(e) I A (f) co P nv + + (g) mech core P P stray P S OLUTION (a) Since this machine has 8 poles, it rotates at a speed of 120 120 ( 50 Hz ) n f e = = 1500 r/ = min P m 4 If the output power is 100 kW, the output torque is ( ) ⎮ ou P t = = 100, 000 W 637 N = m ⊕ load ⎤ m ( ) 1500 r/min 2 r ad 1 min (b) The input power is 1 r 60 s P OUT P 100 kW = = 110 kW = IN ⎜ 0.91 (c) The mechanical speed is m n = 1500 r/min (d) The armature current is P I I = = = 110 kW 156 = A A L ( ) 3 PF 3 ( ) 480 V 0.85 T V I A 156 31.8 = A ° Therefore, E A is = E V I I A A ⎞ A S A R jX ( 277 0 V ) ( 0.08 )( ) 156 31.8 A ( 1.0 )( ) 156 31.8 A E A = ° & ° j & ° E A 375 21 = .8 V ° (e) The magnitude of the armature current is 375 A. = (f) The power converted from electrical to mechanical form is given by the equation conv IN C P P U P 2 ( ) = = 2 ( ) & = CU 3 A A P I 3 1 R 56 A 0.08 5.8 kW = = = conv IN C P P U 11 P 0 kW 5.8 kW 104.2 kW (g) The mechanical, core, and stray losses are given by the equation + + = = = mech core P P stray P conv P OUT 1 P 04.2 kW 100 kW 4.2 kW 6-12. The Y-connected synchronous motor whose nameplate is shown in Figure 6-21 has a per-unit synchronous reactance of 0.90 and a per-unit resistance of 0.02. (a) What is the rated input power of this motor? (b) What is the magnitude of E A at rated conditions? 164 (c) If the input power of this motor is 10 MW, what is the maximum reactive power the motor can simultaneously supply? Is it the armature current or the field current that limits the reactive power output? (d) How much power does the field circuit consume at the rated conditions? (e) What is the efficiency of this motor at full load? (f) What is the output torque of the motor at the rated conditions? Express the answer both in newton- meters and in pound-feet. S OLUTION The base quantities for this motor are: T V ,base = 6600 V 6600 V 3811 V V ⎞ ,base = = 3 = = ,base ,base 14 A L I I 04 A = = ( ) = ( )( ) = base rated S P 3 P T L V F I 3 6600 V 1404 A 1.0 16.05 MW (a) The rated input power of this motor is ( )( = = )( ) = IN 3 P T L F P V I 3 6600 V 1404 A 1.0 16.05 MW (b) At rated conditions, ⎞ V 1.0 0 pu = ° and I ⎞ 1.0 0 pu = ° , so E A is given in per-unit quantities as = E V I I A A ⎞ ( ) 1 0 A S A R jX ( 0.02 )( 1.0 0 ) ( 0.90 )( ) 1 0 E A = ° ° j ° E A 1.33 42 = .6 pu ° The base phase voltage of this motor is 6600 / 3 = 3810 V, so E A is A E ( ) 1.33 42.6 = ( 381 ° 0 V ) 5067 = 42.6 V ° (c) From the capability diagram, we know that there are two possible constraints on the maximum reactive power—the maximum stator current and the maximum rotor current. We will have to check each one separately, and limit the reactive power to the lesser of the two limits. The stator apparent power limit defines a maximum safe stator current. This limit is the same as the rated input power for this motor, since the motor is rated at unity power factor. Therefore, the stator apparent 165 power limit is 16.05 MVA. If the input power is 10 MW, then the maximum reactive power that still protects the stator current is 2 2 Q S P = ( ) 16. = 05 MVA ( 10 MW ) 2 2 12.6 M = VAR Now we must determine the rotor current limit. The per-unit power supplied to the motor is 10 MW / 16.05 MW = 0.623. The maximum E A is 5067 V or 1.33 pu, so with E A set to maximum and the motor consuming 10 MW, the torque angle (ignoring armature resistance) is ( ) ( ) sin 1 1 S X P sin 0.90 0. = = 623 24.9 = ° ™ 3 ⎞ A V E ( ) 1.0 ( 1. ) 33 At rated voltage and 10 MW of power supplied, the armature current will be ⎞ A V E I 1 0 ° 1 .33 2 4. ° 9 0.663 20.2 pu = = = ° + A 0.90 A S R jX j In actual amps, this current is A I ( ) 1404 A ( 0.663 20.2 ) 931 20 = .2 A ° = ° The reactive power supplied at the conditions of maximum E A and 10 MW power is 3 sin 3 ( ) 3811 V ( = = 931 A ) sin ( 20.2 ) 3.68 MV ° AR = ⎞ ⎝ A Q V I Therefore, the field current limit occurs before the stator current limit for these conditions, and the maximum reactive power that the motor can supply is 3.68 MVAR under these conditions. (d) At rated conditions, the field circuit consumes ( ) = = ( ) = field F F P V 12 I 5 V 5.2 A 650 W (e) The efficiency of this motor at full load is ( )( ) ⎜ OUT P 100% = ⋅ 2100 = 0 hp 746 W/hp 100% ⋅ 97.6% = IN P 16.05 MW (f) The output torque in SI and English units is ( )( ) ⎮ OUT P 2 = = 1000 hp 746 W/hp 124, 700 N = m ⊕ load ⎤ m ( ) 1200 r/min 1 min 2 rad 60 s 1 r ⎮ load 5252 P 5252 ( ) 21000 hp ( ) 1200 r/min 91, 910 lb ft = = = ⊕ m n 6-13. A 440-V three-phase Y-connected synchronous motor has a synchronous reactance of 1.5 & per phase. The field current has been adjusted so that the torque angle ™ is 28 ° when the power supplied by the generator is 90 kW. (a) What is the magnitude of the internal generated voltage E A in this machine? (b) What are the magnitude and angle of the armature current in the machine? What is the motor’s power factor? (c) If the field current remains constant, what is the absolute maximum power this motor could supply? 166 S OLUTION (a) The power supplied to the motor is 90 kW. This power is give by the equation P A V E sin = 3 ⎞ ™ X S so the magnitude of E A is ( ) 1. ( 5 90 kW ) E S X P & = = = 3 sin 3 ( ) 254 V sin 28 ⎞ ™ 377 V V A ° (b) The armature current in this machine is given by ⎞ A V E 254 0 V ° I 3 77 28 ° 129 24 A = = = ° A 1.5 jX S j The power factor of the motor is PF = cos 24º = 0.914 leading. (c) The maximum power that the motor could supply at this field current 3 3 ( ) 254 V ( 377 V ) P ⎞ A V E = = = max X S 1.5 & 191.5 kW 6-14. A 460-V, 200-kVA, 0.80-PF-leading, 400-Hz, six-pole, Y-connected synchronous motor has negligible armature resistance and a synchronous reactance of 0.50 per unit. Ignore all losses. (a) What is the speed of rotation of this motor? (b) What is the output torque of this motor at the rated conditions? (c) What is the internal generated voltage of this motor at the rated conditions? (d) With the field current remaining at the value present in the motor in part (c) , what is the maximum possible output power from the machine? S OLUTION (a) The speed of rotation of this motor is 120 120 ( ) 400 Hz n f e = = = sync P 8000 r/min 6 (b) Since all losses are ignored, = = ( ) ⋅ = ( ) = . The output torque of this motor is IN ,rated OUT ,rated P P rated PF S 200 kVA 0.8 160 kW ⎮ OUT P = = 160 kW 191 N = m ⊕ load ⎤ m ( ) 8000 r/min 1 min 2 rad 60 s 1 r (c) The phase voltage of this motor is 460 V / 3 = 266 V. The rated armature current of this motor is P I I = = = 160 kW 251 = A A L ( ) 3 PF 3 ( ) 460 V 0.80 T V Therefore, I A 251 36.87 = 2 A ° . The base impedance of this motor is ( ) 2 Z base 3 V ⎞ ,base 3 266 V 1.06 = = = & S base 200, 000 VA 167 so the actual synchronous reactance is ( ) 0.50 pu ( 1.06 ) 0.53 X S = & = & . The internal generated voltage of this machine at rated conditions is given by = E V I A S ⎞ jX A 266 0 V ( ) 0.53 ( 251 36.87 A ) 362 17.1 V E A = ° j & ° = ° (d) The maximum power that the motor could supply at these conditions is 3 3 ( 266 V )( 362 V ) P ⎞ A V E = = = MAX X S 0.53 & 545 kW 6-15. A 100-hp 440-V 0.8-PF-leading -connected synchronous motor has an armature resistance of 0.22 & and a synchronous reactance of 3.0 & . Its efficiency at full load is 89 percent. (a) What is the input power to the motor at rated conditions? (b) What is the line current of the motor at rated conditions? What is the phase current of the motor at rated conditions? (c) What is the reactive power consumed by or supplied by the motor at rated conditions? (d) What is the internal generated voltage E A of this motor at rated conditions? (e) What are the stator copper losses in the motor at rated conditions? (f) What is P conv at rated conditions? (g ) If E A is decreased by 10 percent, how much reactive power will be consumed by or supplied by the motor? S OLUTION (a) The input power to the motor at rated conditions is ( )( ) P OUT P 100 h = = p 746 W/hp 83.8 kW = IN ⎜ 0.89 (b) The line current to the motor at rated conditions is I P 83.8 kW = = 137 A = L 3 PF 3 ( ) 440 V ( 0.8 ) T V The phase current to the motor at rated conditions is I L 137 A 79.4 A I ⎞ = = = 3 3 (c) The reactive power supplied by this motor to the power system at rated conditions is ( )( = = ) ° = rated 3 ⎞ A sin Q V I ⎝ 3 440 V 79.4 A sin 36.87 62.9 kVAR (d) The internal generated voltage at rated conditions is = E V I I A A ⎞ A S A R jX 440 0 V ( 0.22 )( ) 79.4 36.87 A ( 3.0 )( ) 79.4 36.87 A E A = ° & j ° & ° E A 603 19 = .5 V ° (e) The stator copper losses at rated conditions are 168 . 100 kW = = 110 kW = IN ⎜ 0.91 (c) The mechanical speed is m n = 1500 r/min (d) The armature current is P I I = = = 110 kW. efficiency of this motor at full load is ( )( ) ⎜ OUT P 100 % = ⋅ 2100 = 0 hp 746 W/hp 100 % ⋅ 97.6% = IN P 16.05 MW (f) The output torque in SI. kW 5.8 kW 104 .2 kW (g) The mechanical, core, and stray losses are given by the equation + + = = = mech core P P stray P conv P OUT 1 P 04.2 kW 100 kW 4.2