Polyphse Systems 23 Ojetives eome fmilir with the opertion of threephse genertor nd the mgnitude nd phse reltionship etween the three phse voltges. e le to lulte the voltges nd urrents for three-phse Y-onneted genertor nd Y-onneted lod. Understnd the signifine of the phse sequene on the generted voltges of three-phse Y-onneted or -onneted genertor. e le to lulte the voltges nd urrents for three-phse -onneted genertor nd -onneted lod. Understnd how to lulte the rel, retive, nd pprent power to ll the elements of Y- or -onneted lod nd e le to mesure the power to the lod. 23.1 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 is 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. In 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 periods. On irrft nd ships, the demnd levels permit the use of 400 Hz line frequeny.
1030 POLYPHSE SYSTEMS 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 single-phse nd two-phse inputs re required, they re supplied y one nd two phses of threephse 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. 23.2 THREE-PHSE GENERTOR The three-phse genertor in Fig. 23.1() hs three indution oils pled 120 prt on the sttor, s shown symolilly y Fig. 23.1(). 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 hs 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 re generted simultneously, s shown in Fig. 23.2. 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. 23.1 () 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. This is shown t vt 0 in Fig. 23.2, 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
Y-ONNETED GENERTOR 1031 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. 23.2 Phse voltges of three-phse genertor. 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 in Fig. 23.2 is E N 120 e N E m1n2 sin vt e N E m1n2 sin1vt 120 2 e N E m1n2 sin1vt 240 2 E m1n2 sin1vt 120 2 (23.1) 120 120 E N The phsor digrm of the indued voltges is shown in Fig. 23.3, where the effetive vlue of eh is determined y E N E N 0.707E m(n) E N 0.707E m(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. 23.4 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. 23.3 Phsor digrm for the phse voltges of three-phse genertor. E N E N E N E N E N E N 0 (23.2) 23.3 Y-ONNETED GENERTOR If the three terminls denoted N in Fig. 23.1() re onneted together, the genertor is referred to s Y-onneted three-phse genertor FIG. 23.4 Demonstrting tht the vetor sum of the phse voltges of three-phse genertor is zero.
1032 POLYPHSE SYSTEMS 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. 23.5 Y-onneted genertor. E E E N N E N (phse voltge) E N E (line voltge) (Fig. 23.5). s indited in Fig. 23.5, 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, threephse, four-wire genertor. The funtion of the neutrl will e disussed in detil when we onsider the lod iruit. The three ondutors onneted from,, nd to the lod re lled lines. For the Y-onneted system, it should e ovious from Fig. 23.5 tht the line urrent equls the phse urrent for eh phse; tht is, I L I fg (23.3) FIG. 23.6 Line nd phse voltges of the Y-onneted three-phse genertor. 120 E N E N α = 60 120 E N α = 60 x β = 30 E N E x FIG. 23.7 Determining line voltge for three-phse genertor. 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. 23.6) 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 in Fig. 23.6, 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. 23.7. Sine eh phse voltge, when reversed (E N ), isets the other two, 60. The ngle is 30 sine line drwn from opposite ends of rhomus divides in hlf oth the ngle of origin nd the opposite ngle. Lines drwn etween opposite orners of rhomus lso iset eh other t right ngles. The length x is x E N os 30 13 2 E N nd E 2x 122 13 2 E N 13E N
PHSE SEQUENE (Y-ONNETED GENERTOR) 1033 Noting from the phsor digrm tht u of E 30, the result is E E 30 13E N 30 nd E 13E N 150 E 13E N 270 In words, the mgnitude of the line voltge of Y-onneted genertor is 13 times the phse voltge: E L 13E f (23.4) with the phse ngle etween ny line voltge nd the nerest phse voltge t 30. In sinusoidl nottion, e 12E sin1vt 30 2 e 12E sin1vt 150 2 nd e 12E sin1vt 270 2 The phsor digrm of the line nd phse voltges is shown in Fig. 23.8. If the phsors representing the line voltges in Fig. 23.8() re rerrnged slightly, they will form losed loop [Fig. 23.8()]. Therefore, we n onlude tht the sum of the line voltges is lso zero; tht is, E E E 0 (23.5) E E N 30 120 120 120 30 E N E E E E N 30 () E () E E N Fixed point P FIG. 23.8 () 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. N E N Rottion 23.4 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. 23.9 the phse sequene is. However, sine the fixed point n e hosen nywhere on the phsor digrm, the sequene n E N FIG. 23.9 Determining the phse sequene from the phse voltges of three-phse genertor.
1034 POLYPHSE SYSTEMS E E P E Rottion FIG. 23.10 Determining the phse sequene from the line voltges of three-phse genertor. lso e written s or. The phse sequene is quite importnt in the three-phse distriution of power. In three-phse 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. 23.10, 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 in Fig. 23.10, 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. 23.9. 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. 23.11()] if we wnted the phsor digrm of the line voltges, or E N for the phse voltges [Fig. 23.11()]. For the sequene indited, the phsor digrms would e s in Fig. 23.11. In phsor nottion, E E 0 1referene2 Line voltges E E 120 E E 120 E Phse N E N 0 1referene2 voltges E N E N 120 E N E N 120 E P E N P E E N E E N () () FIG. 23.11 Drwing the phsor digrm from the phse sequene.
Y-ONNETED GENERTOR WITH Y-ONNETED LOD 1035 23.5 Y-ONNETED GENERTOR WITH Y-ONNETED LOD Lods onneted to three-phse supplies re of two types: the Y nd the. 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. 23.12. 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. 23.12 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 then I N will e zero. (This will e demonstrted in Exmple 23.1.) 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 use only one of the phse voltges, nd even if the loding is designed to e lned (s it should e), there is never perfet ontinuous lning sine lights nd other eletril equipment re 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 is 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 (23.6) For lned or n unlned lod, sine the genertor nd lod hve ommon neutrl point, then
1036 POLYPHSE SYSTEMS V f E f (23.7) In ddition, sine I fl V f /Z f, the mgnitude of the urrent in eh phse is equl for lned lod nd unequl for n unlned lod. Rell tht for the Y-onneted genertor, the mgnitude of the line voltge is equl to 13 times the phse voltge. This sme reltionship n e pplied to lned or n unlned four-wire Y-onneted lod: E L 13V f (23.8) 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. 23.13, in whih the stndrd doule susripts for soure of voltge nd voltge drop re indited. E N 120 V 0 EXMPLE 23.1 The phse sequene of the Y-onneted genertor in Fig. 23.13 is.. Find the phse ngles u 2 nd u 3.. Find the mgnitude of the line voltges.. Find the line urrents. d. Verify tht, sine the lod is lned, I N 0. E I I n 3 V n 120 V θ 3 N 120 V θ 2 E N E N E I N I 3 V n I n n 4 4 lned lod 4 Vn 3 I n E I FIG. 23.13 Exmple 23.1. Solutions:. For n phse sequene, u 2 120 nd u 3 120. E L 13E f 11.7321120 V2 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 I fl I n V n 120 V 0 120 V 0 Z n 3 j 4 5 53.13 24 53.13
Y- SYSTEM 1037 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 I N I I I In retngulr form, I n V n 120 V 120 Z n I n V n Z n I 24 53.13 I 24 173.13 I 24 66.87 g 1I I I 2 5 53.13 120 V 120 5 53.13 24 173.13 24 66.87 14.40 j 19.20 22.83 j 2.87 9.43 j 22.07 0 j 0 nd I N is in ft equls to zero, s required for lned lod. 23.6 Y- SYSTEM There is no neutrl onnetion for the Y- system in Fig. 23.14. ny vrition in the impedne of phse tht produes n unlned system simply vries the line nd phse urrents of the system. For lned lod, Z 1 Z 2 Z 3 (23.9) 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 (23.10) 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 I L FIG. 23.14 Y-onneted genertor with -onneted lod.
1038 POLYPHSE SYSTEMS The reltionship etween the line urrents nd phse urrents of lned lod n e found using n pproh very similr to tht used in Setion 23.3 to find the reltionship etween the line voltges nd phse voltges of Y-onneted genertor. For this se, however, Kirhhoff s urrent lw is used insted of Kirhhoff s voltge lw. The results otined re I L 13I f (23.11) 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 -onneted system n e found in Setion 23.7. For lned lod, the line urrents will e equl in mgnitude, s will the phse urrents. EXMPLE 23.2 For the three-phse system in Fig. 23.15:. Find the phse ngles u 2 nd u 3.. Find the urrent in eh phse of the lod.. Find the mgnitude of the line urrents. 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. 23.15 Exmple 23.2. Solutions:. For n sequene, u 2 120 nd u 3 120. V f E L. Therefore, V E V E V E The phse urrents re I V 150 V 0 150 V 0 15 53.13 Z 6 j 8 10 53.13 I V 150 V 120 15 173.13 Z 10 53.13 I V 150 V 120 15 66.87 Z 10 53.13
-ONNETED GENERTOR 1039. I L 13I f 11.732115 2 25.95. Therefore, I I I 25.95 23.7 -ONNETED GENERTOR If we rerrnge the oils of the genertor in Fig. 23.16() s shown in Fig. 23.16(), the system is referred to s three-phse, three-wire, -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 12E N sin vt E E N nd e N 12E N sin1vt 120 2 E E N nd e N 12E N sin1vt 120 2 Phse sequene or E L E fg (23.12) 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. 23.16 -onneted genertor. 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 -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. 23.17 for lned lod. Using the sme proedure to find the line urrent s ws used to find the line voltge of Y-onneted genertor produes the following: In generl: I 13I 30 I 13I 150 I 13I 90 I L 13I fg (23.13)
1040 POLYPHSE SYSTEMS I I 120 I 120 60 30 30 I 3 2 I I 3 2 I I I I 30 120 120 120 30 30 I I I = 3 I I FIG. 23.17 Determining line urrent from the phse urrents of -onneted, three-phse genertor. FIG. 23.18 The phsor digrm of the urrents of three-phse, -onneted genertor. with the phse ngle etween line urrent nd the nerest phse urrent t 30. The phsor digrm of the urrents is shown in Fig. 23.18. Just s for the voltges of Y-onneted genertor, the phsor sum of the line urrents or phse urrents for -onneted systems with lned lods is zero. E Rottion P E Phse sequene: 23.8 PHSE SEQUENE ( -ONNETED GENERTOR) Even though the line nd phse voltges of -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. 23.19. 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 0 E E 120 E E 120 E FIG. 23.19 Determining the phse sequene for -onneted, three-phse genertor. 23.9 -, -Y THREE-PHSE SYSTEMS The si equtions neessry to nlyze either of the two systems ( -, -Y) hve een presented t lest one in this hpter. Following re two desriptive exmples, one with -onneted lod nd one with Y-onneted lod. EXMPLE 23.3 For the - system shown in Fig. 23.20:. Find the phse ngles u 2 nd u 3 for the speified phse sequene.. Find the urrent in eh phse of the lod.. Find the mgnitude of the line urrents.
-, -Y THREE-PHSE SYSTEMS 1041 3-phse, 3-wire Δ-onneted genertor Phse sequene: I E = 120 V 0 V 5 I 5 5 5 V 5 E = 120 V v 3 I I I 5 V I E = 120 V v 2 FIG. 23.20 Exmple 23.3: - system. Solutions:. For n phse sequene, u 2 120 nd u 3 120. V f E L. Therefore, V E V E V E The phse urrents re I V Z 5 j 5 120 V 0 33.9 45 3.54 45 120 V 120 33.9 165 I V Z I V Z 120 V 0 15 0 215 90 2 3.54 45 120 V 120 33.9 75 3.54 45. I L 13I f 11.732134 2 58.82. Therefore, I I I 58.82 120 V 0 25 90 7.071 45 EXMPLE 23.4 For the -Y system shown in Fig. 23.21:. 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
1042 POLYPHSE SYSTEMS I = 2 0 I n 8 V n 6 3-phse, 3-wire Δ-onneted genertor Phse sequene: E E 8 6 n 6 8 I n V n V n I n I = 2 120 E I = 2 120 FIG. 23.21 Exmple 23.4: -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 13V f 11.732120 V2 34.6 V. Therefore, E E E 34.6 V 23.10 POWER Y-onneted lned Lod Plese refer to Fig. 23.22 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. 23.22 Y-onneted lned lod.
POWER 1043 verge Power The verge power delivered to eh phse n e determined y Eq. (23.14). P f V f I f os u If V f I f 2 R f V R 2 R f (wtts, W) (23.14) u If V where f indites tht u is the phse ngle etween V f nd I f. The totl power delivered n e determined y Eq. (23.15) or Eq. (23.16). P T 3P f (W) (23.15) or, sine V f E L 13 nd I f I L then ut Therefore, P T 3 E L 13 I V L os u f If 3 13 112 3 13 13 313 13 13 3 V P T 13E L I L os u f If 3I 2 L R f (W) (23.16) Retive Power retive) is The retive power of eh phse (in volt-mperes V Q f V f I f sin u f If I 2 f X f V X 2 X f (VR) (23.17) The totl retive power of the lod is Q T 3Q f (VR) (23.18) or, proeeding in the sme mnner s ove, we hve V Q T 13E L I L sin u f If 3I 2 L X f (VR) (23.19) pprent Power The pprent power of eh phse is S f V f I f (V) (23.20) The totl pprent power of the lod is S T 3S f (V) (23.21) or, s efore, S T 13E L I L (V) (23.22)
1044 POLYPHSE SYSTEMS Power Ftor The power ftor of the system is given y F p P T V os u f If 1leding or lgging2 S T (23.23) EXMPLE 23.5 For the Y-onneted lod in Fig. 23.23: R = 3 E L = 173.2 V 0 X L = 4 E L = 173.2 V 120 X L = 4 V I R = 3 I V n X L = 4 V I R = 3 E L = 173.2 V 120 FIG. 23.23 Exmple 23.5.. 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 or V P T 13E L I L os u f If 11.73221173.2 V2120 210.62 3600 W. The retive power is or V P f V f I f os u f If 1100 V2120 2 os 53.13 12000210.62 1200 W P f I 2 f R f 120 2 2 13 2 14002132 1200 W P f V R 2 160 V22 R f 3 3600 3 1200 W P T 3P f 13211200 W2 3600 W V Q f V f I f sin u f If 1100 V2120 2 sin 53.13 12000210.82 1600 VR Q f IfX 2 f 120 2 2 14 2 14002142 1600 VR Q T 3Q f 13211600 VR2 4800 VR or V Q T 13E L I L sin u f If 11.73221173.2 V2120 210.82 4800 VR
POWER 1045. 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 13E L I L 11.73221173.2 V2120 2 6000 V d. The power ftor is F p P T 3600 W 0.6 lgging S T 6000 V -onneted lned Lod Plese refer to Fig. 23.24 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. 23.24 -onneted lned lod. verge Power P f V f I f os u If V f I f 2 R f V R 2 R f (W) (23.24) P T 3P f (W) (23.25) Retive Power V Q f V f I f sin u f If I 2 f X f V X 2 X f (VR) (23.26) Q T 3Q f (VR) (23.27) pprent Power S f V f I f (V) (23.28)
1046 POLYPHSE SYSTEMS S T 3S f 13E L I L (V) (23.29) Power Ftor F p P T S T (23.30) EXMPLE 23.6 For the -Y onneted lod in Fig. 23.25, find the totl verge, retive, nd pprent power. In ddition, find the power ftor of the lod. 6 4 8 E L = 200 V 0 E L = 200 V 120 3 8 6 4 3 3 4 6 8 E L = 200 V 120 FIG. 23.25 Exmple 23.6. Solution: onsider the nd Y seprtely. For the : For the Y: Z 6 j 8 10 53.13 I f E L 200 V 20 Z 10 P T 3I 2 f R f 132120 2 2 16 2 7200 W Q T 3I 2 f X f 132120 2 2 18 2 9600 VR 12 S T 3V f I f 1321200 V2120 2 12,000 V Z Y 4 j 3 5 36.87 I f E L> 13 200 V> 13 116 V 23.12 Z Y 5 5 P TY 3I 2 f R f 132123.12 2 2 14 2 6414.41 W Q TY 3I 2 f X f 132123.12 2 2 13 2 4810.81 VR 1L2 S TY 3V f I f 1321116 V2123.12 2 8045.76 V
POWER 1047 For the totl lod: P T P T P TY 7200 W 6414.41 W 13,614.41 W Q T Q T Q TY 9600 VR 12 4810.81 VR 1I2 4789.19 VR 12 S T 2P T 2 Q T 2 2113,614.41 W2 2 14789.19 VR2 2 14,432.2 V F p P T 13,614.41 W 0.943 leding S T 14,432.20 V EXMPLE 23.7 Eh trnsmission line of the three-wire, three-phse system in Fig. 23.26 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. 23.26 Exmple 23.7.. 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 f 1lod2 V L 12,000 V 6936.42 V 13 1.73 P T 1lod2 3V f I f os u nd I f P T 3V f os u 160,000 W 316936.42 V210.862 8.94 Sine u 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. 23.27, where E N I f Z line V f 0
1048 POLYPHSE SYSTEMS I L 15 20 I f Z line I f = 8.94 30.68 E N Z 1 V f FIG. 23.27 The loding on eh phse of the system in Fig. 23.26. 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 13E fg 11.73217143.5 V2 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 or nd os u T P o P T 13V L I L os u T P T 163,596.55 W 13V L I L 11.732112,358.26 V218.94 2 F p 0.856 < 0.86 of lod h P o 160 kw 0.978 P i P o P losses 160 kw 3596.55 W 97.8% 23.11 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. 23.28. 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 (23.31) For the lod (lned or unlned), the wttmeters re onneted s shown in Fig. 23.29. The totl power is gin the sum of the three wttmeter redings: P T P 1 P 2 P 3 (23.32)
TWO-WTTMETER METHOD 1049 Line P 1 1 P 1 Z 1 Line 1 P 1 Neutrl n P 3 P 1 P 2 P 2 2 Line Z 3 Z 2 Z 3 Z 1 P 3 P 2 P 3 3 P 3 Line Line Line 3 Z 2 P 2 2 FIG. 23.28 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. 23.12 TWO-WTTMETER METHOD The power delivered to three-phse, three-wire, - or Y-onneted, lned or unlned lol 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. 23.30. One end of eh potentil oil is onneted to the sme line. The urrent oils re then pled in the remining lines. The onnetion shown in Fig. 23.31 lso stisfies 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 now onsider two 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 re le to find the power ftor (leding or lgging) of ny one FIG. 23.29 Three-wttmeter method for -onneted lod. P 1 1 P 1 P 2 P 2 2 Line Line Line - or Y- onneted lod FIG. 23.30 Two-wttmeter method for - or Y-onneted lod. Line P 1 1 P 1 Line - or Y- onneted lod P 2 2 P 2 Line FIG. 23.31 lterntive hookup for the two-wttmeter method.
1050 POLYPHSE SYSTEMS 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. 23.32 Determining whether the redings otined using the two-wttmeter method should e dded or sutrted. phse of the lod. When this informtion hs een otined, it n e pplied diretly to the urve in Fig. 23.32. The urve in Fig. 23.32 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 lowernd 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 revels tht, when the power ftor is 1 (os 0 1), orresponding to purely resistive lod, P l /P h 1 or P l P h, nd oth wttmeters 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 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 u os 1 0.5 60. t this power ftor, P l /P h 0, so tht P l 0, while P h reds 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 upsle 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. 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.
TWO-WTTMETER METHOD 1051 For lned system, sine P T P h P 1 13E L I L os u If V f the power ftor of the lod (phse) n e found from the wttmeter redings nd the mgnitude of the line voltge nd urrent: V F p os u f If P h P l 13E L I L (23.33) EXMPLE 23.8 For the unlned -onneted lod in Fig. 23.33 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. 23.33 Exmple 23.8.. 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:. I V E 208 V 0 20.8 0 Z Z 10 0 I V E 208 V 120 208 V 120 Z Z 15 j 20 25 53.13 8.32 173.13 I V E 208 V 120 208 V 120 Z Z 12 j 12 16.97 45 12.26 165. 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
1052 POLYPHSE SYSTEMS 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. V P 1 V I os u I V 208 V 0 I 32.79 5.55 1208 V2132.79 2 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 V P 2 V I os u I 1208 V215.5 2 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.) 23.13 UNLNED, THREE-PHSE, FOUR-WIRE, Y-ONNETED LOD For the three-phse, four-wire, Y-onneted lod in Fig. 23.34, 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 (23.34) The phse urrents n therefore e determined y Ohm s lw: I f1 V f 1 Z 1 E f 1 Z 1 nd so on (23.35)
UNLNED, THREE-PHSE, FOUR-WIRE, Y-ONNETED LOD 1053 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. 23.34 Unlned Y-onneted lod. 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 L2 I L3 (23.36) 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 used to ensure stedy voltge levels nd to provide pth for the urrent resulting from n unlned lod. The system in Fig. 23.35 hs three-phse trnsformer dropping the line voltge from 13,800 V to 208 V. ll the lower-power-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 high-demnd equipment, the full three-phse power n e tken diretly off the system, s shown in Fig. 23.35. In the design nd onstrution of ommeril estlishment, the 208 V 0 208 V 120 208 V 120 1φ 120 V 1φ 208 V 1φ 120 V 1φ 208 V 3φ 208 V lned lod Seondry 3 φ trnsformer FIG. 23.35 3f/1f, 208 V/120 V industril supply.
1054 POLYPHSE SYSTEMS Ntionl Eletri ode requires tht every effort e mde to ensure tht the expeted lods, whether they e single- or multiphse, result in totl lod tht is s lned s possile etween the phses, thus ensuring the highest level of trnsmission effiieny. 23.14 UNLNED, THREE-PHSE, THREE-WIRE, Y-ONNETED LOD For the system shown in Fig. 23.36, the required equtions n e derived y first pplying Kirhhoff s voltge lw round eh losed loop to produe E V n V n 0 E V n V n 0 E V n V n 0 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. 23.36 Unlned, three-phse, three-wire, Y-onneted lod. Sustituting, we hve 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 1 123.372 123.372 123.372 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. (23.37) nd (23.37) 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 )] Using determinnts, we hve
UNLNED, THREE-PHSE,THREE-WIRE, Y-ONNETED LOD 1055 I n E ` Z 2 E 1Z 2 Z 3 2 ` Z ` 1 Z 2 Z 2 Z 2 1Z 2 Z 3 2 ` 1Z 2 Z 3 2E E Z 2 Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 Z 2 2 Z 2 2 I n Z 21E E 2 Z 3 E 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 21 E 2 Z 3 E Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 nd I n E Z 3 E Z 2 (23.38) Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 In the sme mnner, it n e shown tht I n E Z 2 E Z 1 Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 (23.39) Sustituting Eq. (23.39) for I n in the right-hnd side of Eq. (23.37), we otin I n E Z 1 E Z 3 Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 (23.40) EXMPLE 23.9 phse-sequene inditor is n instrument tht n disply the phse sequene of polyphse iruit. network tht performs this funtion ppers in Fig. 23.37. The pplied phse sequene is. The ul orresponding to this phse sequene urns more rightly thn the ul inditing the sequene euse greter urrent is pssing through the ul. lulting the phse urrents demonstrtes tht this sitution does in ft exist: Z 1 X 1 v 1 166 1377 rd>s2116 10 6 F2 y Eq. (23.39), I n E Z 2 E Z 1 Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 1200 V 120 21200 0 2 1200 V 120 21166 90 2 1166 90 21200 0 2 1166 90 21200 0 2 1200 0 21200 0 2
1056 POLYPHSE SYSTEMS (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. 23.37 Exmple 23.9. I n Dividing the numertor nd denomintor y 1000 nd onverting oth to the retngulr domin yields y Eq. (23.40), I n E Z 1 E Z 3 Z 1 Z 2 Z 1 Z 3 Z 2 Z 3 1200 V 120 21166 90 2 1200 V 0 21200 0 2 77.52 10 3 58.93 I n 40,000 V 120 33,200 V 30 33,200 90 33,200 90 40,000 0 1 20 j 34.642 128.75 j 16.602 I n 40 j 66.4 33,200 V 210 40,000 V 0 77.52 10 3 58.93 Dividing y 1000 nd onverting to the retngulr domin yields 28.75 j 16.60 40.0 68.75 j 16.60 I n 77.52 58.93 77.52 58.93 8.75 j 18.04 77.52 58.93 I n 0.259 123.06 20.05 64.13 77.52 58.93 70.73 166.43 0.91 225.36 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.
PROLEMS 1057 PROLEMS SETION 23.5 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 in Fig. 23.38 is.. Find the ngles u 2 nd u 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. 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. 7. For the system in Fig. 23.39, find the mgnitude of the unknown voltges nd urrents. 120 V 0 20 120 V 0 θ 3 N 120 V 02 θ 20 20 FIG. 23.38 Prolems 4, 5, 6, nd 31. I I n 3-phse, 3-wire 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. 23.39 Prolems 7, 32, nd 44.
1058 POLYPHSE SYSTEMS *8. ompute the mgnitude of the voltge E for the lned three-phse system in Fig. 23.40. 1 16 12 3-phse, 3-wire Y-onneted genertor E Line resistne 12 n 12 1 16 V = 50 V φ 16 1 FIG. 23.40 Prolem 8. *9. For the Y-Y system in Fig. 23.41:. 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 E = 22 kv 0 N E = 22 kv 120 I 30 40 40 I n 0.4 k V n 1 k 1 k V n 0.4 k I n 1 k I n V n 0.4 k E = 22 kv 120 I 30 40 FIG. 23.41 Prolem 9.
PROLEMS 1059 SETION 23.6 Y- System 10. lned 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- system in Fig. 23.42 is.. Find the ngles u 2 nd u 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 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. 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 in Fig. 23.43, find the mgnitude of the unknown voltges nd urrents. E = 208 V 0 E = 208 V v 3 22 22 N 22 E = 208 V v 2 FIG. 23.42 Prolems 13, 14, 15, 34, nd 45. I 3-phse, 3-wire 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. 23.43 Prolems 16, 35, nd 47.
1060 POLYPHSE SYSTEMS *17. For the -onneted lod in Fig. 23.44:. 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. 23.44 Prolem 17. SETION 23.9 -, -Y Three-Phse Systems 18. lned Y lod hving 30 resistne in eh leg is onneted to three-phse, -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 in Fig. 23.45, find the mgnitude of the unknown voltges nd urrents. I I n 3-phse, 3-wire Δ-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. 23.45 Prolems 21, 22, 23, nd 37.
PROLEMS 1061 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 lod hving 220 resistne in eh leg is onneted to three-phse, -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. 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 - system in Fig. 23.46 is.. Find the ngles u 2 nd u 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 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. 23.46 Prolem 27. 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 23.10 Power 30. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system in Prolem 2. 31. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system in Prolem 4. 32. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system in Prolem 7.
1062 POLYPHSE SYSTEMS 33. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system in Prolem 12. 34. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system in Prolem 14. 35. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system in Prolem 16. 36. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system in Prolem 20. 37. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system in Prolem 22. 38. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system in Prolem 26. 39. Find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the three-phse system in Prolem 28. 40. lned, three-phse, -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 in Fig. 23.47. 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. 23.47 Prolem 42. *43. The Y-Y system in Fig. 23.48 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.
PROLEMS 1063 N I 4 20 E N V n E = 16 kv Z 1 n I n = 80 Z 1 = Z 2 = Z 3 lgging F p 4 20 Z 2 Z 3 4 20 FIG. 23.48 Prolem 43. 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. SETION 23.11 Three-Wttmeter Method 44.. Sketh the onnetions required to mesure the totl wtts delivered to the lod in Fig. 23.39 using three wttmeters.. Determine the totl wttge dissiption nd the reding of eh wttmeter. 45. Repet Prolem 44 for the network in Fig. 23.42. SETION 23.12 Two-Wttmeter Method 46.. For the three-wire system in Fig. 23.49, properly onnet seond wttmeter so tht the two 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. P Wttmeter Δ- or Y- onneted lod FIG. 23.49 Prolem 46. 47. Sketh three different wys tht two wttmeters n e onneted to mesure the totl power delivered to the lod in Prolem 16.
1064 POLYPHSE SYSTEMS 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. 23.50 Prolem 48. *48. For the Y- system in Fig. 23.50:. 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. SETION 23.13 Unlned, Three-Phse, Four-Wire, Y-onneted Lod *49. For the system in Fig. 23.51:. 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 0 E = 208 V 240 E = 208 V 120 2 FIG. 23.51 Prolem 49. 2 n 10 10 SETION 23.14 Unlned, Three-Phse, Three-Wire, Y-onneted Lod 12 12 *50. For the three-phse, three-wire system in Fig. 23.52, find the mgnitude of the urrent through eh phse of the lod, nd E = 200 V 0 E = 200 V 240 E = 200 V 120 20 FIG. 23.52 Prolem 50. n 12 16 3 4 find the totl wtts, volt-mperes retive, volt-mperes, nd F p of the lod. GLOSSRY -onneted genertor three-phse genertor hving the three phses onneted in the shpe of the pitl Greek letter delt ( ). 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 or 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 -onneted genertor.
GLOSSRY 1065 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 - 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.