O, nd the lod phse voltge mgnitude referred to primry V 0 s tul lod is not known, ut n e otined from the phsor digrm of Fig rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 0 tul lod ¼ V send ph y 2 X p tot I 0 rted 2 lod R p tot I 0 rted lod ¼ 1349:07V ¼ V 0 tul lod. The lod phse voltge referred to seondry is ph tul V lod ¼ V 0 tul lod N sy N py ¼ 1349:07=11:54 ¼ 116:90V: The line-to-line voltge mgnitude referred to seondry is with Fig L L tul p V lod ¼ ffiffi 3 V ph tul lod ¼ 202:47V: () At short-iruit V L L lod ¼ V 0 lod ¼ 0 s indited in Fig

17 8.2 Three-Phse Trnsformers 355 Fig Reltion etween phse nd line-to-line voltges Fig Short-iruit on lod side The short-iruit urrent referred to the primry is I p sy ¼ I p L ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1;385:68V ¼ 1; 054:4 A. With Fig the lod urrent ð0:468þ 2 þð1:228þ 2 referred to the seondry eomes I lod ¼ I s ph ¼ I p 1385:68 sy 120 ¼ 1054:4 11:54A ¼ 12:176kA. With Fig the phse urrent in the primry D is I p phd ¼ I p pffiffi L = 3 ¼ I p p sy= ffiffiffi pffiffiffi 3 ¼ 1054:4= 3 ¼ 608:78A Applition Exmple 8.3: D-(Ungrounded Y), Three-Phse Trnsformer with Diode Retifier () Perform PSpie nlysis [8 11] for the iruit of Fig. 8.24, where threephse diode retifier with filter (e.g., pitor C f ) serves lod R lod. You my ssume idel trnsformer onditions. For your onveniene you my ssume (N 1 / ) ¼ 1. The iruitp prmeters re R syst ¼ 0.01 O, X syst ¼ 0.05 f ¼ 60 Hz, v AB ðtþ ¼ ffiffi p 2 600V os ot, vbc ðtþ ¼ ffiffiffi 2 600V osðot 120 Þ, p v CA (t) ¼ ffiffi 2 600V osðot 240 Þ, idel diodes D 1 to D 6,C f ¼ 500 mf, nd R lod ¼ 10 O. Plot one or two periods of oth voltges nd urrents fter stedy-stte hs een rehed s requested in prts ) to e). () Plot the line-to-line voltges v AB (t) nd line-to-neutrl voltges v (t). Why re they different? Sujet v AB (t) nd v (t) to Fourier nlysis. () Plot nd sujet the input line urrent i AL (t) to the D primry to Fourier nlysis. The input line urrents of the primry D do not ontin the 3 rd,6 th,9 th, 12 th,...hrmonis, tht is, there re no hrmoni zero-sequene urrent omponents [12 14]. This is one dvntge of the D-(ungrounded Y) trnsformer onnetion. (d) Plot nd sujet the phse urrent i Aph (t) of the D primry to Fourier nlysis. The phse urrents of the primry D do not ontin the 3 rd,6 th,9 th,12 th,... hrmonis. (e) Plot nd sujet the output urrent i ph (t) of the Y seondry to Fourier nlysis. The output urrents of the seondry Y do not ontin the 3 rd,6 th, 9 th,12 th,...hrmonis.

18 356 8 Three-Phse Power System nd Three-Phse Trnsformers v R syst X syst B v AB (t) i AL (t) A Rsyst X syst BC (t) v CA (t) i Aph (t) N 1 R syst C X syst N 1 N 1 i ph (t) (t) v power system D / Y trnsformer retifier filter lod D 1 D 3 D 5 D 4 D 6 D 2 Cf R lod v lod (t) Fig Connetion of D/(ungrounded Y) three-phse trnsformer with diode retifier, filter, nd lod () The input progrm for the PSpie nlysis with extension.ir is listed elow. *delt/(ungrounded wye) trnsformer with diode retifier, filter, nd lod L L L Lxsys u Lxsys u Lxsys u *R meg Rxsys m Rxsys m Rxsys m L L L D Didel D Didel D Didel D Didel D Didel D Didel Rlod Clod u Rn meg VAN 1 0 sin( ) VB 0 sin( ) VCN 3 0 sin( ) *V_BC 2 3 SIN( ) *V_AB 1 2 SIN( ) *V_CA 3 1 SIN( ) R u.model Didel d(is=1m).trn 100u 200m 160m 100u ui.four V(1,2) V(11,12) I(L1) I(L4) I(Lxsys).options stol=1m hgtol=0.1m reltol=0.1 vntol=1m.options itl4=200 itl5=0.proe K14 L1 L K25 L2 L K36 L3 L end

19 8.2 Three-Phse Trnsformers 357 () Figures 8.25 nd 8.26 show the line-to-line voltges v AB (t), v BC (t), v CA (t), nd the line-to-neutrl voltges v (t), v (t), v (t), respetively. The line-toline voltges v AB (t), v BC (t), v CA (t) re sinusoidl nd re imposed y the power system while the line-to-neutrl voltges v (t), v (t), v (t) re distorted euse of the retifier limiting the voltge mplitude nd using ommuttion spikes. For v AB (t) one otins the Fourier omponents of Tle 8.1. FOURIER COMPONENTS OF TRANSIENT RESPONSE V(1,2). For v (t) one otins the Fourier omponents of Tle 8.2. FOURIER COMPONENTS OF TRANSIENT RESPONSE V(11,12). () Input line urrent i AL (t) of primry D is depited in Fig. 8.27, nd Fig illustrtes its Fourier spetrum For i AL (t) one otins the Fourier omponents of Tle 8.3. FOURIER COMPONENTS OF TRANSIENT RESPONSE I(Lxsys). 1kV 0.5 kv kv 1kV 160 time [ms] Fig Line-to-line voltges v AB (t), v BC (t), v CA (t) 1 kv 0.5 kv kv 1 kv 160 time [ms] Fig Line-to-neutrl voltges v (t), v (t), v (t)

20 358 8 Three-Phse Power System nd Three-Phse Trnsformers Tle 8.1 Fourier nlysis for v AB (t) DC COMPONENT ¼ E 02 V Hrmoni no Frequeny [HZ] Fourier omponent [V] Normlized omponent [pu] Phse [DEG] Normlized phse [DEG] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E01 Totl hrmoni distortion ¼ E 03% Tle 8.2 Fourier nlysis for v (t) DC COMPONENT ¼ E00 V Hrmoni no Frequeny [HZ] Fourier omponent [V] Normlized omponent [pu] Phse [DEG] Normlized phse [DEG] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E02 Totl hrmoni distortion ¼ E01% 300 A 200 A 100 A A 200 A 300 A 160 time [ms] Fig Input line urrent i AL (t) of primry D winding of D-Y trnsformer

21 8.2 Three-Phse Trnsformers A 100 A 0 A 0 5 th hrmoni 7 th hrmoni 11 th hrmoni 13 th hrmoni 0.2 khz 0.4 khz 0.6 khz 0.8 khz 1 khz Fig Fourier spetrum of the input line urrent i AL (t) of primry D Tle 8.3 Fourier nlysis for i AL (t) DC COMPONENT ¼ E 01 A Hrmoni no Frequeny [HZ] Fourier omponent [A] Normlized omponent [pu] Phse [DEG] Normlized phse [DEG] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E03 Totl hrmoni distortion ¼ E01% (d) The phse urrent i Aph (t) of primry D is depited in Fig. 8.29, nd Fig illustrtes its Fourier spetrum For i Aph (t) one otins the Fourier omponents of Tle 8.4. FOURIER COMPONENTS OF TRANSIENT RESPONSE I(L1). (e) The phse urrent i ph (t) of the seondry Y is depited in Fig. 8.31, nd Fig illustrtes its Fourier spetrum For i ph (t) one otins the Fourier omponents of Tle 8.5. FOURIER COMPONENTS OF TRANSIENT RESPONSE I(L4). Applition Exmple 8.4: D/Zigzg, Three-Phse Trnsformer Configurtion Figure 8.33 depits the so-lled D/zigzg onfigurtion [5] of three-phse trnsformer whih is used for supplying power to unlned lods nd threephse retifiers. You my ssume idel trnsformer onditions.

22 360 8 Three-Phse Power System nd Three-Phse Trnsformers 150 A 100 A 50 A 0 50 A 100 A 150 A 160 time [ms] Fig Phse urrent i Aph (t) of primry D winding of D-Y trnsformer 120 A 100 A 75 A 50 A 25 A 0 A 0 5 th hrmoni 7 th hrmoni 11 th hrmoni 13 th hrmoni 0.2 khz 0.4 khz 0.6 khz 0.8 khz 1 khz Fig Fourier spetrum of phse urrent i Aph (t) of primry D Tle 8.4 Fourier nlysis for i Aph (t) DC COMPONENT ¼ E 01 A Hrmoni no Frequeny [HZ] Fourier omponent [A] Normlized omponent [pu] Phse [DEG] Normlized phse [DEG] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E03 Totl hrmoni distortion ¼ E01%

23 8.2 Three-Phse Trnsformers A 100 A 50 A 0 50 A 100 A 150 A 160 time [ms] Fig Phse urrent i ph (t) of seondry Y winding of D-Y trnsformer 100 A 75 A 50 A 25 A 0 A 0 5 th hrmoni 7 th hrmoni 11 th hrmoni 13 th hrmoni 0.2 khz 0.4 khz 0.6 khz 0.8 khz 1 khz Fig Fourier spetrum of phse urrent i ph (t) of seondry Y Tle 8.5 Fourier nlysis for i ph (t) DC COMPONENT ¼ E 02 A Hrmoni no Frequeny [HZ] Fourier omponent [A] Normlized omponent [pu] Phse [DEG] Normlized phse [DEG] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E03 Totl hrmoni distortion ¼ E01%

24 362 8 Three-Phse Power System nd Three-Phse Trnsformers C primry X A 4 A V x3x4 seondry A B A V H1H2 H A 1 H A 2 H C 2 N 1 H B 1 N 1 N 1 C V H1H2 H C 1 H B 2 B V H1H2 B V x3x4 neutrl B V x1x2 X B N 3 2 X A X A 3 2 A V X A x1x2 1 X C 1 N X B X C X B C 2 V x1x2 X C 3 X C 4 C V x3x4 X B 4 Fig Connetion of D /zigzg, three-phse trnsformer with the definition of primry nd seondry voltges C I LC primry X A 4 seondry A B I LA I LB I A H A 1 H A 2 H C 2 N 1 H B 1 N 1 N 1 I C I B H C 1 H B 2 I 2 X B 4 X B N 3 2 neutrl X A 3 X A 2 X A 1 I 1 X B 1 X B 2 I X C 1 X C 2 X C 3 X C 4 I R lod I lod V lod Fig Connetion of D/zigzg, three-phse trnsformer with the definition of urrent for line-to-line lod () Drw phsor digrm of the primry nd seondry voltges when there is no-lod on the seondry side. For your onveniene you my ssume (N 1 / ) ¼ 1. The D/zigzg three-phse onfigurtion is used for feeding unlned lods nd three-phse retifiers. Even when only one line-to-line lod (e. g., R lod )of the seondry is present s indited in Fig the primry line urrentsi LA,I LB, nd I LC will e lned euse the line-to-line lod is distriuted to ll three (single-phse) trnsformers. This is the dvntge of D/zigzg onfigurtion. () If there is resistive line-to-line lod on the seondry side I lod ¼ 10A present s illustrted in Fig drw phsor digrm of the primry nd seondry urrents. For your onveniene you my ssume tht the sme voltge definitions pply, s in Fig nd (N 1 / ) ¼ 1. You my ssume idel trnsformer onditions. () Repet the nlysis of () ifthere is resistive line-to-neutrl lod on the seondry side ¼ 10A present, s illustrted in Fig. 8.35; tht is, drw I lod

25 8.2 Three-Phse Trnsformers 363 C I LC primry X A 4 seondry I lod n A B I LA I LB I A H A 1 H A 2 H C 2 N 1 H B 1 N 1 N 1 I C I B H C 1 H B 2 I X B 4 neutrl X B 3 X B 2 X A 3 X A 2 X A 1 X B 1 X C 1 I X C 2 X C 3 X C 4 R lod I lod V lod Fig Connetion of D/zigzg, three-phse trnsformer with the definition of urrent for line-to-neutrl lod C A B I LC I LA I LB I A H A 1 H A 2 H C 2 N 1 H B 1 primry N 1 N 1 I C I B H C 1 H B 2 I 2 X B 4 neutrl X B N 3 2 X A 4 X A 3 I 2 X A 2 X A I 1 1 I X C 1 1 X B X C 2 1 X B I 1 2 seondry X C 3 X C 4 I 2 I lod I lod neutrl I lod power system delt / zig-zg trnsformer symmetri lod Fig Connetion of D/zigzg, three-phse trnsformer with the definition of urrents for unlned line-to-neutrl lods phsor digrm of the primry nd seondry urrents s defined in Fig For your onveniene you my ssume tht the sme voltge definitions pply s in Fig nd (N 1 / ) ¼ 1. In this se the lod is distriuted to two (single-phse) trnsformers. (d) Repet the nlysis of () if there is resistive unlned lod on the seondry side I lod ¼ 30A; I lod ¼ 20A; I lod ¼ 10A present s illustrted in Fig. 8.36; tht is, drw phsor digrm of the primry nd seondry urrents s defined in Fig For your onveniene you my ssume tht the sme voltge definitions pply s in Fig nd (N 1 / ) ¼ 1. Wht onlusion n you drw from the phsor digrms? (e) Perform PSpie nlysis for the iruit of Fig where three-phse diode retifier with filter (e.g., pitne C f ¼ 500 mf) serves the lod R lod. You my ssume idel trnsformer onditions. For your onveniene you my

26 364 8 Three-Phse Power System nd Three-Phse Trnsformers B v AB (t) R syst X syst primry X A 4 X A 3 seondry i ph (t) A R X H A syst syst 1 v BC (t) v CA (t) i Aph (t) R X H A syst syst 2 C H C 2 N 1 N 1 N 1 H B 1 H C 1 H B 2 X B 3 X A 2 X A 1 X C 1 C X X B 2 1 X B 2 X C 3 X C 4 v (t) D 1 D3 D5 D4 D6 D2 C f R lod v lod (t) X B 4 power system delt / zig-zg trnsformer retifier filter lod Fig Connetion of D/zigzg, three-phse trnsformer with diode retifier, filter, nd lod R lod ssume (N 1 / ) ¼ 1. The iruit p prmeters re R syst ¼ 0.01 O, X syst ¼ 0.05 f¼ 60 Hz, v AB (t) ¼ ffiffi p 2 600V os ot, vbc ðtþ ¼ ffiffiffi 2 600V osðot 120 Þ, p v CA (t) ¼ ffiffi 2 600V osðot 240 Þ, idel diodes D 1 to D 6, nd R lod ¼ 10 O. Plot one or two periods of either voltge or urrent fter stedy stte hs een rehed s requested in the following prts. (f) Plot nd sujet the line-to-line voltges v AB (t) nd v (t) to Fourier nlysis. Why re they different? (g) Plot nd sujet the input line urrent i AL (t) of the D primry to Fourier nlysis. The input line urrents of the primry D do not ontin the 3 rd,6 th, 9 th,12 th,... (h) Plot nd sujet the phse urrent i Aph (t) of the D primry to Fourier nlysis. Explin why the phse urrents of the primry D do not ontin the 3 rd,6 th, 9 th,12 th,... (i) Plot nd sujet the output urrent i ph (t)) of the zigzg seondry to Fourier nlysis. The output urrents of the seondry zigzg do not ontin the 3 rd,6 th, 9 th,12 th,... (j) Plot nd sujet the output voltge v lod (t) to Fourier nlysis. Solution () Phsor digrm of the primry nd seondry voltges t no-lod is shown in Fig () Phsor digrm of the primry nd seondry urrents if there is resistive lineto-line lod on the seondry side I lod ¼ 10A is depited in Fig Note, I LA ¼ I A I C, I LB ¼ I B I A, I LC ¼ I C I B, I A ¼ I, I B ¼ I 1 þ I 2, nd I C ¼ I. () Phsor digrm of the primry nd seondry urrentsif there is resistive lineto-neutrl lod on the seondry side I lod ¼ 10A is depited in Fig Note, I LA ¼ I A, I LB ¼ I B I A, I LC ¼ I B, I A ¼ I, I B ¼ I, nd I C ¼ I ¼ 0. (d) Phsor digrm of the primry nd seondry urrents if there is resistive unlned lod on the seondry side I lod ¼ 30A; I lod ¼ 20A;

27 8.2 Three-Phse Trnsformers 365 A A V H1H2 B referene diretion of phse A C V H1H2 C B V H1H2 referene diretion of phse C V A 4 V x3x C 2 V x1x B 2 V x1x n A 2 V x1x V C 4 V x3x referene diretion of phse B B 4 V x3x V Fig Phsor digrm of the primry nd seondry voltges t no-lod. You my use hexgonl pper referene diretion of phse referene diretion of phse I I C I LB I 2 I B I 1 n I A I LC I I LA referene diretion of phse Fig Phsor digrm of the primry nd seondryurrents if there is resistive line-to-line lod on the seondry side I lod ¼ V =Rlod ¼ 10A I I LB I B I I A I LC n I LA Fig Phsor digrm of the primry nd seondry urrents if there is resistive line-toneutrl lod on the seondry side I lod ¼ V n =Rlod ¼ 10A I lod ¼ 10AÞ is shown in Fig Note, I 1 ¼ I lod ¼ 20A ¼ 4 units, I 2 ¼ I lod ¼ 30A ¼ 6 units, I 1 ¼ I lod ¼ 10A ¼ 2 units, I 2 ¼ I lod ¼ 20A ¼ 4 units, I 1 ¼ I lod ¼ 30A ¼ 6 units, I 2 ¼ I lod ¼ 10A ¼ 2 units, I LA ¼ I A I C ¼ I 1 þ I 2, I LB ¼ I B I A ¼ I 1 þ I 2, nd I LC ¼ I C I B ¼ I 1 þi 2.

28 366 8 Three-Phse Power System nd Three-Phse Trnsformers I LC I C I 2 I 2 I A I B I 1 I LB I LA I 2 Fig Phsor digrm of the primry nd seondry urrents if thereis resistive unlned lod on the seondry side I lod ¼ 30A; I lod ¼ 20A; I lod ¼ 10A (e) The input progrm for the PSpie nlysis with extension.ir is listed elow. I 1 *delt/zigg three-phse trnsformer onfigurtion with diode retifier, filter, nd lod L L L R u Lxsys u Lxsys u Lxsys u Rxsys m Rxsys m Rxsys m L L L L L L D Didel D Didel D Didel D Didel D Didel D Didel Rlod Clod u Rn meg VAN 1 0 sin( ) VB 0 sin( ) VCN 3 0 sin( ) *V_BC 2 3 SIN( ) *V_AB 1 2 SIN( ) *V_CA 3 1 SIN( ).model Didel d(is=1u).trn 100u 200m 160m 100u ui.four V(1,2) V(12,16) I(L1) I(L5) I(Lxsys).options stol=100u hgtol=0.1m reltol=0.1 vntol=100u.options itl4=200 itl5=0.proe K145 L1 L4 L K389 L3 L8 L K267 L2 L6 L end I 1

29 8.2 Three-Phse Trnsformers kv 0.5 kv kv 1 kv 160 time [ms ] Fig Line-to-line voltges v AB (t), v BC (t), v CA (t) 3 kv 2 kv 1 kv 0 1 kv 2 kv 3 kv 160 time [ms] Fig Line-to-line voltges v (t), v (t), v (t) (f) Figures 8.42 nd 8.43 show the line-to-line voltges v AB (t), v BC (t), v CA (t), nd the line-to-line voltges v (t), v (t), v (t), respetively. The line-to-line voltges v AB (t), v BC (t), v CA (t) re sinusoidl nd re imposed y the power system while the line-to-line voltges v (t), v (t), v (t) re distorted euse of the retifier limiting the voltge mplitude. For v AB (t) one otins the Fourier omponents of Tle 8.6. FOURIER COMPONENTS OF TRANSIENT RESPONSE V(1,2). For v (t) one otins the Fourier omponents of Tle 8.7. FOURIER COMPONENTS OF TRANSIENT RESPONSE V(12,16). (g) Input line urrent i AL (t) of primry D is depited in Fig For i AL (t) one otins the Fourier omponents of Tle 8.8. FOURIER COMPONENTS OF TRANSIENT RESPONSE I(Lxsys). (h) The phse urrent i Aph (t) of primry D is depited in Fig For i Aph (t) one otins the Fourier omponents of Tle 8.9. FOURIER COMPONENTS OF TRANSIENT RESPONSE I(L1). (i) The phse urrent i ph (t) of the seondry zigzg-y winding is depited in Fig For i ph (t) one otins the Fourier omponents of Tle FOURIER COMPONENTS OF TRANSIENT RESPONSE I(L5).

30 368 8 Three-Phse Power System nd Three-Phse Trnsformers Tle 8.6 Fourier nlysis for v AB (t) DC COMPONENT ¼ E 02 V Hrmoni no Frequeny [HZ] Fourier omponent [V] Normlized omponent [pu] Phse [DEG] Normlized phse [DEG] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E00 Totl hrmoni distortion ¼ E -03% Tle 8.7 Fourier nlysis for v (t) DC COMPONENT ¼ E 01 V Hrmoni no Frequeny [HZ] Fourier omponent [V] Normlized omponent [pu] Phse [DEG] Normlized phse [DEG] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E02 Totl hrmoni distortion ¼ E01% 800 A 400 A A 800 A 160 time [ms] Fig Input line urrent i AL (t) of primry D winding of D-Y trnsformer

31 8.2 Three-Phse Trnsformers 369 Tle 8.8 Fourier nlysis for i AL (t) DC COMPONENT ¼ E00 A Hrmoni no Frequeny [HZ] Fourier omponent [A] Normlized omponent [pu] Phse [DEG] Normlized phse [DEG] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E03 Totl hrmoni distortion ¼ E01% 400 A 200 A A 400 A 160 time [ms] Fig Phse urrent i Aph (t) of primry D winding of D-Y trnsformer Tle 8.9 Fourier nlysis for i Aph (t) DC COMPONENT ¼ E00 A Hrmoni no Frequeny [HZ] Fourier omponent [A] Normlized omponent [pu] Phse [DEG] Normlized phse [DEG] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E03 Totl hrmoni distortion ¼ E01%

32 370 8 Three-Phse Power System nd Three-Phse Trnsformers 300 A 200 A 100 A A 200 A 300 A 160 time [ms] Fig Phse urrent i ph (t) of the seondry zigzg-y winding Tle 8.10 Fourier nlysis for i ph (t) DC COMPONENT ¼ E00 A Hrmoni no Frequeny [HZ] Fourier omponent [A] Normlized omponent [pu] Phse [DEG] Normlized phse [DEG] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E02 Totl hrmoni distortion ¼ E01% 2.5 kv 2 kv 1.5 kv 1 kv 0.5 kv 0 kv 160 time[ms] Fig Output DC voltge v out (t) of retifier (j) Output DC voltge v out (t) of retifier is depited in Fig For v out (t) one otins the Fourier omponents of Tle FOURIER COMPONENTS OF TRANSIENT RESPONSE V(17,18).

33 8.2 Three-Phse Trnsformers 371 Tle 8.11 Fourier nlysis for v out (t) DC COMPONENT ¼ E03 V Hrmoni no Frequeny [HZ] Fourier omponent [V] Normlized omponent [pu] Phse [DEG] Normlized phse [DEG] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E02 Totl hrmoni distortion ¼ E02% Applition Exmple 8.5: Deomposition of Unlned Lod into Positive/ Negtive/Zero (/ /0) Sequene Components Determine (/ /0) sequene omponents of the urrent set I ¼ 5Aff 0, I ¼ 3Aff 150,I ¼ 3Aff 210. Cn this set of urrents exist in three-wire system? View positive/negtive/zero sequene omponents t Internet ddresses [18,19] Solution Aording to [13] the zero-sequene omponent is defined s I ð0þ ¼ 1 3 ð I þ I þ I Þ¼ 1 3 ð5ff 0 þ 3ff 150 þ 3ff 210 ÞA ¼ 0:067ff 0 A ¼ 0:067ff 180 A: (8.13) The zero-sequene omponent is not zero nd therefore there must e onnetion to ground nd this set of urrents nnot exist in 3-wire system. The positive-sequene system is defined s I ð1þ ¼ 1 3 ð I þ I þ 2 I Þ ¼ 1 3 ð5ff 0 þ 3ff ð150 þ 120 Þþ3ff ð210 þ 240 Þ¼1:67ff 0 A: (8.14) The negtive-sequene system is defined s I ð2þ ¼ 1 3 ð I þ 2 I þ I Þ ¼ 1 3 ð5ff 0 þ 3ff ð150 þ 240 Þþ3ff ð210 þ 120 ÞA ¼ 3:4ff 0 A: (8.15)

34 372 8 Three-Phse Power System nd Three-Phse Trnsformers These results n e heked euse the following reltion must hold for I I ¼ðI ð0þ þ I ð1þ þ I ð2þ Þ¼ð 0:067ff 0 þ 1:67ff 0 þ 3:4ff 0 Þ A ¼ 5ff 0 A: Now we n express the - nd -phse omponents in terms of the -phse omponent s follows: I ð0þ ¼ I ð0þ ¼ 0:067ff 0 A, I ð1þ ¼ 2 I ð1þ ¼ 1:67ff 120 A, I ð2þ ¼ I ð2þ ¼ 3:4ff 120 A, (8.16) nd I ð0þ ¼ I ð0þ ¼ 0:067ff 0 A, I ð1þ ¼ I ð1þ ¼ 1:67ff 120 A, I ð2þ ¼ 2 I ð2þ ¼ 3:4ff 120 A: (8.17) Relultion of the given urrents I, nd I from their omponents I ¼ I ð0þ þ 2 I ð1þ ¼ 3ff 150 A; þ I ð2þ ¼ 0:067ff 0 A þ 1:67ff 240 A þ 3:4ff 120 A I ¼ I ð0þ þ I ð1þ ¼ 3ff 210 A: þ 2 I ð2þ ¼ 0:067ff 0 A þ 1:67ff 120 A þ 3:4ff 240 A Applition Exmple 8.6: Per Unit (pu) Anlysis of Unlned Three-Phse Lod It is esy to mesure the line-to-line voltge mgnitudes of the voltges V, V, nd V with voltmeter, tht is V, V, nd V (see Fig. 8.48). The mesurement of the phse ngles of three-phse system requires n osillosope. Alterntively, one n determine the phse ngles of the voltges V, V, nd V y using the osine lw. The pprent se power is S totl se ¼ 500 kva, nd the se voltge is V L L ¼ 2; 300 V. se Fig Unlned threephse lod voltges, not to sle, lokwise phse rottion,, V = 3000V V V = 2000V = 2300V

35 8.2 Three-Phse Trnsformers 373 () Express ll voltges of Fig in pu. () Determine the phse ngles of V, V, nd V using pu vlues for the mgnitudes. This voids the use of lrge vlues (e.g., 2,300 V). () Find the symmetril omponents of V, V, V. (d) Determine the symmetril omponents of the phse voltges V n, V n,ndv n. (e) Determine the symmetril omponents of the phse urrents I, I, nd I,if the 3 lods onsist of the se impednes/resistnes R se or R pu. Solution () There re severl wys how the per-unit vlues n e defined. Aording to [13] the line-to-line voltge is tken s the se s well s the totl 3-phse pprent power. Therefore, I se ¼ S totl se V L L se ¼ V L L se Z se ¼ R se ¼ I se nd the line-to-line voltge mgnitudes re in per unit V pu 500 kva ¼ 217:4A; (8.18) 2; 300 V 2; 300 V ¼ ¼ 10:6 O; (8.19) 217:4A 2; 300 V ¼ 2; 300 V ¼ 1:0 pu, V pu 3; 000 V ¼ 2; 300 V ¼ 1:3 pu, nd V pu ¼ 2; 000 V ¼ 0:87 pu: (8.20) 2; 300 V () Figure 8.48 shows the line-to-line voltge digrm where the ngles A, B, nd C re unknown (see Fig. 8.49). One n otin these ngles y pplying the osine lw. If the voltge V pu ¼ 1ff 0 pu is tken s referene then one gets V osa ¼ pu 2 þ V pu 2 V pu 2 V pu V pu Fig Phsor digrm for the determintion of the lineto-line voltges, mgnitudes nd phse ngles, not to sle, lokwise phse rottion,, 2 ¼ 1 þ 0:872 1:3 2 ¼ 0:038 or A ¼ 87:8 : (8.21) 2 1 0:87 V V 92.2 V B=41.97 C=50.23 A=87.8 V V Re (1)

36 374 8 Three-Phse Power System nd Three-Phse Trnsformers V osb ¼ pu 2 þ V pu 2 V pu 2 V pu V pu 2 V osc ¼ pu 2 þ V pu 2 V pu 2 V pu V pu 2 ¼ 1:32 þ1 2 0:87 2 ¼ 0:744 or B ¼ 41:97 : (8.22) 21:31 ¼ 1:32 þ 0: ¼ 0:64 or C ¼ 50:23 : (8.23) 2 1:3 0:87 Knowing these ngles one n determine the line-to-line voltges from phsor digrm Fig From Figure E8:6:2 follows V pu ¼ 1ff 0 pu, V pu ¼ 1:3ff 138:03 pu; V pu ¼ 0:87ff 92:2 pu (8.24) () Clultion of the symmetril omponents of V pu, V pu, nd V pu. For the positive-sequene omponents one otins: V ð1þpu ¼ 1 h V 3 i pu þ V pu þ 2 V pu ¼ 1 ½ 3 1ff 0 þ 1:3ff ð 138:03 þ 120Þ þ 0:87ff ð92:2 þ 240Þ Š V ð1þpu V ð1þpu ¼ 1:037ff 15:03 pu; (8.25) ¼ 1 h V 3 i pu þ V pu þ 2 V pu ¼ 1 ½ 3 1:3ff 138:03 þ 0:87ff ð92:2 þ 120Þ þ 1ff 240 Š V ð1þpu V ð1þpu ¼ 1:037ff 224:96 pu; (8.26) ¼ 1 h V 3 i pu þ V pu þ 2 V pu ¼ 1 ½ 3 0:87ff 92:2 þ 1ff 120 þ 1:3ff ð 138:03 þ 240Þ Š V ð1þpu ¼ 1:037ff 104:97 pu: (8.27) For the negtive-sequene omponents one otins: V ð2þpu ¼ 1 h V 3 i pu þ 2 V pu þ V pu ¼ 1 ½ 3 1ff 0 þ 1:3ff ð 138:03 þ 240Þ þ 0:87ff ð92:2 þ 120Þ Š V ð2þpu ¼ 0:269ff 90:43 pu; (8.28)

37 8.2 Three-Phse Trnsformers 375 V ð2þpu ¼ 1 h V 3 i pu þ 2 V pu þ V pu ¼ 1 ½ 3 1:3ff 138:03 þ 0:87ff ð92:2 þ 240Þ þ 1ff 120 Š V ð2þpu V ð2þpu ¼ 0:269ff 210:38 pu; (8.29) ¼ 1 h V 3 i pu þ 2 V pu þ V pu ¼ 1 ½ 3 0:87ff 92:2 þ 1ff 240 þ 1:3ff ð 138:03 þ 120Þ Š V ð2þpu ¼ 0:269ff 29:61 pu: (8.30) For the zero-sequene omponents one otins: V ð0þpu ¼ 1 h V 3 i pu þ V pu þ V pu ¼ 0: (8.31) Note there re no zero-sequene omponents euse of the 3-wire system. (d) Determintion of the symmetril omponents of the phse voltges V pu n, V pu n, nd V pu n from the phsor digrms. By inspetion one gets from phsor digrm Fig the positive-sequene omponents of the line-to-neutrl voltges s follows: V ð1þpu n ¼ V ð1þpu ff 30 pu ¼ 1:037ff 45:03 pu; (8.32) with the se voltge of V L N V se ¼ L L se pffiffi ¼ 1328V: 3 Correspondingly, V ð1þpu n ¼ V ð1þpu ff 30 pu ¼ 1:037ff 165:04 pu; (8.33) Fig Phsor digrm for the positive-sequene omponents of the line-to-line nd line-to-neutrl voltges, not to sle, lokwise phse rottion,,