Mdule B Prblem The -hase lads are cnnected n arallel. One s a urely resste lad cnnected n wye. t cnsumes 00kW. The secnd s a urely nducte 00kR lad cnnected n wye. The thrd s a urely caacte 00kR lad cnnected n wye. The lne-t-lne ltage at the lad s 5k. -hase dstrbutn lne sulyng ths lad has an medance f 0j5 hms er hase. (a Calculate the currents drawn by each lad (magntude and hase. (b ndcate the wer factr f each lad. Remember that nn-unty wer factrs must als nclude whether they are laggng r leadng. (c What s the wer factr f the entre lad? That s, what s the wer factr seen by the transmssn lne at the lad end? (d Calculate the real and reacte wer suled at the sendng end f the dstrbutn lne. lutn t rblem 0 j5ω N 886. 8 5k (a ad 886. 8 0 Nte that:, Therefre: φ 00 0 W, j00 0 R, j00 0 R. fr urely resste lad: fr urely nducte lad: fr urely caacte lad: 00 0 4. 64 0 ad 886. 8 j00 0 4. 64 90 ad 886. 8 ad j00 0 886. 8 4. 64 90
Nte n the abe that fr the resste lad, the current and ltage are n hase, fr the nducte, the current lags by 90 deg, and fr the caacte, the current leads by 90 deg. (b ad : f. 0 ad : ad : f 0 laggng f 0 leadng (c Need current angle wth resect t 886.98 0 ad ad 4. 64 0 j4. 64 j4. 64 4 64 0. s ad and ad f ad.0 are n hase! (d s ad ad ( Zt 886. 8 ( 4. 64 0 ( 0 j5 7. 8. 07 ( 7. 8. 07 ( 4. 64 0 6, 47. 07 φ s ad P φ 5. 99kW, Q φ 8. 0kR lternately, ne culd cmute lsses and add t lad : lss ( φ ( ad Zt ( 4. 64 ( 0 j5 5, 999. 9 j7, 998. 9 φ lad ( φ lss( φ 00 0 P φ 5.99kW Q φ 8.0kR 5,997.9 j7,998.9 Prblem three hase lad has a er hase medance, cnnected n Y, f 00 j 0Ω. The lne-t-lne ltage magntude at the lad s 500. The three-hase dstrbutn lne sulyng ths lad has an medance f 0 j 5Ω /φ. (a Calculate the lne-t-lne ltage magntude at the sendng end f the dstrbutn lne. (b Calculate the real and reacte wer suled at the sendng end f the dstrbutn lne. lutn t rblem (a 500 N 866. 05 N 866. 05 7. 945 j. 84 8. 95 6. 7 Z 00 j0 sendng,n 957. 9 N endng,b Z ne OD 866. 05 0 j5. 89 957. 5 0. 95 957. 5 658. 5 ( 8. 95 6. 7 ( 0 j5 (b ( 957. 5 0. 95 ( 8. 95 6. 7, 706. 5 j74. sendng,n a 7 P.706kW, Q 7. 5kR
Prblem three-hase lad cnsumes 00k at 0.7 f laggng. The lne-t-lne ltage magntude at the lad s 500. The three-hase dstrbutn lne sulyng ths lad has an medance f 0 j 5Ω / φ (a Calculate the lne-t-lne ltage magntude at the sendng end f the dstrbutn lne. (b Calculate the real and reacte wer suled at the sendng end f the dstrbutn lne. lutn t rblem (a Nte that θcs - (0.745.57 deg (the angle s ste because the f s laggng, and sn(45.70.74. 500 00 0 an 866. 05 0, φ ( 0. 7 j0. 74 φ. 0 ( 0. 7 j0. 74.. 8 45 45 57 an 866. 05 Z 866. 05 0 8. 45 45. 57 ( 0 j5 79. 6. sendng,an sendng,b an 79. 5. (b ( ( ( 79. 6. ( 8. 45 45. 57 47. 556 9. 6 sendng an,sendng Psendng 4. 09kW, Qsendng 9. 58kR Prblem 4 The cmlex wer absrbed by a three-hase lad s 500k at 0.8 f lag P Q / φ f the ne ltage at the lad n rblem s 8660.540, what s the ltage magntude acrss each hase f the lad, f the lad s cnnected as fllws, d What s the magntude f lne current drawn by ths lad? / φ y lutn t rblem 4 The cmlex wer absrbed by a three-hase lad s 500k at 0.8 f lag
Nte that θcs - (0.86.87 deg (the angle s ste because the f s laggng, and sn(6.870.6. Then P φ 500(0.8/400 kw, Q φ 500(0.6/00 kw. PØ 400 kw Q/Ø 00 kr f the ne ltage at the lad n rblem s 8660.540, what s the ltage magntude acrss each hase f the lad, f the lad s cnnected as fllws, Δ 8660.54 y 5000 8660.54 What s the magntude f lne current drawn by ths lad? 00 500x0 8660. 54 Prblem 5 n the crcut shwn belw, an,000 j 0 (rms. ssume ste hase sequence. The balanced surce sules.5 MW and 0. MR t the three hase balanced lad. Fnd: a The rms lne current. b Z a 5 cn Nn an bn Bb 5 B Z P B BC Z P C ZP C 5 lutn t rblem 5 6 (a φ (.5 j0. 0
6 φ (. 5 j0. 0 anan φ, 000an 6 (. 5 j0. x0 an 4. 49., 000 4. 49. an (b Z Z y Z y an a N an,75.68 0.696 Z Py Z Py a 5, 000 4. 49. 5,75.68 0.696 67.75.99 4.49. x67.75.99 80.405.99 Ω Prblem 6 three hase surce s sulyng a balanced three hase lad er a transmssn lne hang medance f Z j0 hms er hase. The ltage at the surce end f the transmssn lne s 887 0 lts lne t neutral. The current suled thrugh the transmssn lne s 00-0 ameres.. Determne the wer factr seen by the surce, and secfy whether t s leadng r laggng.. Determne the ltage (lne t neutral at the lad.. Determne the wer factr f the lad, and secfy whether the lad s a. leadng r laggng b. nducte r caacte 4. Determne the real and reacte wer cnsumed by the lad. lutn t rblem 6. fcs(00.8660, and t s laggng.. OD 887 0-00 -0(j07.8-j666.5-4.6. f angleangle at whch ltage leads the current-4.6-(-0-.6, s fcs(-.60.97, and the current s leadng the ltage! Ths means the wer factr s leadng (art a and the lad must be caacte (art b. 4. OD ( (66.5-4.6(00 0690044-j6699709.950 -.6 k
Prblem 7 balanced, three-hase lad hang a wer factr f 0.8 laggng s suled by a transmssn lne carryng 00 ams at 5 k lne-t-lne. Cmute the three-hase real and reacte wer delered t the lad. lutn t rblem 7 φ ( ( 5 0 ( 00 59.756 M f 0.8 θ cs ( 0.8 6.9 sn ( θ 0.6 P (59.756 (0.8 47.8MW Q (59.756 (0.6 5.879 MR Prblem 8 balanced, three-hase, delta-cnnected lad cnsumes 50-j0 k at a lne-t-lne ltage f.8 k. Cmute the er-hase medance f ths lad assumng a seres cnnectn between R and X. lutn t rblem 8 (. 8 0 Z 8 jω Z ( 50 j0 0 Z Δ ZY Z Δ ( 8 j 9849 j99ω R 9849Ω X 99Ω( Caacte Prblem 9 three-hase wye-cnnected lad hang medance f Z 00j50 hms er hase s cnnected n arallel wth a three hase delta-cnnected lad hang medance f Z 600j00 hms er hase. The lad s suled by a three-hase wye-cnnected generatr that s drectly ntercnnected wth the lads (.e., there s n transmssn lne between the generatr and the lads. The ltage magntude f the generatr s.8 k lne-t-lne. ssume that the hase t neutral ltage at the generatr s the angle reference.. Draw the three-hase crcut. Clearly dentfy the numercal alues f ne lne t neutral surce ltage hasr and ne-hase medance fr each f lads and.. Draw the er-hase crcut. Clearly dentfy the numercal alues f the surce ltage hasr and the er-hase medances f lads and.. Cmute the three-hase cmlex wer cnsumed by each lad and the ttal, cmlex three-hase wer cnsumed by the tw lads. 4. hw that the ttal, cmlex three-hase wer cnsumed by the tw lads can be cmuted usng the lne current and the lne-t-lne alue f the surce ltage.
lutn t rblem 9. N 7967 Z 00j50 Z 600j00. N 7967 Z 00j50 Z 00j00. We culd use N /Z, N /Z, r we culd get the current nstead. et s d t by gettng the current. N /Z 7967/(00j50 7.498 - j9.79, N ( (7967( 7.498 j9.79 (896.09 j4.0ka P 896. kw, Q 4.0 kr N /Z 7967/(00j00.8680 -j5.940 N ( (7967(.8680 j5.940 (76.68ej80.84ka P 76.7 kw, Q 80.8 kr Ttal 657.8j604.86 P Ttal 657.8 kw, Q Ttal 604.9 kr 4. T 69.598 -j5.069 7.8-0.05, ne,800 Ttal ( (,800(7.8{cs(0.05jsn(0.05} 657.8j604.9 Prblem 0 Cnsder a balanced three-hase surce sulyng a balanced Y- r Δ- cnnected lad wth the fllwng nstantaneus ltages and currents. cs( ωt θ cs( ωt θ an bn cn cs( ωt θ 0 cs( ωt θ 40 a b c cs( ωt θ 0 cs( ωt θ 40 where and are the magntudes f the rms hase ltage and current, resectely. hw that the ttal nstantaneus wer rded t the lad, as the sum f the nstantaneus wers f each hase, s a cnstant. lutn fr Prblem 0
Cnsder a balanced three-hase surce sulyng a balanced Y- r Δ- cnnected lad wth the fllwng nstantaneus ltages cs ω t θ an ( cs( ω t θ 0 bn ( ω t θ 40 cn cs Fr a balanced lad the hase currents are cs ω t θ where ( a b cs( t θ 0 (.4 ω ( ω t θ 40 c cs and are the magntudes f the rms hase ltage and current, resectely. The ttal nstantaneus wer s the sum f the nstantaneus wer f each hase, gen by φ an a bn b cn c ubsttutng fr the nstantaneus ltages and currents cs ω t θ cs ω t θ ( ( φ ( ω t θ 0 cs( ω t θ 0 ( ω t θ 40 cs( ω t θ 40 cs cs Usng the trgnmetrc dentty csx csy cs(x-y cs(xy cs θ θ cs ω t θ θ [ ( ( ] φ [ cs( θ cs( 40 ] θ ω t θ θ [ cs( θ θ cs( ω t θ θ 480 ] The three duble frequency csne terms are ut f hase wth each ther by 0 and add u t zer, and the three-hase nstantaneus wer s φ csθ θ s the angle between hase ltage and hase current r the medance angle. θ θ Prblem three-hase lne has an medance f j4 hms/hase, and the lne feeds tw balanced three-hase lads that are cnnected n arallel. The frst lad s Y-cnnected and has an medance f 0j40 hms/hase. The secnd lad s delta-cnnected and has an medance f 60-j45 hms/hase. The lne s energzed at the sendng end frm a three-hase balanced suly f lne ltage 07.85 lts. Takng the hase ltage a as reference, determne: a. The current, real wer, and reacte wer drawn frm the suly. b. The lne ltage at the cmbned lads. c. The current er hase n each lad. d. The ttal real and reacte wers n each lad and the lne. lutn fr Prblem (a The Δ-cnnected lad s transfrmed nt an equalent Y. The medance er hase f the equalent Y s
60 j45 Z 0 j5 Ω The hase ltage s 07.85 0 The sngle-hase equalent crcut s shwn n the fllwng fgure. The ttal medance s ( 0 j40( 0 j5 Z j4 ( 0 j40 ( 0 j5 j 4 j4 4 Ω wth the hase ltage an 0 0 5 Z 4 The three-hase wer suled s 0 0 5 0 (b The hase ltage at the lad termnal s 0 0 j4 5 0 as reference, the current n hase a s ( ( 800 W ( ( 0 0 j The lne ltage at the lad termnal s ab 0.8 0. ( 9.7 9.64 9.7.8 (c The current er hase n the Y-cnnected lad and n the equalent Y f the Δ lad s 0 j0 j.6 6.4 Z 0 j40 0 j0 4 j 4.47 6.56 Z 0 j5 The hase current n the rgnal Δ-cnnected lad,.e., ab 0 4.47 6.56 0 ab s gen by.58 56.56 (d The three-hase wer absrbed by each lad s.8 0..6 6.4 450 W ( ( 600 ar ( 0. ( 4.47 6.56 00 W - 900 ar j.8 j The three-hase wer absrbed by the lne s
( R jx ( j4( 5 50 W j00 ar t s clear that the sum f lad wers and lne lsses s equal t the wer delered frm the suly,.e., ( 450 j600 ( 00 j900 ( 50 j00 800 W j0 ar Prblem three-hase lne has an medance f 0.4j.7 hms er hase. The lne feeds tw balanced three-hase lads that are cnnected n arallel. The frst lad s absrbng 560.k at 0.707 wer factr laggng. The secnd lad absrbs kw at unty wer factr. The lne-t-lne ltage at the lad end f the lne s 80.5 lts. Determne: a. The magntude f the lne ltage at the surce end f the lne. b. Ttal real and reacte wer lss n the lne. c. Real wer and reacte wer suled at the sendng end f the lne. lutn fr Prblem (a The hase ltage at the lad termnals s 80.5 00 The ttal cmlex wer s R 560. 0.707 j0.707 58 ( φ ( 96 j 660 6.87 k Wth the hase ltage as reference, the current n the lne s R( φ 660,000 6.87 00 6.87 ( 00 0 The hase ltage at the sendng end s 00 0 0.4 j.7 00 6.87 40.7 4.58 The magntude f the lne ltage at the sendng end f the lne s ( 40.7 460 (b The three-hase wer lss n the lne s ( ( ( ( ( φ R jx 0.4 00 j.7 00 kw j8 kar (c The three-hase sendng wer s ( ( φ ( 40.7 4.58 ( 00 6.87 540 kw j477 kar t s clear that the sum f lad wers and the lne lsses s equal t the wer delered frm the suly,.e., 58 j96 j8 540 kw 477 φ j ( φ R( φ ( ( ( kar