Polyphase Systems 22.1 INTRODUCTION

22 Polyphse Systems 22.1 INTRODUTION n genertor designed to develop single sinusoidl voltge for eh rottion of the shft (rotor) is referred to s single-phse genertor. If the numer of oils on the rotor is inresed in speified mnner, the result is polyphse genertor, whih develops more thn one phse voltge per rottion of the rotor. In this hpter, the three-phse system will e disussed in detil sine it is the most frequently used for power trnsmission. In generl, three-phse systems re preferred over single-phse systems for the trnsmission of power for mny resons, inluding the following: 1. Thinner ondutors n e used to trnsmit the sme kv t the sme voltge, whih redues the mount of opper required (typilly out 25% less) nd in turn redues onstrution nd mintenne osts. 2. The lighter lines re esier to instll, nd the supporting strutures n e less mssive nd frther prt. 3. Three-phse equipment nd motors hve preferred running nd strting hrteristis ompred to single-phse systems euse of more even flow of power to the trnsduer thn n e delivered with single-phse supply. 4. In generl, most lrger motors re three phse euse they re essentilly self-strting nd do not require speil design or dditionl strting iruitry. The frequeny generted is determined y the numer of poles on the rotor (the rotting prt of the genertor) nd the speed with whih the shft is turned. Throughout the United Sttes the line frequeny is 60 Hz, wheres in Europe the hosen stndrd is 50 Hz. oth frequenies were hosen primrily euse they n e generted y reltively effiient nd stle mehnil design tht is sensitive to the size of the generting systems nd the demnd tht must e met during pek

978 POLYPHSE SYSTEMS periods. On irrft nd ships the demnd levels permit the use of 400-Hz line frequeny. The three-phse system is used y lmost ll ommeril eletri genertors. This does not men tht single-phse nd two-phse generting systems re osolete. Most smll emergeny genertors, suh s the gsoline type, re one-phse generting systems. The two-phse system is ommonly used in servomehnisms, whih re self-orreting ontrol systems ple of deteting nd djusting their own opertion. Servomehnisms re used in ships nd irrft to keep them on ourse utomtilly, or, in simpler devies suh s thermostti iruit, to regulte het output. In mny ses, however, where singlephse nd two-phse inputs re required, they re supplied y one nd two phses of three-phse generting system rther thn generted independently. The numer of phse voltges tht n e produed y polyphse genertor is not limited to three. ny numer of phses n e otined y sping the windings for eh phse t the proper ngulr position round the sttor. Some eletril systems operte more effiiently if more thn three phses re used. One suh system involves the proess of retifition, whih is used to onvert n lternting output to one hving n verge, or d, vlue. The greter the numer of phses, the smoother the d output of the system. 22.2 THE THREE-PHSE GENERTOR The three-phse genertor of Fig. 22.1() hs three indution oils pled 120 prt on the sttor, s shown symolilly y Fig. 22.1(). Sine the three oils hve n equl numer of turns, nd eh oil rottes with the sme ngulr veloity, the voltge indued ross eh oil will hve the sme pek vlue, shpe, nd frequeny. s the shft of the genertor is turned y some externl mens, the indued voltges e N, e N, nd e N will e generted simultneously, s shown in Fig. 22.2. Note the 120 phse shift etween wveforms nd the similrities in pperne of the three sinusoidl funtions. e N 120 120 N N N e N N e N 120 () () FIG. 22.1 () Three-phse genertor; () indued voltges of three-phse genertor. In prtiulr, note tht t ny instnt of time, the lgeri sum of the three phse voltges of three-phse genertor is zero.

THE THREE-PHSE GENERTOR 979 e e N e N e N 0.866 E m(n) 0.866 E m(n) 0 60 p 2 p 0.5 E m(n) 3 2 p 2p 5 2 p 3p 0.5 E m(n) 7 2 p 4p qt 120 120 FIG. 22.2 Phse voltges of three-phse genertor. This is shown t qt 0 in Fig. 22.2, where it is lso evident tht when one indued voltge is zero, the other two re 86.6% of their positive or negtive mximums. In ddition, when ny two re equl in mgnitude nd sign (t 0.5E m ), the remining indued voltge hs the opposite polrity nd pek vlue. The sinusoidl expression for eh of the indued voltges of Fig. 22.2 is e N E m(n) sin qt e N E m(n) sin(qt 120 ) e N E m(n) sin(qt 240 ) E m(n) sin(qt 120 ) (22.1) E N 120 The phsor digrm of the indued voltges is shown in Fig. 22.3, where the effetive vlue of eh is determined y 120 120 E N E N 0.707E m(n) E N 0.707E m(n) E N E N 0.707E m(n) nd E N E N 0 E N E N 120 E N E N 120 y rerrnging the phsors s shown in Fig. 22.4 nd pplying lw of vetors whih sttes tht the vetor sum of ny numer of vetors drwn suh tht the hed of one is onneted to the til of the next, nd tht the hed of the lst vetor is onneted to the til of the first is zero, we n onlude tht the phsor sum of the phse voltges in three-phse system is zero. Tht is, FIG. 22.3 Phsor digrm for the phse voltges of three-phse genertor. E N E N E N E N E N E N 0 (22.2) FIG. 22.4 Demonstrting tht the vetor sum of the phse voltges of three-phse genertor is zero.

980 POLYPHSE SYSTEMS 22.3 THE Y-ONNETED GENERTOR If the three terminls denoted N of Fig. 22.1() re onneted together, the genertor is referred to s Y-onneted three-phse genertor (Fig. 22.5). s indited in Fig. 22.5, the Y is inverted for ese of nottion nd for lrity. The point t whih ll the terminls re onneted is lled the neutrl point. If ondutor is not tthed from this point to the lod, the system is lled Y-onneted, three-phse, three-wire genertor. If the neutrl is onneted, the system is Y-onneted, three-phse, four-wire genertor. The funtion of the neutrl will e disussed in detil when we onsider the lod iruit. Line I L E N I fg E N I fg N E N I fg Neutrl Line L O D I L Line I L FIG. 22.5 Y-onneted genertor. The three ondutors onneted from,, nd to the lod re lled lines. For the Y-onneted system, it should e ovious from Fig. 22.5 tht the line urrent equls the phse urrent for eh phse; tht is, I L I fg (22.3) where f is used to denote phse quntity nd g is genertor prmeter. The voltge from one line to nother is lled line voltge. On the phsor digrm (Fig. 22.6) it is the phsor drwn from the end of one phse to nother in the ounterlokwise diretion. pplying Kirhhoff s voltge lw round the indited loop of Fig. 22.6, we otin E E N E N 0 or E E N E N E N E N The phsor digrm is redrwn to find E s shown in Fig. 22.7. Sine eh phse voltge, when reversed (E N ), will iset the other two, 60. The ngle is 30 sine line drwn from opposite ends of rhomus will divide in hlf oth the ngle of origin nd the opposite ngle. Lines drwn etween opposite orners of rhomus will lso iset eh other t right ngles.

THE Y-ONNETED GENERTOR 981 E E N E E N N E N (phse voltge) E (line voltge) 120 E N α = 60 120 E N α = 60 x β = 30 E N E x E N FIG. 22.6 Line nd phse voltges of the Y-onneted three-phse genertor. FIG. 22.7 Determining line voltge for three-phse genertor. The length x is x E N os 30 3 2 E N 3 nd E 2x (2) E N 3 E 2 N Noting from the phsor digrm tht v of E 30, the result is E E 30 3 E N 30 nd E 3 E N 150 E 3 E N 270 In words, the mgnitude of the line voltge of Y-onneted genertor is 3 times the phse voltge: E L 3 E f (22.4) with the phse ngle etween ny line voltge nd the nerest phse voltge t 30. In sinusoidl nottion, e 2 E sin(qt 30 ) e 2 E sin(qt 150 ) nd e 2 E sin(qt 270 ) The phsor digrm of the line nd phse voltges is shown in Fig. 22.8. If the phsors representing the line voltges in Fig. 22.8() re rerrnged slightly, they will form losed loop [Fig. 22.8()]. Therefore, we n onlude tht the sum of the line voltges is lso zero; tht is, E E E 0 (22.5)

982 POLYPHSE SYSTEMS E E N 30 120 120 120 30 E N E E E E N 30 E () () E FIG. 22.8 () Phsor digrm of the line nd phse voltges of three-phse genertor; () demonstrting tht the vetor sum of the line voltges of three-phse system is zero. E N E N N Fixed point P E N Rottion FIG. 22.9 Determining the phse sequene from the phse voltges of three-phse genertor. E E P E Rottion FIG. 22.10 Determining the phse sequene from the line voltges of three-phse genertor. 22.4 PHSE SEQUENE (Y-ONNETED GENERTOR) The phse sequene n e determined y the order in whih the phsors representing the phse voltges pss through fixed point on the phsor digrm if the phsors re rotted in ounterlokwise diretion. For exmple, in Fig. 22.9 the phse sequene is. However, sine the fixed point n e hosen nywhere on the phsor digrm, the sequene n lso e written s or. The phse sequene is quite importnt in the three-phse distriution of power. In threephse motor, for exmple, if two phse voltges re interhnged, the sequene will hnge, nd the diretion of rottion of the motor will e reversed. Other effets will e desried when we onsider the loded three-phse system. The phse sequene n lso e desried in terms of the line voltges. Drwing the line voltges on phsor digrm in Fig. 22.10, we re le to determine the phse sequene y gin rotting the phsors in the ounterlokwise diretion. In this se, however, the sequene n e determined y noting the order of the pssing first or seond susripts. In the system of Fig. 22.10, for exmple, the phse sequene of the first susripts pssing point P is, nd the phse sequene of the seond susripts is. ut we know tht is equivlent to, so the sequene is the sme for eh. Note tht the phse sequene is the sme s tht of the phse voltges desried in Fig. 22.9. If the sequene is given, the phsor digrm n e drwn y simply piking referene voltge, pling it on the referene xis, nd then drwing the other voltges t the proper ngulr position. For sequene of, for exmple, we might hoose E to e the referene [Fig. 22.11()] if we wnted the phsor digrm of the line voltges, or E N for the phse voltges [Fig. 22.11()]. For the sequene indited, the phsor digrms would e s in Fig. 22.11. In phsor nottion, Line voltges E E 0 E E 120 E E 120 (referene)

THE Y-ONNETED GENERTOR WITH Y-ONNETED LOD 983 E P E N P E E N E E N () () FIG. 22.11 Drwing the phsor digrm from the phse sequene. Phse voltges E N E N 0 E N E N 120 E N E N 120 (referene) 22.5 THE Y-ONNETED GENERTOR WITH Y-ONNETED LOD Lods onneted to three-phse supplies re of two types: the Y nd the D. If Y-onneted lod is onneted to Y-onneted genertor, the system is symolilly represented y Y-Y. The physil setup of suh system is shown in Fig. 22.12. I L I fg E f E L V f Z 1 I fl E f I fg N E f I fg E L I L I fl I N n Z 3 Z 2 V f V f I fl E L I L FIG. 22.12 Y-onneted genertor with Y-onneted lod. If the lod is lned, the neutrl onnetion n e removed without ffeting the iruit in ny mnner; tht is, if Z 1 Z 2 Z 3

984 POLYPHSE SYSTEMS then I N will e zero. (This will e demonstrted in Exmple 22.1.) Note tht in order to hve lned lod, the phse ngle must lso e the sme for eh impedne ondition tht ws unneessry in d iruits when we onsidered lned systems. In prtie, if ftory, for exmple, hd only lned, three-phse lods, the sene of the neutrl would hve no effet sine, idelly, the system would lwys e lned. The ost would therefore e less sine the numer of required ondutors would e redued. However, lighting nd most other eletril equipment will use only one of the phse voltges, nd even if the loding is designed to e lned (s it should e), there will never e perfet ontinuous lning sine lights nd other eletril equipment will e turned on nd off, upsetting the lned ondition. The neutrl is therefore neessry to rry the resulting urrent wy from the lod nd k to the Y-onneted genertor. This will e demonstrted when we onsider unlned Y-onneted systems. We shll now exmine the four-wire Y-Y-onneted system. The urrent pssing through eh phse of the genertor is the sme s its orresponding line urrent, whih in turn for Y-onneted lod is equl to the urrent in the phse of the lod to whih it is tthed: I fg I L I fl (22.6) For lned or n unlned lod, sine the genertor nd lod hve ommon neutrl point, then V f E f (22.7) In ddition, sine I fl V f /Z f,the mgnitude of the urrent in eh phse will e equl for lned lod nd unequl for n unlned lod. You will rell tht for the Y-onneted genertor, the mgnitude of the line voltge is equl to 3 times the phse voltge. This sme reltionship n e pplied to lned or n unlned four-wire Y-onneted lod: E L 3 V f (22.8) For voltge drop ross lod element, the first susript refers to tht terminl through whih the urrent enters the lod element, nd the seond susript refers to the terminl from whih the urrent leves. In other words, the first susript is, y definition, positive with respet to the seond for voltge drop. Note Fig. 22.13, in whih the stndrd doule susripts for soure of voltge nd voltge drop re indited. EXMPLE 22.1 The phse sequene of the Y-onneted genertor in Fig. 22.13 is.. Find the phse ngles v 2 nd v 3.. Find the mgnitude of the line voltges.. Find the line urrents. d. Verify tht, sine the lod is lned, I N 0. Solutions:. For n phse sequene, v 2 120 nd v 3 120

THE Y-D SYSTEM 985 I I n 3 Ω E N 120 V 0 E V n 120 V θ 3 N 120 V θ 2 E N E N E I N I 3 Ω V n I n n 4 Ω 4 Ω lned lod 4 Ω Vn 3 Ω I n E I FIG. 22.13 Exmple 22.1.. E L 3 E f (1.73)(120 V) 208 V. Therefore, E E E 208 V. V f E f. Therefore, V n E N V n E N V n E N V I fl I n n 120 V 0 120 V 0 Zn 3 j 4 5 53.13 24 53.13 V I n n 120 V 120 Zn 24 173.13 5 53.13 V 120 V 120 I n n Zn 24 66.87 5 53.13 nd, sine I L I fl, I I n 24 53.13 I I n 24 173.13 I I n 24 66.87 d. pplying Kirhhoff s urrent lw, we hve In retngulr form, I N I I I I 24 53.13 14.40 j 19.20 I 24 173.13 22.83 j 2.87 I 24 66.87 09.43 j 22.07 Σ(I I I ) 0 j 0 nd I N is in ft equl to zero, s required for lned lod. 22.6 THE Y-D SYSTEM There is no neutrl onnetion for the Y-D system of Fig. 22.14. ny vrition in the impedne of phse tht produes n unlned system will simply vry the line nd phse urrents of the system.

986 POLYPHSE SYSTEMS I fg E f I L I fl I fg E f N E f I fg E L E L I L V f Z 3 Z 1 V f I fl I fl Z 2 E L V f FIG. 22.14 Y-onneted genertor with D-onneted lod. I L For lned lod, Z 1 Z 2 Z 3 (22.9) The voltge ross eh phse of the lod is equl to the line voltge of the genertor for lned or n unlned lod: V f E L (22.10) The reltionship etween the line urrents nd phse urrents of lned D lod n e found using n pproh very similr to tht used in Setion 22.3 to find the reltionship etween the line voltges nd phse voltges of Y-onneted genertor. For this se, however, Kirhhoff s urrent lw is employed insted of Kirhhoff s voltge lw. The results otined re I L 3 I f (22.11) nd the phse ngle etween line urrent nd the nerest phse urrent is 30. more detiled disussion of this reltionship etween the line nd phse urrents of D-onneted system n e found in Setion 22.7. For lned lod, the line urrents will e equl in mgnitude, s will the phse urrents. EXMPLE 22.2 For the three-phse system of Fig. 22.15:. Find the phse ngles v 2 nd v 3.. Find the urrent in eh phse of the lod.. Find the mgnitude of the line urrents. Solutions:. For n sequene, v 2 120 nd v 3 120

THE D-ONNETED GENERTOR 987 I R = 6 X L = 8 3-phse, 3-wire, Y-onneted genertor Phse sequene: E = 150 V 0 V I X L = 8 E = 150 V v 3 R = 6 I I V I R = 6 I X L = 8 V E = 150 V v 2 FIG. 22.15 Exmple 22.2.. V f E L.Therefore, V E V E V E The phse urrents re V 150 V 0 150 V 0 I Z 15 53.13 6 j8 10 53.13 V 150 V 120 I Z 15 173.13 10 53.13 V 150 V 120 I Z 15 66.87 10 53.13. I L 3 I f (1.73)(15 ) 25.95. Therefore, I I I 25.95 22.7 THE D-ONNETED GENERTOR If we rerrnge the oils of the genertor in Fig. 22.16() s shown in Fig. 22.16(), the system is referred to s three-phse, three-wire, e N E I E I N N e N N e N I E N E N Lod E N I E I I () () FIG. 22.16 D-onneted genertor.

988 POLYPHSE SYSTEMS -onneted genertor. In this system, the phse nd line voltges re equivlent nd equl to the voltge indued ross eh oil of the genertor; tht is, E E N nd e N 2 E N sin qt Phse E E N nd e N 2 E N sin(qt 120 ) sequene E E N nd e N 2 E N sin(qt 120 ) or E L E fg (22.12) Note tht only one voltge (mgnitude) is ville insted of the two ville in the Y-onneted system. Unlike the line urrent for the Y-onneted genertor, the line urrent for the D-onneted system is not equl to the phse urrent. The reltionship etween the two n e found y pplying Kirhhoff s urrent lw t one of the nodes nd solving for the line urrent in terms of the phse urrents; tht is, t node, I I I or I I I I I The phsor digrm is shown in Fig. 22.17 for lned lod. I 120 120 60 30 30 3 2 I I 3 2 I I I I I = 3 I FIG. 22.17 Determining line urrent from the phse urrents of D-onneted, threephse genertor. Using the sme proedure to find the line urrent s ws used to find the line voltge of Y-onneted genertor produes the following: I 3 I 30 I 3 I 150 I 3 I 90 In generl: I L 3 I fg (22.13) with the phse ngle etween line urrent nd the nerest phse urrent t 30. The phsor digrm of the urrents is shown in Fig. 22.18.

THE D-D, D-Y THREE-PHSE SYSTEMS 989 I I 30 120 120 I I 30 120 30 I I FIG. 22.18 The phsor digrm of the urrents of three-phse, D-onneted genertor. It n e shown in the sme mnner employed for the voltges of Y-onneted genertor tht the phsor sum of the line urrents or phse urrents for D-onneted systems with lned lods is zero. 22.8 PHSE SEQUENE (D-ONNETED GENERTOR) Even though the line nd phse voltges of D-onneted system re the sme, it is stndrd prtie to desrie the phse sequene in terms of the line voltges. The method used is the sme s tht desried for the line voltges of the Y-onneted genertor. For exmple, the phsor digrm of the line voltges for phse sequene is shown in Fig. 22.19. In drwing suh digrm, one must tke re to hve the sequene of the first nd seond susripts the sme. In phsor nottion, E E E 0 E E 120 E E 120 P Rottion E 22.9 THE D-D, D-Y THREE-PHSE SYSTEMS The si equtions neessry to nlyze either of the two systems (D-D, D-Y) hve een presented t lest one in this hpter. We will therefore proeed diretly to two desriptive exmples, one with D-onneted lod nd one with Y-onneted lod. E Phse sequene: FIG. 22.19 Determining the phse sequene for D-onneted, three-phse genertor. EXMPLE 22.3 For the D-D system shown in Fig. 22.20:. Find the phse ngles v 2 nd v 3 for the speified phse sequene.. Find the urrent in eh phse of the lod.. Find the mgnitude of the line urrents.

990 POLYPHSE SYSTEMS 3-phse -onneted genertor Phse sequene: I E = 120 V 0 E = 120 V v 3 I I V 5 I I 5 5 5 V 5 V 5 I E = 120 V v 2 FIG. 22.20 Exmple 22.3: D-D system. Solutions:. For n phse sequene, v 2 120 nd v 3 120. V f E L. Therefore, V E V E V E The phse urrents re V I 120 V 0 120 V 0 Z (5 0 )(5 90 ) 25 90 5 j 5 7. 071 45 120 V 0 33.9 45 3.54 45 V I 120 V 120 Z 33.9 165 3.54 45 V 120 V 120 I Z 33.9 75 3.54 45. I L 3 I f (1.73)(34 ) 58.82. Therefore, I I I 58.82 EXMPLE 22.4 For the D-Y system shown in Fig. 22.21:. Find the voltge ross eh phse of the lod.. Find the mgnitude of the line voltges. Solutions:. I fl I L. Therefore, I n I 2 0 I n I 2 120 I n I 2 120

POWER 991 I = 2 0 I n 8 V n 6 3-phse -onneted genertor E E 8 6 n 6 8 I n V n V n I n I = 2 120 Phse sequene: E I = 2 120 FIG. 22.21 Exmple 22.4: D-Y system. The phse voltges re V n I n Z n (2 0 )(10 53.13 ) 20 V 53.13 V n I n Z n (2 120 )(10 53.13 ) 20 V 173.13 V n I n Z n (2 120 )(10 53.13 ) 20 V 66.87. E L 3 V f (1.73)(20 V) 34.6 V. Therefore, E E E 34.6 V 22.10 POWER Y-onneted lned Lod Plese refer to Fig. 22.22 for the following disussion. I L I Z Z = R ± jx V E L E L n V Z Z V I L I I E L I L FIG. 22.22 Y-onneted lned lod.

992 POLYPHSE SYSTEMS verge Power The verge power delivered to eh phse n e determined y ny one of Eqs. (22.14) through (22.16). P f V f I f os v V f If I 2 fr f V R f 2 R (wtts, W) (22.14) where v V f If indites tht v is the phse ngle etween V f nd I f. The totl power to the lned lod is P T 3P f (W) (22.15) or, sine V f nd I f I L 3 E L then P T 3 I L os v V f If 3 ut (1) 3 3 3 Therefore, 3 3 E L 3 3 3 3 3 P T 3 E L I L os v V f If 3I 2 LR f (W) (22.16) Retive Power retive) is The retive power of eh phse (in volt-mperes Q f V f I f sin v V f If I 2 fx f V X f 2 X (VR) (22.17) The totl retive power of the lod is Q T 3Q f (VR) (22.18) or, proeeding in the sme mnner s ove, we hve Q T 3 E L I L sin v V f I f 3I 2 LX f (VR) (22.19) pprent Power The pprent power of eh phse is S f V f I f (V) (22.20) The totl pprent power of the lod is S T 3S f (V) (22.21) or, s efore, S T 3 E L I L (V) (22.22)

POWER 993 Power Ftor The power ftor of the system is given y F p P S T T os v V f If (leding or lgging) (22.23) EXMPLE 22.5 For the Y-onneted lod of Fig. 22.23: R = 3 Ω E L = 173.2 V 0 X L = 4 Ω E L = 173.2 V 120 X L = 4 Ω V I R = 3 Ω I V n X L = 4 Ω V I R = 3 Ω E L = 173.2 V 120 FIG. 22.23 Exmple 22.5.. Find the verge power to eh phse nd the totl lod.. Determine the retive power to eh phse nd the totl retive power.. Find the pprent power to eh phse nd the totl pprent power. d. Find the power ftor of the lod. Solutions:. The verge power is P f V f I f os v V f If (100 V)(20 ) os 53.13 (2000)(0.6) 1200 W P f IfR 2 f (20 ) 2 (3 ) (400)(3) 1200 W V 2 R (60 V) 2 3600 P f 1200 W Rf 3 3 P T 3P f (3)(1200 W) 3600 W or P T 3 E L I L os v V f If (1.732)(173.2 V)(20 )(0.6) 3600 W. The retive power is Q f V f I f sin v V f If (100 V)(20 ) sin 53.13 (2000)(0.8) 1600 VR or Q f I 2 fx f (20 ) 2 (4 ) (400)(4) 1600 VR Q T 3Q f (3)(1600 VR) 4800 VR or Q T 3 E L I L sin v V f If (1.732)(173.2 V)(20 )(0.8) 4800 VR

994 POLYPHSE SYSTEMS. The pprent power is S f V f I f (100 V)(20 ) 2000 V S T 3S f (3)(2000 V) 6000 V or S T 3 E L I L (1.732)(173.2 V)(20 ) 6000 V d. The power ftor is P T 3600 W F p ST 0.6 lgging 6000 V D-onneted lned Lod Plese refer to Fig. 22.24 for the following disussion. E L E L I L V Z I Z V Z = R ± jx I L I Z I E L V I L FIG. 22.24 D-onneted lned lod. verge Power P f V f I f os v V f If I 2 fr f V R f 2 R (W) (22.24) P T 3P f (W) (22.25) Retive Power Q f V f I f sin v V f If I 2 fx f V X f 2 X (VR) (22.26) Q T 3Q f (VR) (22.27) pprent Power S f V f I f (V) (22.28)

POWER 995 S T 3S f 3 E L I L (V) (22.29) Power Ftor F p P S T T (22.30) EXMPLE 22.6 For the D-Y onneted lod of Fig. 22.25, find the totl verge, retive, nd pprent power. In ddition, find the power ftor of the lod. 6 4 8 E L = 200 V 120 E L = 200 V 0 3 8 6 4 3 3 4 6 8 E L = 200 V 120 FIG. 22.25 Exmple 22.6. Solution: onsider the D nd Y seprtely. For the D: Z D 6 j 8 10 53.13 E I f L 200 V ZD 20 10 P TD 3IfR 2 f (3)(20 ) 2 (6 ) 7200 W Q TD 3IfX 2 f (3)(20 ) 2 (8 ) 9600 VR () S TD 3V f I f (3)(200 V)(20 ) 12,000 V For the Y: Z Y 4 j 3 5 36.87 E L / 3 200 V/ 3 116 V I f 23.12 ZY 5 5 P TY 3IfR 2 f (3)(23.12 ) 2 (4 ) 6414.41 W Q TY 3IfX 2 f (3)(23.12 ) 2 (3 ) 4810.81 VR (L) S TY 3V f I f (3)(116 V)(23.12 ) 8045.76 V

996 POLYPHSE SYSTEMS For the totl lod: P T P TD P TY 7200 W 6414.41 W 13,614.41 W Q T Q TD Q TY 9600 VR () 4810.81 VR (I) 4789.19 VR () S T P T 2 Q 2 T (1 3,6 1 4.4 1 W ) 2 ( 4 7 8 9.1 9 V R ) 2 14,432.2 V P T 13,614.41 W F p ST 0.943 leding 14,432.20 V EXMPLE 22.7 Eh trnsmission line of the three-wire, three-phse system of Fig. 22.26 hs n impedne of 15 j 20. The system delivers totl power of 160 kw t 12,000 V to lned three-phse lod with lgging power ftor of 0.86. 15 20 Z 1 = Z 2 = Z 3 Z 1 N E 12 k V n Z 2 Z 3 15 20 15 20 FIG. 22.26 Exmple 22.7.. Determine the mgnitude of the line voltge E of the genertor.. Find the power ftor of the totl lod pplied to the genertor.. Wht is the effiieny of the system? Solutions: V L 12,000 V. V f (lod) 6936.42 V 3 1.73 P T (lod) 3V f I f os v nd P T 160,000 W I f 3Vf os v 3(6936.42 V)(0.86) 8.94 Sine v os 1 0.86 30.68, ssigning V f n ngle of 0 or V f V f 0, lgging power ftor results in I f 8.94 30.68 For eh phse, the system will pper s shown in Fig. 22.27, where E N I f Z line V f 0

THE THREE-WTTMETER METHOD 997 I L 15 20 I f Z line I f = 8.94 30.68 E N Z 1 V f FIG. 22.27 The loding on eh phse of the system of Fig. 22.26. or E N I f Z line V f (8.94 30.68 )(25 53.13 ) 6936.42 V 0 223.5 V 22.45 6936.42 V 0 206.56 V j 85.35 V 6936.42 V 7142.98 V j 85.35 V 7143.5 V 0.68 Then E 3 E fg (1.73)(7143.5 V) 12,358.26 V. P T P lod P lines 160 kw 3(I L ) 2 R line 160 kw 3(8.94 ) 2 15 160,000 W 3596.55 W 163,596.55 W nd P T 3 V L I L os v T P T 163,596.55 W or os v T 3 VL I L (1.73)(12,358.26 V)(8.94 ) nd F p 0.856 < 0.86 of lod P o P o 160 kw. h 0.978 Pi Po P losses 160 kw 3596.55 W 97.8% 22.11 THE THREE-WTTMETER METHOD The power delivered to lned or n unlned four-wire, Y-onneted lod n e found y the three-wttmeter method, tht is, y using three wttmeters in the mnner shown in Fig. 22.28. Eh wttmeter mesures the power delivered to eh phse. The potentil oil of eh wttmeter is onneted prllel with the lod, while the urrent oil is in series with the lod. The totl verge power of the system n e found y summing the three wttmeter redings; tht is, P TY P 1 P 2 P 3 (22.31) For the lod (lned or unlned), the wttmeters re onneted s shown in Fig. 22.29. The totl power is gin the sum of the three wttmeter redings: P TD P 1 P 2 P 3 (22.32)

998 POLYPHSE SYSTEMS P 1 1 P 1 Line Line 1 P 1 Z 1 P 3 Z 3 Z 1 P 1 Neutrl n P 3 P 2 P 2 P 2 2 Line P 3 3 P 3 Z 3 Z 2 Line Line Line 3 FIG. 22.29 Three-wttmeter method for D-onneted lod. Z 2 P 2 2 FIG. 22.28 Three-wttmeter method for Y-onneted lod. If in either of the ses just desried the lod is lned, the power delivered to eh phse will e the sme. The totl power is then just three times ny one wttmeter reding. 22.12 THE TWO-WTTMETER METHOD The power delivered to three-phse, three-wire, D- or Y-onneted, lned or unlned lod n e found using only two wttmeters if the proper onnetion is employed nd if the wttmeter redings re interpreted properly. The si onnetions of this two-wttmeter method re shown in Fig. 22.30. One end of eh potentil oil is onneted to the sme line. The urrent oils re then pled in the remining lines. P 1 1 P 1 Line Line Line - or Y- onneted lod P 1 1 P 1 Line - or Y- onneted lod P 2 2 P 2 Line P 2 2 P 2 Line FIG. 22.30 Two-wttmeter method for D- or Y-onneted lod. FIG. 22.31 lterntive hookup for the two-wttmeter method. The onnetion shown in Fig. 22.31 will lso stisfy the requirements. third hookup is lso possile, ut this is left to the reder s n exerise. The totl power delivered to the lod is the lgeri sum of the two wttmeter redings. For lned lod, we will now onsider two

THE TWO-WTTMETER METHOD 999 methods of determining whether the totl power is the sum or the differene of the two wttmeter redings. The first method to e desried requires tht we know or e le to find the power ftor (leding or lgging) of ny one phse of the lod. When this informtion hs een otined, it n e pplied diretly to the urve of Fig. 22.32. Power ftor Led or lg F p 1.0 0.8 0.6 0.5 0.4 0.2 P l / P h 1.0 0.75 0.5 0.25 0 0.25 0.5 0.75 1.0 P T = P h P l P T = P l P h FIG. 22.32 Determining whether the redings otined using the two-wttmeter method should e dded or sutrted. The urve in Fig. 22.32 is plot of the power ftor of the lod (phse) versus the rtio P l /P h, where P l nd P h re the mgnitudes of the lower- nd higher-reding wttmeters, respetively. Note tht for power ftor (leding or lgging) greter thn 0.5, the rtio hs positive vlue. This indites tht oth wttmeters re reding positive, nd the totl power is the sum of the two wttmeter redings; tht is, P T P l P h. For power ftor less thn 0.5 (leding or lgging), the rtio hs negtive vlue. This indites tht the smller-reding wttmeter is reding negtive, nd the totl power is the differene of the two wttmeter redings; tht is, P T P h P l. loser exmintion will revel tht, when the power ftor is 1(os 0 1), orresponding to purely resistive lod, P l /P h 1or P l P h, nd oth wttmeters will hve the sme wttge indition. t power ftor equl to 0 (os 90 0), orresponding to purely retive lod, P l /P h 1 or P l P h, nd oth wttmeters will gin hve the sme wttge indition ut with opposite signs. The trnsition from negtive to positive rtio ours when the power ftor of the lod is 0.5 or v os 1 0.5 60. t this power ftor, P l /P h 0, so tht P l 0, while P h will red the totl power delivered to the lod. The seond method for determining whether the totl power is the sum or differene of the two wttmeter redings involves simple lortory test. For the test to e pplied, oth wttmeters must first hve n up-sle defletion. If one of the wttmeters hs elow-zero indition, n up-sle defletion n e otined y simply reversing the leds of the urrent oil of the wttmeter. To perform the test: 1. Tke notie of whih line does not hve urrent oil sensing the line urrent. 2. For the lower-reding wttmeter, disonnet the led of the potentil oil onneted to the line without the urrent oil. 3. Tke the disonneted led of the lower-reding wttmeter s potentil oil, nd touh onnetion point on the line tht hs the urrent oil of the higher-reding wttmeter.

1000 POLYPHSE SYSTEMS 4. If the pointer deflets downwrd (elow zero wtts), the wttge reding of the lower-reding wttmeter should e sutrted from tht of the higher-reding wttmeter. Otherwise, the redings should e dded. For lned system, sine P T P h P l 3 E L I L os v V f If the power ftor of the lod (phse) n e found from the wttmeter redings nd the mgnitude of the line voltge nd urrent: F p os v V P f h Pl If 3 E L I L (22.33) EXMPLE 22.8 For the unlned D-onneted lod of Fig. 22.33 with two properly onneted wttmeters: W 1 E = 208 V 0 E = 208 V 120 I I X I I 12 10 R 3 12 R 2 I R 1 X L 15 20 E = 208 V 120 W 2 I FIG. 22.33 Exmple 22.8.. Determine the mgnitude nd ngle of the phse urrents.. lulte the mgnitude nd ngle of the line urrents.. Determine the power reding of eh wttmeter. d. lulte the totl power sored y the lod. e. ompre the result of prt (d) with the totl power lulted using the phse urrents nd the resistive elements. Solutions: V. I E 208 V 0 Z Z 20.8 0 10 0 V E 208 V 120 I Z Z 15 j20 8.32 173.13 V 208 V 120 I E Z Z 12 j12 12.26 165 208 V 120 25 53.13 208 V 120 16.97 45. I I I 20.8 0 12.26 165 20.8 ( 11.84 j 3.17 ) 20.8 11.84 j 3.17 32.64 j 3.17 32.79 5.55

UNLNED, THREE-PHSE, FOUR-WIRE, Y-ONNETED LOD 1001 I I I 8.32 173.13 20.8 0 ( 8.26 j 1 ) 20.8 8.26 20.8 j 1 29.06 j 1 29.08 178.03 I I I 12.26 165 8.32 173.13 ( 11.84 j 3.17 ) ( 8.26 j 1 ) 11.84 8.26 j (3.17 1) 3.58 j 4.17 5.5 130.65. P 1 V I os v V I V 208 V 0 I 32.79 5.55 (208 V)(32.79 ) os 5.55 6788.35 W V E 208 V 120 ut V E 208 V 120 180 208 V 60 with I 5.5 130.65 P 2 V I os v V I P 2 (208 V)(5.5 ) os 70.65 379.1 W d. P T P 1 P 2 6788.35 W 379.1 W 7167.45 W e. P T (I ) 2 R 1 (I ) 2 R 2 (I ) 2 R 3 (20.8 ) 2 10 (8.32 ) 2 15 (12.26 ) 2 12 4326.4 W 1038.34 W 1803.69 W 7168.43 W (The slight differene is due to the level of ury rried through the lultions.) 22.13 UNLNED, THREE-PHSE, FOUR-WIRE, Y-ONNETED LOD For the three-phse, four-wire, Y-onneted lod of Fig. 22.34, onditions re suh tht none of the lod impednes re equl hene we hve n unlned polyphse lod. Sine the neutrl is ommon point etween the lod nd soure, no mtter wht the impedne of eh phse of the lod nd soure, the voltge ross eh phse is the phse voltge of the genertor: V f E f (22.34) The phse urrents n therefore e determined y Ohm s lw: f I f1 V 1 E 1 Z Z 1 f 1 nd so on (22.35) The urrent in the neutrl for ny unlned system n then e found y pplying Kirhhoff s urrent lw t the ommon point n: I N I f1 I f2 I f3 I L1 I L 2 I L 3 (22.36)

1002 POLYPHSE SYSTEMS Line I L1 V f1 Z 1 I fl1 I N Neutrl I fl2 I fl3 E L E L Z 3 Vf3 Z 2 V f2 I L2 Line E L Line I L3 FIG. 22.34 Unlned Y-onneted lod. euse of the vriety of equipment found in n industril environment, oth three-phse power nd single-phse power re usully provided with the single-phse otined off the three-phse system. In ddition, sine the lod on eh phse is ontinully hnging, fourwire system (with neutrl) is normlly employed to ensure stedy voltge levels nd to provide pth for the urrent resulting from n unlned lod. The system of Fig. 22.35 hs three-phse trnsformer dropping the line voltge from 13,800 V to 208 V. ll the lowerpower-demnd lods suh s lighting, wll outlets, seurity, et., use the single-phse, 120-V line to neutrl voltge. Higher power lods, suh s ir onditioners, eletri ovens or dryers, et., use the single-phse, 208 V ville from line to line. For lrger motors nd speil highdemnd equipment, the full three-phse power n e tken diretly off the system, s shown in Fig. 22.35. In the design nd onstrution of ommeril estlishment, the Ntionl Eletri ode requires tht every effort e mde to ensure tht the expeted lods, whether they e single- 208 V 0 208 V 120 208 V 120 1φ 120 V 1φ 280 V 1φ 120 V 1φ 120 V 3φ 208 V lned lod Seondry 3 φ trnsformer FIG. 22.35 3f/1f, 208-V/120-V industril supply.

UNLNED, THREE-PHSE, THREE-WIRE, Y-ONNETED LOD 1003 or multiphse, result in totl lod tht is s lned s possile etween the phses, thus ensuring the highest level of trnsmission effiieny. 22.14 UNLNED, THREE-PHSE, THREE-WIRE, Y-ONNETED LOD For the system shown in Fig. 22.36, the required equtions n e derived y first pplying Kirhhoff s voltge lw round eh losed loop to produe I n E E V n Z 1 E E V n Z 3 n Z 2 I n I n V n E E FIG. 22.36 Unlned, three-phse, three-wire, Y-onneted lod. Sustituting, we hve E V n V n 0 E V n V n 0 E V n V n 0 V n I n Z 1 V n I n Z 2 V n I n Z 3 E I n Z 1 I n Z 2 E I n Z 2 I n Z 3 E I n Z 3 I n Z 1 (22.37) (22.37) (22.37) pplying Kirhhoff s urrent lw t node n results in I n I n I n 0 nd I n I n I n Sustituting for I n in Eqs. (22.37) nd (22.37) yields E I n Z 1 [ (I n I n )]Z 2 E (I n I n )Z 2 I n Z 3 whih re rewritten s E I n (Z 1 Z 2 ) I n Z 2 E I n ( Z 2 ) I n [ (Z 2 Z 3 )]

1004 POLYPHSE SYSTEMS Using determinnts, we hve E Z 2 E (Z 2 Z 3 ) I n Z 1 Z 2 Z 2 Z 2 (Z 2 Z 3 ) (Z 2 Z 3 )E E Z 2 Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 Z 2 2 Z 2 2 Z 2 (E E ) Z 3 E I n Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 pplying Kirhhoff s voltge lw to the line voltges: E E E 0 or E E E Sustituting for (E E ) in the ove eqution for I n : I n Z 2 ( E ) Z 3 E Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 E Z 3 E Z 2 nd I n (22.38) Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 In the sme mnner, it n e shown tht E Z 2 E Z 1 I n (22.39) Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 Sustituting Eq. (22.39) for I n in the right-hnd side of Eq. (22.37), we otin E Z 1 E Z 3 I n (22.40) Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 EXMPLE 22.9 phse-sequene inditor is n instrument tht n disply the phse sequene of polyphse iruit. network tht will perform this funtion ppers in Fig. 22.37. The pplied phse sequene is. The ul orresponding to this phse sequene will urn more rightly thn the ul inditing the sequene euse greter urrent is pssing through the ul. lulting the phse urrents will demonstrte tht this sitution does in ft exist: 1 1 Z 1 X 166 q (377 rd/s)(16 10 6 F) y Eq. (22.39), I n E Z 2 E Z 1 Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 (200 V 120 )(200 0 ) (200 V 120 )(166 90 ) (166 90 )(200 0 ) (166 90 )(200 0 ) (200 0 )(200 0 ) I n

UNLNED, THREE-PHSE, THREE-WIRE, Y-ONNETED LOD 1005 (1) f = 60 Hz E = 200 V 0 E = 200 V 120 Z 3 Z 1 16 mf uls (150 W) 200 internl resistne n Z2 (3) (2) 200 200 E = 200 V 120 FIG. 22.37 Exmple 22.9. Dividing the numertor nd denomintor y 1000 nd onverting oth to the retngulr domin yields I n y Eq. (22.40), E Z 1 E Z 3 I n Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 (200 V 120 )(166 90 ) (200 V 0 )(200 0 ) 77.52 10 3 58.93 33,200 V 210 40,000 V 0 I n 77.52 10 3 58.93 Dividing y 1000 nd onverting to the retngulr domin yields I n 40,000 V 120 33,200 V 30 33,200 90 33,200 90 40,000 0 8.75 j 18.04 20.05 64.13 77.52 58.93 77.52 58.93 I n 0.259 123.06 28.75 j 16.60 40.0 77.52 58.93 70.73 166.43 77.52 58.93 ( 20 j 34.64) (28.75 j 16.60) 40 j 66.4 0.91 225.36 68.75 j 16.60 77.52 58.93 nd I n > I n y ftor of more thn 3:1. Therefore, the ul inditing n sequene will urn more rightly due to the greter urrent. If the phse sequene were, the reverse would e true.

1006 POLYPHSE SYSTEMS PROLEMS SETION 22.5 The Y-onneted Genertor with Y-onneted Lod 1. lned Y lod hving 10- resistne in eh leg is onneted to three-phse, four-wire, Y-onneted genertor hving line voltge of 208 V. lulte the mgnitude of. the phse voltge of the genertor.. the phse voltge of the lod.. the phse urrent of the lod. d. the line urrent. 2. Repet Prolem 1 if eh phse impedne is hnged to 12- resistor in series with 16- pitive retne. 3. Repet Prolem 1 if eh phse impedne is hnged to 10- resistor in prllel with 10- pitive retne. 4. The phse sequene for the Y-Y system of Fig. 22.38 is.. Find the ngles v 2 nd v 3 for the speified phse sequene.. Find the voltge ross eh phse impedne in phsor form.. Find the urrent through eh phse impedne in phsor form. d. Drw the phsor digrm of the urrents found in prt (), nd show tht their phsor sum is zero. e. Find the mgnitude of the line urrents. f. Find the mgnitude of the line voltges. 120 V 0 20 Ω 120 V 0 θ 3 N 120 V 02 θ 20 Ω 20 Ω FIG. 22.38 Prolems 4, 5, 6, nd 31. 5. Repet Prolem 4 if the phse impednes re hnged to 9- resistor in series with 12- indutive retne. 6. Repet Prolem 4 if the phse impednes re hnged to 6- resistne in prllel with n 8- pitive retne.

PROLEMS 1007 7. For the system of Fig. 22.39, find the mgnitude of the unknown voltges nd urrents. I I n 3-phse Y-onneted 4-wire genertor E = 220 V 0 10 10 V n Phse sequene: N E = 220 V 120 I n V n V n 10 10 10 10 I n I n E = 220 V 120 I FIG. 22.39 Prolems 7, 32, nd 44. *8. ompute the mgnitude of the voltge E for the lned three-phse system of Fig. 22.40. 1 16 12 3-phse Y-onneted genertor E Line resistne 12 n 12 16 16 1 V = 50 V φ 1 FIG. 22.40 Prolem 8.

1008 POLYPHSE SYSTEMS *9. For the Y-Y system of Fig. 22.41:. Find the mgnitude nd ngle ssoited with the voltges E N, E N, nd E N.. Determine the mgnitude nd ngle ssoited with eh phse urrent of the lod: I n, I n, nd I n.. Find the mgnitude nd phse ngle of eh line urrent: I, I, nd I. d. Determine the mgnitude nd phse ngle of the voltge ross eh phse of the lod: V n, V n, nd V n. I 30 40 E = 22 kv 0 V n I n 0.4 k N 1 k 1 k V n 0.4 k E = 22 kv 120 I 30 40 I n 1 k V n 0.4 k I n E = 22 kv 120 I 30 40 FIG. 22.41 Prolem 9. SETION 22.6 The Y-D System 10. lned D lod hving 20- resistne in eh leg is onneted to three-phse, three-wire, Y-onneted genertor hving line voltge of 208 V. lulte the mgnitude of. the phse voltge of the genertor.. the phse voltge of the lod.. the phse urrent of the lod. d. the line urrent. 11. Repet Prolem 10 if eh phse impedne is hnged to 6.8- resistor in series with 14- indutive retne. 12. Repet Prolem 10 if eh phse impedne is hnged to n 18- resistne in prllel with n 18- pitive retne. 13. The phse sequene for the Y-D system of Fig. 22.42 is.. Find the ngles v 2 nd v 3 for the speified phse sequene.. Find the voltge ross eh phse impedne in phsor form.. Drw the phsor digrm of the voltges found in prt (), nd show tht their sum is zero round the losed loop of the D lod. d. Find the urrent through eh phse impedne in phsor form. e. Find the mgnitude of the line urrents. f. Find the mgnitude of the genertor phse voltges.

PROLEMS 1009 E = 208 V 0 E = 208 V v 3 22 22 N 22 E = 208 V v 2 FIG. 22.42 Prolems 13, 14, 15, 34, nd 45. 14. Repet Prolem 13 if the phse impednes re hnged to 100- resistor in series with pitive retne of 100. 15. Repet Prolem 13 if the phse impednes re hnged to 3- resistor in prllel with n indutive retne of 4. 16. For the system of Fig. 22.43, find the mgnitude of the unknown voltges nd urrents. I 3-phse Y-onneted 4-wire genertor E = 220 V 0 E = 220 V 120 V 10 I I 10 V Phse sequene: 10 10 I 10 I 10 V I E = 220 V 120 FIG. 22.43 Prolems 16, 35, nd 47.

1010 POLYPHSE SYSTEMS *17. For the D-onneted lod of Fig. 22.44:. Find the mgnitude nd ngle of eh phse urrent I, I, nd I.. lulte the mgnitude nd ngle of eh line urrent I, I, nd I.. Determine the mgnitude nd ngle of the voltges E, E, nd E. E E E I I I 10 10 10 20 20 20 1 k 0.3 k 1 k I I I 0.3 k 1 k 0.3 k V = 16 kv 0 V = 16 kv 120 V = 16 kv 120 FIG. 22.44 Prolem 17. SETION 22.9 The D-D, D-Y Three-Phse Systems 18. lned Y lod hving 30- resistne in eh leg is onneted to three-phse, D-onneted genertor hving line voltge of 208 V. lulte the mgnitude of. the phse voltge of the genertor.. the phse voltge of the lod.. the phse urrent of the lod. d. the line urrent. 19. Repet Prolem 18 if eh phse impedne is hnged to 12- resistor in series with 12- indutive retne. 20. Repet Prolem 18 if eh phse impedne is hnged to 15- resistor in prllel with 20- pitive retne. *21. For the system of Fig. 22.45, find the mgnitude of the unknown voltges nd urrents. 22. Repet Prolem 21 if eh phse impedne is hnged to 10- resistor in series with 20- indutive retne. 23. Repet Prolem 21 if eh phse impedne is hnged to 20- resistor in prllel with 15- pitive retne. 24. lned D lod hving 220- resistne in eh leg is onneted to three-phse, D-onneted genertor hving line voltge of 440 V. lulte the mgnitude of. the phse voltge of the genertor.. the phse voltge of the lod.. the phse urrent of the lod. d. the line urrent. 25. Repet Prolem 24 if eh phse impedne is hnged to 12- resistor in series with 9- pitive retne.

PROLEMS 1011 I I n 3-phse -onneted genertor E = 120 V 0 V n 24 Phse sequene: E = 120 V 120 V n n V n 24 24 I n I n I E = 120 V 120 I FIG. 22.45 Prolems 21, 22, 23, nd 37. 26. Repet Prolem 24 if eh phse impedne is hnged to 22- resistor in prllel with 22- indutive retne. 27. The phse sequene for the D-D system of Fig. 22.46 is.. Find the ngles v 2 nd v 3 for the speified phse sequene.. Find the voltge ross eh phse impedne in phsor form.. Drw the phsor digrm of the voltges found in prt (), nd show tht their phsor sum is zero round the losed loop of the D lod. d. Find the urrent through eh phse impedne in phsor form. e. Find the mgnitude of the line urrents. E = 100 V 0 E = 100 V v 3 20 20 20 E = 100 V v 2 FIG. 22.46 Prolem 27.

1012 POLYPHSE SYSTEMS 28. Repet Prolem 25 if eh phse impedne is hnged to 12- resistor in series with 16- indutive retne. 29. Repet Prolem 25 if eh phse impedne is hnged to 20- resistor in prllel with 20- pitive retne. SETION 22.10 Power 30. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system of Prolem 2. 31. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system of Prolem 4. 32. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system of Prolem 7. 33. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system of Prolem 12. 34. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system of Prolem 14. 35. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system of Prolem 16. 36. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system of Prolem 20. 37. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system of Prolem 22. 38. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system of Prolem 26. 39. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system of Prolem 28. 40. lned, three-phse, D-onneted lod hs line voltge of 200 nd totl power onsumption of 4800 W t lgging power ftor of 0.8. Find the impedne of eh phse in retngulr oordintes. 41. lned, three-phse, Y-onneted lod hs line voltge of 208 nd totl power onsumption of 1200 W t leding power ftor of 0.6. Find the impedne of eh phse in retngulr oordintes. *42. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the system of Fig. 22.47. E = 125 V 0 E = 125 V 120 20 3 15 3 4 4 n 4 20 15 3 15 20 E = 125 V 120 FIG. 22.47 Prolem 42.

PROLEMS 1013 *43. The Y-Y system of Fig. 22.48 hs lned lod nd line impedne Z line 4 j 20. If the line voltge t the genertor is 16,000 V nd the totl power delivered to the lod is 1200 kw t 80, determine eh of the following:. The mgnitude of eh phse voltge of the genertor.. The mgnitude of the line urrents.. The totl power delivered y the soure. d. The power ftor ngle of the entire lod seen y the soure. e. The mgnitude nd ngle of the urrent I if E N E N 0. f. The mgnitude nd ngle of the phse voltge V n. g. The impedne of the lod of eh phse in retngulr oordintes. h. The differene etween the power ftor of the lod nd the power ftor of the entire system (inluding Z line ). i. The effiieny of the system. N I 4 20 E N V n E = 16 kv Z 1 n I n = 80 Z 1 = Z 2 = Z lgging F p 4 20 Z 2 Z 3 4 20 FIG. 22.48 Prolem 43. SETION 22.11 The Three-Wttmeter Method 44.. Sketh the onnetions required to mesure the totl wtts delivered to the lod of Fig. 22.39 using three wttmeters.. Determine the totl wttge dissiption nd the reding of eh wttmeter. 45. Repet Prolem 44 for the network of Fig. 22.42. SETION 22.12 The Two-Wttmeter Method 46.. For the three-wire system of Fig. 22.49, properly onnet seond wttmeter so tht the two will mesure the totl power delivered to the lod.. If one wttmeter hs reding of 200 W nd the other reding of 85 W, wht is the totl dissiption in wtts if the totl power ftor is 0.8 leding?. Repet prt () if the totl power ftor is 0.2 lgging nd P l 100 W. 47. Sketh three different wys tht two wttmeters n e onneted to mesure the totl power delivered to the lod of Prolem 16. P Wttmeter FIG. 22.49 Prolem 46. - or Y- onneted lod

1014 POLYPHSE SYSTEMS *48. For the Y-D system of Fig. 22.50:. Determine the mgnitude nd ngle of the phse urrents.. Find the mgnitude nd ngle of the line urrents.. Determine the reding of eh wttmeter. d. Find the totl power delivered to the lod. I W 1 I E = 208 V 120 E = 208 V 0 I E = 208 V 120 W 2 R 1 10 R 3 10 R 2 X I 10 10 I 10 I X L FIG. 22.50 Prolem 48. E = 208 V 0 E = 208 V 240 2 10 10 n 12 2 12 SETION 22.13 Unlned, Three-Phse, Four-Wire, Y-onneted Lod *49. For the system of Fig. 22.51:. lulte the mgnitude of the voltge ross eh phse of the lod.. Find the mgnitude of the urrent through eh phse of the lod.. Find the totl wtts, volt-mperes retive, voltmperes, nd F p of the system. d. Find the phse urrents in phsor form. e. Using the results of prt (), determine the urrent I N. E = 208 V 120 E = 200 V 0 E = 200 V 240 FIG. 22.51 Prolem 49. 20 SETION 22.14 Unlned, Three-Phse, 12 Three-Wire, Y-onneted Lod *50. For the three-phse, three-wire system of Fig. 22.52, find the mgnitude of the urrent through eh phse of the n 16 3 lod, nd find the totl wtts, volt-mperes retive, voltmperes, nd F p of the lod. 4 E = 200 V 120 FIG. 22.52 Prolem 50.

GLOSSRY 1015 GLOSSRY -onneted genertor three-phse genertor hving the three phses onneted in the shpe of the pitl Greek letter delt (D). Line urrent The urrent tht flows from the genertor to the lod of single-phse or polyphse system. Line voltge The potentil differene tht exists etween the lines of single-phse or polyphse system. Neutrl onnetion The onnetion etween the genertor nd the lod tht, under lned onditions, will hve zero urrent ssoited with it. Phse urrent The urrent tht flows through eh phse of single-phse or polyphse genertor lod. Phse sequene The order in whih the generted sinusoidl voltges of polyphse genertor will ffet the lod to whih they re pplied. Phse voltge The voltge tht ppers etween the line nd neutrl of Y-onneted genertor nd from line to line in D-onneted genertor. Polyphse genertor n eletromehnil soure of power tht genertes more thn one sinusoidl voltge per rottion of the rotor. The frequeny generted is determined y the speed of rottion nd the numer of poles of the rotor. Single-phse genertor n eletromehnil soure of power tht genertes single sinusoidl voltge hving frequeny determined y the speed of rottion nd the numer of poles of the rotor. Three-wttmeter method method for determining the totl power delivered to three-phse lod using three wttmeters. Two-wttmeter method method for determining the totl power delivered to D- or Y-onneted three-phse lod using only two wttmeters nd onsidering the power ftor of the lod. Unlned polyphse lod lod not hving the sme impedne in eh phse. Y-onneted three-phse genertor three-phse soure of power in whih the three phses re onneted in the shpe of the letter Y.