Collection of Problems. BALANCED POLYPHASE CIRCUITS Ew E Le" € 2 TROTIT 0 b . ‘0 € co THOTT 0d \ Eg = Elo" (a) (b) Fig. 4 coils having induced emt"s shown in part (b) E ai ? b —— ortantoo le LE (a) Fig. 5. Resultant emf shown in (b) for connection of coils shown in (a) ) Problem 1. In Fig. 4.a connect terminal b to terminal ¢ and compare the resultant voltage E,qwith voltge E. of Fig. Sb. Ans : Exe= Ess Problem 2. (a) Connect terminal d to terminal b in Fig. 4a and find the voltage E., if E = 20 Volts, Ey, and Es have the same vector relation as shown in Fig. 4b, Ans : E., = 120 /-60* Volts. (b) With terminal d connected to terminal b as above, find Ey. Ans : Exc= 120/120° Volts,2 5 c E og = 100 (30 voits . s at Ey 100 (0 vots b 4 Fig 6. See Problem 3 Problem 3, Find the magnitude and vector position of voltage Es in Fig. 6a if Exy and Eos are displaced from each other by 30° in time phase as shown in Fig. 65 Ans > E.. = 51.76 /105° Oscillogram 1. Illustrating the 30° angular displacement between the phase voltages and the systematically labeled line-to-line voltages in a balanced, tuee-phase, wye-comnected load. Effective value of each line-to-line voltage is 100 volts Problem 4. (a) Draw a polar (or single-origin) vector diagram which will represent the same phase voltages and the same line voltages as shown in Oscillogram 1 using Ves as reference. Specify the effective magnitude of the phase voltages, the sequence of the phase voltages, and the seqthe sequence of the line voltages.Aus : Viphase = $7.7 Volts. Phase voltage sequence : an-be-ca. Line voltage sequence : ab-be-ca. (b) Draw a polar (or single-origin) vector diagram which will represent the same phase voltages as shown in Oscillogram 1, namely V.,, Vis, and V., . together with the line voltages Viw Van, and Vee using Veq as reference. Specify the sequence of these line voltages. Ans : Line voltage sequence : ba-cb-ac. x XK 2 LAAN RN AROS at Oscillogram 2. Oscillographic study of a balanced, delta-connected, unity- power-factor load. Problem 5, Refer to Oscillogram 2. Draw a compete vector diagram of Vs, Vic. Vea: Tats Tees La Ten Ibu. Le employing Vy. as reference. From the scaled ordinates given on Oscilliogram 2, determine the effective values of line (or phase) voltage, phase current, and line current. Ans : V= 100 Volts; Ip =3.5 Ampere; I= 6 Amperes. Problem 6, Find the magnitude of the line currents issuing from a balanced six-phase, mesh connected generator if the phase currents are known to be 100 amperes in magnitude. Illustrate solution by means of a vector diagram. Aus: =1,= 100 amperes.Problem 7. Find the voltage between adjacent lines of a balanced twelve-phase, star- connected system if the phase voltages are 50 volts in magnitude. Illustrate solution by means of a vector diagram. Ans : 25.88 volts. Problem 8. Find the voltage between alternate lines of a balanced six-phase, star-connected system if the phase voltages are 132.8 volts in magnitude, Ans : 230 volts. Problem 9, Three-phase line voltages of 2300 volts magnitude are impressed on a balanced wye-connected load which consists of 100 olins resistance per phase in series with 173.2 ols inductive reactance per phase. Find the line eurrent and the total power taken by the three-phase load. Calculate Py as 31,°Ry as 3 Volp C08 Opt a...d as FTP fy 605 6. Ans: I, =Ip 6.64 amperes, P, = 13.22 kw. Problem 10. Repeat Problem 9, assuming that the three impedances are connected in delta (rather than in wye) across the same line voltages. Ans : I, = 19. 92 ampere, P, = 39.66 kw. Problem 11. Three-phase line voltages of 410 volts are impressed on a balanced delta- connected load which consists of 8 ohms resistance in series with 6 chms inductive reactance per phase (a)Find the volt-amperes per phase, the reactive volt-amperes per phase, and the reactive factor of each phase. Ans : va, = 19.360, rva, = 11.616. rf. = 0.6. (b)Find the total volt-amperes of the system, the total reactive volt-amperes of the system, and the reactive factor of each phase. Ans : va, = 58,080, rva, = 34.848, rf = 0.6Oscillogram 4. Oscillographic representation of all voltages and currents involved in the two-wattmeter method of measuring balanced three-phase power at 0.5 p.f lag, the condition under which one wattmeter reads zero. Problem 12. Refer to Oscillogram 3. (a) If the line-to-line voltages have instantaneous maximum values of 155,5 volts and the dlta-line currents have instantaneous maximum values of 14,14 amperes, find the average power readings of the wattmeters Ways and Ware'e- (b) Draw a vector diagram indicated all currents and voltages show on Oscillogram 3. Use ‘Vay as reference, and include the delta-phase currents Ty, Ibo. and Icy Which are not shown on the oscillogram but which combine to form the delta-line currents Iy,and Ie. Ans : (a) Wave Werce= 952.6 watts. (b)_ab-be-ea sequence of line-to-line voltages; Ixy in time phase with Vas; Irs lags Vas by 30°: Ire leads Vig by 30°, Problem 13. Refer to Fig 49. Find the ratio of the copper required for two-phase, three-wire transmission to that required for three-phase, three-wire transmission under the following conditions, all imposed simultaneously.(a) A fixed amount of power transmitted. (b) The same distance. (©) With the same total line losss. Hint From condition (a) : Pa: = 2V,alsc0s 6 = Pa: = 3Vpalscos 8 From condition (d) From condition (e) : 2 From condition (e) : Area of Ry wire = V2 x area of Ry wire From condition (b) : R: 4. 16. 17. 18. 19. -& z Ans: 1,94 JNANSWERED PROBLEMS ‘What is the phase voltage and also the voltage berween adjacent line of two-phase star connection if the greatest voltage between any pair of lines is 156 volts. The voltage between adjacent lines of a twelve-phase star is 400 volts. Find voltage to neuiral, the voltage between alternate lines, and the greatest voltage between any pair of lines Find the phase current in a six-phase mesh if the line current is 10 amperes; for a twelve-phase mesh for the same line current Given six coils each having an induced voltage of 63.5 volts. Adjacent coil voltages are 60° apart. In how many ways may you connect these coils to form a need three- phase wye system of voltages if all coils must be used for each system and if the magnitude of the line voltages of each system must be different? What is the line voltages for each wye system? A generator has six coils, adjacent coils being displaced 30 electrical degrees, each coil voltage is 114 volts, show how to connect them and alculat the line terminal voltage for three-phase star. Repeat for three-phase mesh. Repeat two-phase, where line voltage is taken as the phase voltage. A generator has six coils, adjacent coils being displaced 30 electrical degrees. All coils are used to form a three-phase mesh, what inust be the emf of each coil yield balancedthree-phase voltages of 230 volts each? If all coils are connected three-phase star, what must he the emf of each coil to give an emf between lines 230 volts? 20. Draw vector diagrams which represent the currents and voltages shown in Oscillogram 3 and 4, pages 355 and 356, and label them in accordance with the beling on the oscillogram. 21. Three-phase line voltages of 230 volts are impressed on a balanced wye load having 1G ohms resistance and 12 ohms reactance in series in each phase. Find be line current and total power. If the three impedances are reconnected in delta and placed across the same line voltages, what are the line and phase currents and be total power? 22. A current of 10 amperes flows in the lines to a twelve-phase mesh-connected having 5 ohms resistance and 8 oluns capacitive reactance in series in each phase. What is the voltage between altemate lines on the load? Draw the vector diagram of the voltages and phase currents of two adjacent phases, and also show the line current from the junetion of the two phase A balanced wye load consists of 3 ohms resistance and 4 ohms capacitive reactance in series per phase. Balanced three-phase voltages of 100 volts each phase impressed across the lines at the load. Ifthe load is connectd to a generator through three lines of waual impedance, each line containing a resistance of 1 ohm and an inductive reactance of 4 ohms, find the voltage at the generator terminals. 24. A balanced wye load having 8 ohms resistance and 6 ohms inductive reactance a series in each phase is supplied through lines each having 1 ohm resistance and 2 obins inductive reactance. If the sending-end voltage between lines is 250 volts, what will be the voltage between lines at the load? 25. A balanced delta load contains a resistance of 12 ohms and a capacitive reactance of 16 ohms in series in each phase. If the balanced impressed line voltages the load are 115 volts each, calculate the line and phase currents, 26. A balanced delta load having 18 ohms resistance and 24 olims capacitive reactance in series in each phase is supplied though lines each having 1 ohm resistance and 2 chms inductive reactance. If the line-to-line voltage at the sending end is 250 volts, find the line-to-line voltage at the load terminals. Also find the total power consumed by the load. 27. A balanced wye inductive load takes 5.4 kw at 0.6 power factor at a line voltage of 200 volts, It is in parallel with a pure resistive balanced wye load taking 5 kw. Find the resultant line current supplied the combination.28. 29. The total power supplied two balanced three-phase load in parallel is 12 kw at 0.8 power factor lagging. One of the load takes 10 kva at 0.8 power factor load. The second load is a delta-connected balance load. Find the resistance and the reactance per phase of the delta load if the line voltage is 230 volts. If the unknown load were wye- connected, what would be the resistance and reactance per phase. Each phase of a delta load has 6 ohms resistance and 9 ohms capacitive reactance in series. Each phase of a wye load has 8 ohms resistance and 6 ohms inductive reactance in series. The two loads are connected in parallel across three-phase line voltages of 100 volts. Calculate the resultant line current, the total power consumed, and the power factor of the combination. . A three-phase, 5 hp, 220 volt induction motor (balanced load has an efficiency of 86 per cent and operate at 86.6 per cent lagging power factor. It is paralleled with a three- phase resistance furnace consisting of three 36 ohm resistance connected in delta. Find the kilo voli-amperes demanded by the combination, the power factor , and the line current. A three-phase generator supplies balanced voltages of 230 volts. each at its terminals when it carries a load which requires 10 amperes. If the power factor at the generator terminals is 0.8 leading, calculate the voltage at the load if the load is connected though lines each having 1 ohm resistance and 5 ohms inductive resistance, A balanced three-phase load requires 10 kva at 0.5 lagging power factor. Find the kva size of the condenser bank which may be paralleled with the load to bring the power factor of the combination to 0.866 lag, and also to 0.866 lead. If the line voltage for problem 32 is 230 volts and the frequency 60 cycles, find the capacitance in microfarads of capacitors required in each phase of the capacitor bank if they are delta-connected. What capacitance is required if they are wye-connected. Three 15 /60° ohm load impedances are connected in delta and supplied by lines. each ine containing 1 chm resistance and 1 ohm induetive reactance. If the line voltages on the supply side of the line impedances are balanced three-phase of 115 volts each, find the voltage across the load impedances. Also calculate the power loss in the supply lines and the power dissipated by load it self. . If the current through each of the load impedances in problem 34 is 20 amperes, find the required voltage on the supply side of the line impedances . A three-phase line has three capacitors, each having a reactance of 300 ohms connected in delta across the lines at the source. Three similar capacitor are so connected between37. the line at the load, Between these two sets of capacitors each line a series inductive reactance of 10 chm. If a balanced three-phase load of 100 kva at 0.6 power-factor lag requires 2300 volts between lines. What voltage between lines will be required at the source? What will be the power input to the lines and the power factor at the source ? ostizn ostiza 1000.0 or os+j2a wig | | Sei Fig 54. See problem 37 and 38 The motor M in Fig.54 has 2300 volts balanced three-phase voltage impressed at each terminal and takes 120 kva at 0.6 leading power factor. Calculate the line volts, power input, and the power factor at a, b,c. If the motor in Fig.54 is removed from the circuit and balanced three-phase line voltages of 2300 volts each are impressed at a, b, ¢ how many volt will appear between, lines at the motor end of the line? 39. A three-phase resonant shunt is connected to three-phase, 2300 volts lines to furnish a 40. low impedance for a certain frequency so as to reduce the inductive interference with a telephone line. The shunt consist of three 10 kva, 60 cycle, 2300 volts capacitors connected in delta. In series with each line terminal from the delta is an inductance of 2.5 millihenrys. At what frequency does this three-phase combination resonate. That is offer minimum impedance? Assume that resistances of capacitors aud inductances are negligible. (a) Three coils each having 36 chms resistance and 100 millihenrys inductance are connected in delta. Find the microfarad capacitance of each capacitor which may be placed in each of the three lines from the delta to pruduce resonance (unity p.f. ) of the system as a whole for a frequency of 800 cycles. This is a type of resonant shunt sometimes connected to power lines to reduce inductive interference with telephone circuits. 10AL 44. 46. 47 (b) Assume the capacitors calculated for each line in (a) are removed and connected in delta, Find how many henrys of inductance would be required in each line from this delta to bring the power factor of the combinations to unity at 800 cycles. a Fig See problems 41, 43, and 44 . Find the readings of Ws and Wy in Fi dissipated in each phase. . A balanced three-phase load takes 5 kw and 20 reactive kva. Find the readings of two 55 for the sequence Vs, Ves, Veo. Find the power wattmeters properly connected to measure the total power . In Fig. 55 find the reading of Wa Also calculate the total reactive volt-amperes taken by the load. What is the ratio of the total reactive volt-ampere taken to the reading of Wr? Prove that the ratio of the reading of Wa of Fig. 55 to the total reactive volt-amperes obtained in Problem 43 will obtain for all balanced loads when the impressed voltages are sinusoidal balance three-phase. 5. (a) Caleulate analytically the power-factor angle for a balanced three-phase cireuit in which two wattmeters properly connected to measure three-phase power read + 1000 and + 800 watts, respectively. (b) Also calculate the angle if the meters read +1000 and -800 watts respectively . Two wattmeters measuring power to a balanced three-phase load read 1200 and -100 watts respectivel y. How many volt-amperes does the load take? At what power factor? . The power to a balanced three-phase leading-power-factor load is measure by two wattmeters. The wattmeter having its current coil in line A and each potential coil from line A to line C indicates +1000 watts the other wattmeter with it’s cuzrent coil in line anB and it’s potential coil fiom line B to line C indicates +400 watts. What is the voltage sequence? What is the power factor of the load? 48. Each phase of balanced twelve-phase star-connected load consists of 3 ohms resistance and 4 ohms inductive reactance in series. Balanced twelve-phase line voltages of 51.76 volts between adjacent lines are applied to the load. Calculate the line current, power factor, and total power consumed by the load. 49. The voltage induced in phase na of a three-phase wye-connected generator is @,¢ = 127 sin ol +50 sin(3et — 30°) +20 sin(Seot + 40°) If the sequence is €,,.. €.4. 2x. find the equation with respect to time of the line voltage e,,. Note : Phase voltages of polyphase generators differ only in phase angle. If the phases of generator in Problem 49 are reconnected in delta, what will be the equation with respect to time of the line voltage across phase na. ? 51. A wye-connected generator has a generated voltage per phase which contains only the fundamental, third, fifth, and seventh harmonies. The line voltage to as measured by a voltmeter is 230 volts; the voltage to neutral is 160 volts. Calculate the magnitude of the third harmonies in the generated voltage. Fig 56. See problems 52 and 53 52. The induced emf of a delta generator with one comer of the delta open as shown in Fig. 56 contains only odd harmonies up to the seventh, A voltmeter across ac reads 2500 volts, and, across bb’ when negligible current flows, 1800 volts, Find the reading ofa voltmeter connected from @ to D’ . The induced phase voltage of a delta generator with one comer open as shown in Fig. 56 contains only odd harmonies up to the seventh. A voltmeter connected from @ to b reads 2500 volts, and from « to a it reads 2200 volts when negligible current flows. ‘What should it read from b to 6°? 2W600 watts 1a tae Fig 57. See problem 54 Figure 57 shows a generator connected to a balanced pure resistance load. An ammeter in the reads 15 amperes, and the wattmeter shown reads 600 watts. A voltmeter shows a balanced line voltage of 230 volts. Find the line currents to the load and the voltage from the line to neutral at the load. assuming that the generated voltage contains only fimdamental and third-harmonic components. 2BUNBALANCED POLYPHASE CIRCUITS Problem 1. Determine the values of Van, Veo. and Ven in example 2. Aus : Vag = 36.6 /15® Veg = 205.6 /-80.12 ; Veg = 239.6 /-23.8° Problem 2. Determine the power dissipated in each of the three phases (an, bn, and en) of example 2. Ans : Pay = 134: Ppa = 2120: Poa = 0 watts. a Fig 2. Conversion from a wye-comnected load to an equivalent delta- connected load 212 (90°, Vie = 212 30°, and V.q = 212 /150° volts. As in example 2. Zia = (10+j0). Zia = (10+)10), and Zan = (0 i20) ohms. Problem 3. Find the magnitudes of Iv, nv, and [eg in Fig. 2 if Ans : Ip, = 13.65; Iyyp = 6.20; Ine=7.54 amperes, Problem 4. Solve equation (S) explicitly for Iyrs and state in words how to find Ir, Ips, and Iecafter Innhas been evaluated. ‘Ans ; [yyy = Su 2e%e*Enn Sela* Ene Zap Fakpdethy Eate*eha*ea ty) Problem 5. Show by means of a qualitative vector diagram that the voltmeter (V,) of Fig. 11a reads below line voltage for voltagesequence ab-ca-be 4Ye 2 ‘ Vor ae Lach \ . Vee Nee (@) () Fig 11. A voltmeter method of checking phase sequence in three-phase systems. See example 6 and problems 5 and 6. Problem 6. What is the magnitude of the voltmeter reading in Fig. 11a if X.= 100 ohms, R yy, 100 ohms, and V, Ve 41.4 volts if the voltage sequence is ab-ca-be? Ans : 51.8 volts Problem 7. Calculate the readings of Wix-Wo and WeeWee in the above example and compare the sum if the wattmeter readings this found with the total commected load, Ans Woa-Wo's = 5685; WearWe'e = 4315 watts, Wager a Fig 22. A particular unbalanced three-phase load Problem 8. If the reactive volt-ampere meters shown in Fig. 22 are placed so that the current coils cay Iy, and Ie, what will be the individual meter readings in vars? It is assumed that the potential circuits of the meters are connected in such as manner that the algebraie sum of the readings will be equal to EV Ieos@? Aus : Meter a reads -1464 vars; meter a reads zero. 45Problem 9. What is the power factor of the unbalanced load shown in Fig.22 as determined from EV Isin and EV Ieos6? Ans : 0.939 Problem 10. Find the magnitudes of fy, Ip», and ee in Fig, 26 utilizing the caleulationsof example 13 in so far as they are helpful. Ans : Tya= 55.6: Ty» 6; and Ive = 111.2 amperes UNANSWERED PROBLEM 11, An unbalanced delta system labeled abc at the comers consists of Z,,= 10/-60°, Z..= 50° Z.,= 10/-60%0hms. If V..=100/0 ° and the voltage sequenee is cb-ba-ac, find the vector expressions for the currents entering the terminals a, b, and ¢. The three-phase supply voltages are balanced. Also solve for the opposite sequence. 12. An unbalanced load labeled abe at the comers contains of Z,,= S/19° .Z,.= 10 /-30% Z. 8 /45° ohms. Three phase balanced line voltages 115 volts each are applied. If the sequence is ch-ac-ba, calculated the complex expressions for the line currents leaving 5/0 ° volts terminals a, b, and ¢ for V.=: w= 23/0" ohms Fig 27, See problem 13 13. Refer the Fig. 27. V5 and Vo; represent a balanced two-phase system of voltage drops, the magnitude of each being 115 volts. The voltage phase sequence is 4B-CB. Vy is to be used as reference. Find J,5, cy.J 9. and draw a vector diagram of the voltages and currents. 1614. A wye-connected set of impedances contains of Z,,=5 ohms. Find the equivalent delta-connected impedances Z,, Zs. Zi Which can be used to replace the wye-connected set of impedances 15. Refer to Fig. 23 The terminals a’b’c’ represent a balanced three-phase system voltages the sequence of wihich is ’c’-a’b -c’a’. The magnitude of each line-to-line voltage is 230 volts. Find the readings of ammeters placed in the a'a, bb, and c’c lines. Undstanced toad Teacing ay Ie Fig 7. A three-wire three-phase network 16. In Fig 7. page 380, it will be assumed that the generated voltages are Z,,.=100/0 °, Zp =100/:120°,_Z,, =100/-240 ° volts and that Z pian = (2 — fljohms Zvi = (A= f3)ohms Z, 4)olms Find the lines currents 7Z,,, [4s. and Draw a vector diagram of line-to-line voltages and the line currents Fig 8. A four-wire three-phase system wv17. Refer to Fig. 8, page 381. let it be assumed that the following quantities are known: Exe = 1000 + j0 = 1000 /0° volts Egy ~ 500 — j866 = 1000 /-120° volts Eye =-500 + j866 = 1000 /120° volts Zoey = 20 — j20 = 28.28 /-45° Zee = 50 + j0 = 50.0 /0° ohms 30 + j52 = 60.00 /60° ohms + j8 = 8.25 /76° ohms + jl = 141/45* ohms 5 + j1 = 2.70/21.8° ohms ohms Write the expressions for Ing; Ibs Ices employing determinants and the numerical values of the E’s and Z’s specified above. Use loop currents Ih = Inia Ii = Ip, dan Ts = ee all returning through line vm’. (Results may left in the for of the ratio of two matrices.) 18. A delta-connected set of impedance consists of Z,,~5 10°, Z,.~5 160°, dan Z,.—5 f= 60° ohms. Find the equivalent wye-connected impedances 2475108, Z,,=5/60 ® and Z,,=5/-60 ° which can be employed to replace the above delta-connected impedances. Fig 9. A wye-delta cireuit arrangement 19. Refer to Fig. 9, page 382. Assume that the generator is capable of maintaining a balances system three-phase of voltages Ey .: Ea. Ez»: the sequence of which is ba” — a’c’ ~c’b’. The magnitude of each line voltage is 100 volts. Z..= Zs: AOS chm, Z,=5 (0°, Z,.=5 60°, dan Z.=5 /-60° ohms. Find Lys, Lyi, Lea, dan Ise =Z2.= 05+ dan I. with respect to Va9'as a reference. 1820. Explain, by means of qualitative vector diagrams, the operation of a three-phas- sequence indicator that employs an inductance coil in place of the condenser shown in Fig 11a, page 386. Does the voltmeter read above or below line voltage for sequene ab-ca-be? 21. Devise some cheme for checking the phase sequence of two-phase-voltages 22. Find the reading of wattmeter which has its current coil current coil in the 4 “4 line and its potential coil across the voltage Vac in Problem 13 and Fig. 27. Fig 13. The three-wattmeter method of measuring individual phase powers in a delta-comnected load 23. Refer to Fig 13, page 386. Vaz = 200, Vac= 141.4, and Veq= 141.4 volts. Sequence ab- be-ca. Zap = Zie = Zea = (S ~ j6) olums. Find the reading of each of the watermetters, Find the reading of a wattmeter with its current coil in line @ and potential coil from a also one with current coil in line ¢ and potential coil from ¢ to b. Fig 1. Unbalanced delta load as24. 26. (a) if wattmeter HW has its current coil in line « and its potential coil form line a to b of Fig. 1, page 373, what will it read for a sequence Vas-Va-Vic? If another wattmeter 1, has its current coil in line 6 and its potential coil connected form line 6 to ¢, what will it read? (b) If 17, and 1; were wattmeters what would they read? (a) Find readings of wattmeter 1, and iV, with its current coils in lines a and 6, respectively, supplying load of Problem 11 if the potential coils are properly connected. so that the sum of the readings will give the total power consumed by the load. (b) Find the readings if W7.and Ware varmeters. Fig 29. See problem 26 Refer to Fig. 29. Vey -Ves-Voe: represent a balanced three-phase system of voltage drops, the magnitude of each being 200 volts. The voltage sequence is a’b'-b'c'-c'a Two balanced three-phase loads indicated by the circles are connected to the terminals abc as shown in Fig. 29. In addition to the two balanced load. a single-phase, 4-kw, unity-power-factor load is placed across the bc terminals as indicated (a) Find the reading of Wasa and Worcs (b) If reactive volt-ampere meters replaced Wy.y-co and Weeco,. find their respective readings. (c) Find the combined veetor power factor of the composite load. 20one Rvarmeter ¢ Fig 21. The two reactive volt-ampere meter method of measuring VI Sin @ in a three-wire, three-phase system 27. In Fig. 21 page 386, it will be assumed that Vy, Vee, and Vey, represent a balanced three-phas sytem of voltages the sequence of which is a’b-c’a’-bc;. Z,,= 10 0°, Z,,= 10 -60°, and Z,,= 10 /90° ohms. Assume that line-to-line voltage of the volts. (a) Find the readings of the reactive volt-ampere meters shown in Fig. 21 (5) Find the readings of wattmeters placed at similar posistions in the cireuit, namely, at the a‘a-ad and c’c-cb positions. (c) Find the vector power factor of the unbalanced load as recognized by ’e ALE.E. 28. In Fig. 30 Vos’. Vee, and Ver, are balanced three-phase voltages each having a magnitude of 200 volts and a phase sequence of ab-be-ca, Determine the readings of the two wattmeters shown in the figure. Wetes — Fig 30. See Problems 28 29. In Fig. 31, Eye: Exry:, Eve: are balanced three-phase voltages with magnitudes of 115.4 volts and a phase sequence of ”'a'-n'b'-n'c . Find the following quantities and express all complex quantities with reference to Vis (a) Var, Vie, Vea () Tats Taos Toe 2131. 33 34. (© lawl ke (d) The sum of the readings of wattmeters IV,, 1, W when they are connected as shown, (e) The individual readings of wattmeters W, , WV», W. if the common point O is connected to line bb. Fig. 31. See Problem 29. . The line-to-line voltages of a three phase system are Vis = 200, Vas = 150, dan Ve, 120 volts. Write the polar expressions for Vyp, Vie, dan Voz with respect to V3 as reference for both phase sequences. . Refer to Fig 2. In a particular case measurements yield V2s= 140, Vse= 120, dan Vos = 150, Vier = 200, Vig = 200, Vin = 0, and Vx = 104.2 volts. Draw the qualitative phasor diagram of the voltages for sequence abc, and determine analytically the complex expressions for each of the currents with respect to V2. ~ as a reference. . Calculate the line currents in Prooblem 16 by the loop-current method. Refer to example 13, page 402-403, including Fig, 26. Solve for Iy, Ip, and I; by the loop-current method, neglecting the resistive components of ail branch impedances for a voltage sequence Eyy-E,-Eq) (Resutls may be left in the form of the ratio of two matrices.) In Fig. 32, Zap = Las = 0,01 henry and the coefficient of coupling is 0.5. Assume no resistance or inductances except as indicated driving voltages is n’a’-n'b'-n'c’, and Ey’ = 57.7/90 volts. For @ = 1000 radians per seconds calculate the line and phase currents for the load. Use Maxwell’s cyclic-current method. 22Fig. 32. See Problems 35 and 36 36. Set up the determinant form of the solution for Ja." in Problem 35 if 3 chms pure resistance is inserted in each line to the load and the same sequence and reference as specified in Problem 35 are employed. For uniformity in checking results, use loop currents as follows: Loop current f= J,,%0 37. Solve for Jia = In dan Lee, in Fig. 33 if Ey. = 1350 + j0 volts, Ey»°= -675 —j1170 volts, and £,,°= -675 + j1170 volts 1039 #j0.5).0 > cootinn F050 Fig 33. See problem 37 23