Synchronous Machine Problems

ENEE3521 – Synchronous Machine Problems

c. Calculate the AC frequency of the generator output voltages when the generator is driven at rated speed. (50Hz) d. The generator is operated at rated speed with an excitation current of 20A, and a Y-connected load with per phase impedance of 21 ∠25°Ω is connected its output terminals. Calculate the real and reactive power output in MW and MVAR being supplied to the load by the generator. (14.962451MW, 6.977106MVAR) 6. A 3φ AC synchronous motor is rated 1000HP, 60Hz, 2300V, 0.8 leading power factor. The stator has 2poles per phase, is Y-connected, has per phase winding resistance of 0.4 Ω, and has per phase winding synchronous reactance of 2.8 Ω. While operating under normal conditions, the machine has 24KW of friction and windage loss, and has 18KW of core heating loss. The excitation supply for the machine operates at 200V, and the open circuit magnetization characteristics for the machine at rated speed are given in the table below. IEX (A) ELLA (V)

0 0

1.0 690

2.0 1255

3.0 1813

3.5 2026

4.0 2210

4.5 2378

5.0 2500

5.5 2608

6.0 2643

7.0 2788

8.0 2860

9.0 2874

10.0 2880

Assume that rated voltage is applied to the motor input terminals and that the motor input real power operates at a value equal to the motor’s rated mechanical output power value converted to Watts. a. Calculate the motor’s induced internal armature voltage (1358.231V L-N, 2352.525V L-L) b. Calculate the motor excitation current. (4.424A) c. Calculate the motor efficiency. (88.624%) d. The motor excitation current is changed to 4.645A, and the motor torque angle doesn’t change from its previous value. Calculate the motor input current, power factor, and reactive power. (190.697A, 0.998004 leading, 47.9695KVAR) e. Calculate the motor drive shaft output power in HP and torque, and the motor efficiency for the operating condition in part d. (901.876HP, 1783.938N-m, 88.596%) 7. A 3φ AC synchronous generator is connected to and supplies a 3 φ AC synchronous motor. The generator is rated 375KVA, 480V, 0.8 lagging power factor, and has a Y-connected stator with per phase winding synchronous reactance of 0.4 Ω and negligible winding resistance. The motor is rated 80KW, 480V, 0.8 leading power factor, and has a Y-connected stator with per phase winding synchronous reactance of 1.1 Ω and negligible winding resistance. The motor input real power operates at a value equal to the motor’s rated mechanical output power, and the generator excitation is adjusted to create rated voltage at the generator/motor terminals. a. Assuming that the terminal voltage between the two machines in the equivalent circuit is the 0 ° phase angle reference in the circuit, calculate the generator model and the motor model internal armature induced voltage phasors. (281.274 ∠9.849°V, 307.092∠-25.521°V) b. Assuming that the motor torque angle doesn’t change from its previous value, calculate the terminal voltage magnitude increase if the motor excitation increases by 10% which causes the motor internal induced armature voltage magnitude to increase by 10%. (284.54V L-N, 492.838V L-L) c. Calculate the new power factor at the generator/motor interface terminals. (0.988082 leading) 8. A 3φ AC synchronous motor is rated 100KW, 50Hz, 480V, 0.8 leading power factor. The stator has 6-poles per phase, is Y-connected, has negligible winding resistance, and has per phase winding synchronous reactance of 1.5Ω. The motor’s friction, windage, and core heating losses are negligible. The motor is supplied from a variable frequency drive (VFD). Assuming that the motor operates at rated speed, rated voltage, rated power factor, and the input real power is the same as the motor’s rated mechanical output power, calculate the following quantities. a. Motor input current and induced internal armature voltage. (141.507A, 428.753V L-N, 742.622V L-L) b. Reactive power being supplied by the VFD to the motor. (-61.9744KVAR) c. Output drive shaft power in HP. (134.102HP) Prepared by Henri Alciatore for the University of New Orleans ENEE3521 Course © 2007 Version 03-31-07

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d. Calculate the frequency range that the VFD must output in order to operate the motor over a speed range of 300rpm to 1000rpm. (15Hz to 50Hz) 9. A 3φ AC synchronous generator is rated 1000KVA, 60Hz, 2300V, 0.8 lagging power factor. The stator has 2-poles per phase, is Y-connected, has per phase winding resistance of 0.15 Ω, and has per phase winding synchronous reactance of 1.1 Ω. While operating under normal conditions, the machine has 24KW of friction and windage loss, and has 18KW of core heating loss. The excitation supply for the machine operates at 200V, and the open circuit magnetization characteristics for the machine at rated speed are given in the table below. IEX (A) ELLA (V)

0 0

1.0 690

2.0 1287

3.0 1800

3.5 2049

4.0 2231

4.5 2383

5.0 2535

5.5 2596

6.0 2689

7.0 2802

8.0 2867

9.0 2895

10.0 2910

a. Compute the excitation current required to produce a no-load terminal L-L voltage of 2300V. (4.227A ) b. If the machine operates under rated conditions, calculate the machine’s internal armature induced voltage. (1536.553V L-N , 2661.388V L-L) c. Compute the excitation current required to operate the machine under rated conditions. (5.852A) d. Compute the drive shaft speed, power in HP, and torque that the prime mover must provide operate the machine under rated conditions. (3600rpm, 1279.811HP, 2531.506N-m) e. Compute the generator torque angle while operating the machine under rated conditions. (7.415 °) f. Compute the generator efficiency while operating the machine under rated conditions. (83.724%) 10. The same generator specified in Problem #9 is driven at rated speed and is operated with an excitation current of 4.5A while a ∆-connected 3φ load impedance with value 20 ∠30°Ω is connected to the generator output terminals. (Hint: the load impedance must be divided by 3 in the equivalent circuit) a. Calculate the voltage and current at the generator output terminals. (2147.193V, 185.952A) b. Calculate the generator torque angle and the generator input drive shaft torque. (6.812 °, 1741.351N-m) c. Calculate the generator efficiency and percent loading. (91.107%, 69.157%) d. A second identical ∆-connected 3φ load impedance is connected to the generator output terminals in parallel with the first load impedance. Calculate the generator output terminals voltage and current, torque angle, input drive shaft torque, efficiency, and percent loading. (1933.589V, 334.907A, 12.335 °, 2821.905N-m, 91.230%, 112.163%) e. Calculate the excitation current required when both loads are connected to produce the same terminal voltage that was present when only one load was connected. (5.770A) f. Continuing part e, also calculate the generator output current, torque angle, input drive shaft power in HP, input drive shaft torque, real & reactive power output, efficiency, and percent loading. (371.905A, 12.335°, 1746.102HP, 3453.843N-m, 1.197827MW, 0.691566MVAR, 91.913%, 138.313%) 11. The same generator specified in Problem #9 has its excitation current adjusted so that the generator operates at rated voltage, rated current, and unity power factor. a. Compute the generator torque angle. (11.431 °) b. Calculate the generator excitation current, output current, input drive shaft power in HP, input drive shaft torque, and efficiency. (4.599A, 251.022A, 1435.37HP, 2839.206N-m, 93.347%) 12. A 3φ synchronous motor is rated 400HP, 480V, 60Hz, 0.8 leading power factor. The stator has 6-poles per phase, is ∆-connected, has negligible winding resistance, and has per phase winding synchronous reactance of 1.1Ω. Assume that the excitation energy required to operate the motor is negligible. Also assume that the friction, windage, and core heating losses are all negligible. Rated voltage is applied to the motor input terminals, the motor is operating with an input power factor of 0.8 lagging, and the motor input real power operates at a value equal to the motor’s rated mechanical output power value converted to Watts.

Prepared by Henri Alciatore for the University of New Orleans ENEE3521 Course © 2007 Version 03-31-07

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a. Assuming the terminal voltage phasor in the motor model has a zero degree angle, determine the motor model induced armature voltage phasor and armature current phasor. (221.710 ∠-36.395°V, 448.469∠36.870°A) b. Calculate the torque being produced by the motor and the torque angle. (2373.637N-m, -36.395 °). c. Calculate the maximum possible torque (called the pull-out torque) that this motor can produce at this excitation level. (4000.425N-m) d. If the motor’s excitation energy and thus armature voltage magnitude is increased by 15%, then calculate the motor’s resulting armature current and power factor. (456.811A, 0.903199 lagging) 13. A 3φ AC synchronous generator is rated 470KVA, 60Hz, 480V, 0.85 lagging power factor. The stator has 4poles per phase, is Y-connected, and has per phase winding resistance of 0.016 Ω, and has per phase winding synchronous reactance of 0.125 Ω. While operating under normal conditions, the machine has 8KW of friction and windage loss, and has 7KW of core heating loss. A load is connected to the generator output terminals, and the generator operates at rated conditions. a. If the load is suddenly removed, what will the generator output voltage change to? (326.755V L-N , 565.957KV L-L) b. What is the generator electrical power loss (this is also called the generator copper loss and is the total real power consumed by the generator armature winding resistance)? (15.3403KW) c. Calculate the generator input drive shaft torque. (2280.373N-m) d. Calculate the generator torque angle. (9.744 °) 14. A 3φ AC synchronous motor is rated 100HP, 440V, 0.8 leading power factor, 89% efficiency. The stator is ∆-connected, has per phase winding resistance of 0.22 Ω, and has per phase winding synchronous reactance of 3.0Ω. The motor’s friction, windage, and core heating losses are negligible. Assume that the motor excitation energy is negligible. If the motor operates under rated conditions, calculate the following motor quantities. a. Input current magnitude. (137.427A) b. Motor model internal armature induced voltage phasor is 348.307 ∠-19.451°V. c. Input real and reactive power. (83.7865KW, -62.8399KVAR) d. Copper losses (total real power consumed by the armature winding resistances). (4154.936W) e. Internally induced mechanical power in HP. (117.002HP) 15. A gasoline combustion engine is coupled to and drives a 1 φ AC synchronous generator rated 4.5KVA, 240V, 1500rpm, 5-poles, 2Ω armature resistance, 9Ω synchronous reactance, 175W friction & windage losses, 80W core loss. An impedance load is connected to the output terminals of the generator. While operating under a certain loading condition, the following observations were made on this machine system. The drive shaft speed was 1350rpm. The generator operated with 230V, 10A, and a 0.89459 leading power factor at its output. The generator’s DC excitation voltage and current were 24V, 2.254A. The drive shaft coupling between the two machines should be assumed to have a power loss of 40W at 1350rpm. The gasoline engine runs at 33% efficiency, and the type of fuel used has a HHV of 115,000 BTU/gallon. a. b. c. d. e. f. g. h. i. j.

Draw a wiring diagram for this system, clearly showing all components, connections, and variables. Calculate the frequency in Hz of the AC voltage produced by the generator. (56.250Hz) Calculate the generator real and reactive output powers. (2.0576KW, -1.0278KVAR) Calculate the impedance of the generator’s load. (23 ∠-26.544°Ω) Calculate the generator torque angle. (23.303 °) Calculate the generator input drive shaft power in HP and torque. (3.369HP, 17.773N-m) Calculate the generator efficiency and percent loading. (80.165%, 51.111%) Calculate the gasoline engine output drive shaft power in HP and torque. (3.423HP, 18.056N-m) Calculate the gasoline fuel consumption rate in gallons per hour. (0.2295gph) Calculate the machine system efficiency. (26.416%)


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16. A machine system for operating a water pump from a DC source using a power inverter and a 3 φ AC synchronous motor controlled by a VFD (variable frequency drive) is shown in the diagram below.. PS1

350V DC Source

NS1

3φ AC Power Inverter

A1

A2

B1

B2

C1 N1

VFD

C2 N2

3φ AC Synchronous Motor

Water Pump

PS2 NS2

The 350V DC source supplies both the input power to the DC to AC power inverter and the excitation energy for the synchronous machine. The DC to AC power inverter can be assumed to be 90% efficient. The VFD can be assumed to be 87% efficient. The motor is rated 15HP, 480V, 60Hz, has 6-poles per phase, has an armature resistance of 2Ω per phase, has a synchronous reactance of 22 Ω per phase, is ∆-connected, has mechanical bearing and windage loss of 350W, and has core heat loss of 190W. The drive shaft coupling between the motor and the pump has 60W power loss at the motor’s rated speed, and decreases by 5W for every 100rpm below the motor’s rated speed. The water pump can be assumed to be 88% efficient at its rated speed of 1500rpm, and this efficiency decreases by 5% for every 200rpm below the rated speed. The system was operated under a certain loading condition, and the following observations were made. The power inverter output operated at 60Hz, 500V, 11.353A, and a 0.899453 lagging power factor. The VFD output operated 48Hz, 482V, and a 0.882519 leading power factor. The synchronous machine excitation current was 3.875A. The pump operated with 20psi suction port pressure, and 80psi discharge port pressure. Calculate the following machine system quantities. a. Real and reactive power output of the power inverter. (8843.410W, 4296.750VAR) b. Current and power output of the 350V DC source. (31.949A, 11.1823KW) c. Real and reactive power output of the VFD. (7693.766W, -4099.854VAR) d. Motor model internal armature induced voltage magnitude. (316.194V) e. Torque angle of the motor. (-12.949 °) f. Motor drive shaft speed, power in HP, and torque. (960rpm, 9.301HP, 68.990N-m) g. Motor efficiency and percent loading. (76.637%, 62.006%) h. Drive shaft coupling loss. (48W) i. Mechanical power in HP and torque on the pump drive shaft. (9.237HP, 68.513N-m) j. Pump efficiency. (74.500%) k. Water flow rate in gallons per minute produced by the pump. (181.158gpm) l. Machine system efficiency. (49.423%)


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Synchronous Machine Problems

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