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Chapter 7 Additional Problems X7.1 A three-phase, 1200 HP, 2300 V, 60 Hz, RR synchronous machine has Rs 0.2 and X s 5.6 . When operated with the mechanical load disconnected with rated voltage and frequency, the field current is adjusted until line current has a minimum value. At this point, I a 22.1 A , the measured input power is PT 17.5 kW , and the measured field circuit values are V f 276 V and I f 53.2 A . (a) Determine the rotational losses (core losses plus friction and windage) for this machine. (b) If the field current, frequency, and impressed terminal voltage are unchanged, but a mechanical load requiring 600 HP is attached, predict the power factor, line current, and efficiency. (a) Using the no-load data, PFW PT 3I a2 Rs 17,500 3 22.1 0.2 17,793.05 W 2 (b) Since the no-load current was a minimum value, the machine was operating at unity power factor. The excitation voltage corresponding to I f 53.2 A can be determined. E f Van I a Rs jX s 2300 22.1 0.2 j 5.6 1329.28 5.34 V 3 Since Rs X s , little error is introduced if Rs is neglected in calculation of the torque for the loaded condition. From energy balance, Ps PFW 3Van E f Xs sin or Ps PFW X s 1 600 746 17,793.05 5.6 sin 31.84 sin 3 2300 / 3 1329.28 3Van E f 1 Based on [7.36], 2300 0 1329.28 31.84 3 Ia 130.07 13.77 A Rs jX s 0.2 j 5.6 Van E f Hence, I a 130.07 A PFin cos Van I a cos 11.35 0.97 lagging The losses are Losses PFW 3I a2 Ra V f I f Losses 17,793.05 3130.07 0.2 276 53.2 42,627.17 W 2 1 100 600 746 100 Ps 91.30% Ps losses 600 746 42,627.17 X7.2 If the synchronous motor of Problem X7.1 still operates the 600 HP load with terminal voltage and frequency unchanged, but the field current is increased to 80 A, predict (a) the input power factor and (b) the leading VARs supplied to the three-phase grid. Assume that core losses (lumped with friction and windage losses) of Problem X7.1 are valid. (a) For the new field current, the excitation voltage is Ef 80 1329.28 1998.92 V 53.2 Assuming Rs negligible, Ps PFW X s 1 600 746 17,793.05 5.6 sin 19.10 3 2300 / 3 1998.92 3Van E f sin 1 2300 0 1998.92 19.10 3 Ia 153.45 42.86 A Rs jX s 0.2 j 5.6 Van E f PFin cos Van I a cos 42.86 0.733 leading (b) QT 3 VL I a sin 3 2300 153.45 sin 42.86 QT 415.80 kVARs X7.3 If the synchronous motor of Problem X7.1 now supplies rated power to a coupled mechanical load with rated voltage and frequency applied to the stator and I f 80 A , calculate (a) line current and (b) efficiency. Assume that Rs can be neglected in torque angle determination. Also, assume that core losses have changed negligibly from Problem X7.1. (a) The excitation voltage for I f 80 A is determined in Problem X7.2 to be E f 1998.92 V . Ps PFW X s 1 1200 746 17,793.05 5.6 39.97 sin 3 2300 / 3 1998.02 3Van E f sin 1 2 2300 0 1998.02 39.97 3 Ia 229.46 11.75 A Rs jX s 0.2 j 5.6 Van E f or I a 229.46 A (b) From the values of V f and I f given in Problem X7.1, Rf Vf If 276 5.19 53.2 Losses PFW 3I a2 Rs I 2f R f Losses 17,793.05 3 229.46 0.2 80 5.19 82,600.18 W 2 2 100 1200 746 100 Ps 91.55% Ps losses 1200 746 82,600.18 X7.4 The synchronous machine of Problem X7.1 is now operated as a 2300 V, 60 Hz alternator supplying 800 kW to a 0.8 PF lagging load. Field current I f 53.2 A . Determine the required input mechanical power to drive the rotor and coupled exciter if the exciter has an efficiency of 91% when supplying the field voltage and current given in Problem X7.1. Assume the lumped friction and windage losses and core losses of Problem X7.1 are valid. Since I f is unchanged from Problem X7.1, E f 1329.28 V is still valid. Assume Van on the reference. Ia PT 3 VL PF 800,000 251.02 A 3 2300 0.8 Ps PT 3I a2 Rs PFW V f I f / ex Ps 800,000 3 251.02 0.2 17,793.05 276 53.2 / 0.91 2 Ps 871.73 kW X7.5 Fig. X7.1 shows the phase a circuit of a three-phase synchronous generator that has a per phase load consisting of a 5 pure inductive load. The machine is operating at zero power factor lagging. Determine the ratio E f / Van . 3 By voltage division, Van jX L j5 Ef Ef j XL Xs j 5 j10 or Ef Van 3 X7.6 Fig. X7.2 shows the phase a circuit of a three-phase synchronous generator that has a per phase load consisting of a 15 pure capacitive load. The machine is operating at zero power factor leading. Determine the ratio E f / Van . 4 By voltage division, jX C j15 Ef Ef j XC X s j15 j10 Van or Ef Van 1 3 X7.7 A 3600 rpm, 60 Hz, 13.8 kV synchronous generator has a synchronous reactance of 20 . The generator is operating at rated voltage and speed with the excitation voltage E f 11.5 kV and the torque angle 15 . Calculate (a) stator current, (b) power factor, and (c) total output power. (a) Ia E f Van jX s 13,800 0 3 216.35 46.54 A 2090 11,50015 I a 216.35 A (b) (c) PF cos Van I a cos 46.54 0.688 lagging PT 3VL I a PF 3 13,800 216.35 0.688 3557.8 kW X7.8 For the synchronous generator of Problem X7.7, determine (a) the maximum power that can be converted from mechanical to electrical form without loss of synchronism if the field current is unchanged and (b) the value of current I a for this condition. (a) Since 90 , 3Pd max E f Van Xs 11,500 13,800 / 20 3 4.58 MW (b) Ia E f Van jX s 13,800 0 3 699.5234.72 A 2090 11,50090 I a 699.52 A 5 X7.9 A three-phase synchronous generator, rated at 240 V, 50 A, 60 Hz, has the stator winding schematic of Fig. X7.3. All 12 phase group leads ( a1 , a1 , c2 , c2 ) are available at the generator terminals. In addition to the given rating, list the balanced, three-phase voltage-current ratings available from the generator by coil reconnection without mix of phase coils – that is, use only phase a coils in phase a connections. Three additional ratings are possible under the stated constraint. 1. Series connect phase coils and use wye configuration: 480 V, 25 A 2. Series connect phase coils and use delta configuration: 277 V, 43.3 A 3. Leave phase coils parallel-connected and use delta configuration: 139 V, 86.6 A X7.10 The three-phase synchronous generator of Problem X7.9 is to be reconnected as a single-phase alternator. What is the largest possible voltage that can be obtained? Connect the phase coils in series with additive polarity to give V1 Va1 Va 2 Vb1 Vb 2 Vc1 Vc 2 V1 2770 277 120 277120 5540 V Or the largest voltage possible is V1 554 V . 6 X7.11 A 500 MVA, 24 kV, 60 Hz three-phase alternator is operating at rated voltage and frequency with a terminal power factor of 0.8 lagging. The leakage reactance X 0.15 and the synchronous reactance X s 0.8 . Stator coil resistance is negligible. The excitation voltage E f 18 kV . Determine (a) the torque angle , (b) the total output power, (c) the line current I a , and (d) the resultant voltage Er . (a) The power converted must equal the output power. 3E f I a cos 3Van I a cos or Van cos E f cos1 24,000 0.8 3 cos 1 cos 1 0.8 15.12 18,000 (b) PT 3E f Van Xs sin 3 18,000 24,000 / 3 0.8 sin 15.12 244 MW (c) Ia PT 3VL PF 244 106 7337.4 A 3 24,000 0.8 (d) Er Van jX I a Er 24,000 0 0.1590 7337.4 36.87 14,5443.47 V 3 Er 14,544 V X7.12 A 250 MVA, 24 kV, 60 Hz, three-phase alternator is operating at rated voltage, rated frequency, and rated apparent power with a terminal power factor of 0.8 lagging. Stator coil resistance is negligible. The excitation voltage E f 20 kV . Determine (a) the torque angle and (b) the value of synchronous reactance X s . (a) Assume Van on the reference. Ia ST 3VL cos 1 PF 250 106 cos 1 0.8 6014.2 36.87 A 3 24,000 7 Power converted must equal power output. 3E f I a cos 3Van I a cos or Van cos E f cos1 24,000 0.8 3 cos 1 cos 1 0.8 19.47 20,000 (b) E f Van j I a X s or Xs E f Van j Ia 24,000 0 3 1.38 190 6014.2 36.87 20,00019.47 X7.13 A 100 MVA, 13.8 kV, 60 Hz, salient-pole, synchronous generator with 8 poles has X d 1.9 and X q 1.1 . The stator coil resistance is negligible. The generator is supplying rated apparent power at a power factor of 0.866 lagging. Determine the value of excitation voltage E f and torque angle . Ia ST 3 VL cos 1 PF 100 106 cos 1 0.866 4183.8 30 A 3 13,800 By [3] of Example 7.10, E f Van jX q I a 13,800 0 1.190 4183.8 30 3 E f 11,01521.21 V Thus, E f 21.21 30 21.21 51.21 I d I a sin 4183.8sin 51.21 3261 A I d I d A 8 E f E f j X d X q Id E f 11,01521.21 0.890 3261 68.79 E f 13,623.821.21 V E f 13,623.8 V X7.14 For the salient-pole synchronous generator of Problem X7.13, calculate the developed torque by two methods. s 2 2 2 60 30 rad / s p 8 By [7.53] with values from solution of Problem X7.13, 3Td 3Van E f s X d sin 3Van2 X d X q 2 s X d X q sin 2 2 13,800 13,800 3 3 13,623.8 1.9 1.1 3 3 3Td sin 21.21 sin 2 21.21 30 1.9 2 30 1.9 1.1 3Td 918.8 kN m Alternatively for this machine with Rs 0 , 3Td 3Pd s PT s 100 10 0.866 918.8 kN m 6 30 X7.15 Determine the maximum power that the salient-pole synchronous generator of Problem X7.13 can convert if the field circuit were open circuit. In [7.53], set E f 0 , 45 , and multiply by s to give 3Pd max 3Van2 X d X q 2Xd Xq 2 3Pd max 13,800 3 1.9 1.1 3 36.45 MW 2 1.9 1.1 9