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# PROBLEMS - chapter 6

Module by: NGUYEN Phuc. E-mail the author

PROBLEMS

This lecture note is based on the textbook # 1. Electric Machinery - A.E. Fitzgerald, Charles Kingsley, Jr., Stephen D. Umans- 6th edition- Mc Graw Hill series in Electrical Engineering. Power and Energy

6.1 The nameplate on a 460-V, 50-hp, 60-Hz, four-pole induction motor indicates that its speed at rated load is 1755 r/min. Assume the motor to be operating at rated load.

a. What is the slip of the rotor?

b. What is the frequency of the rotor currents?

c. What is the angular velocity of the stator-produced air-gap flux wave with respect to the stator? With respect to the rotor?

d. What is the angular velocity of the rotor-produced air-gap flux wave with respect to the stator? With respect to the rotor?

6.2 Stray leakage fields will induce rotor-frequency voltages in a pickup coil mounted along the shaft of an induction motor. Measurement of the frequency of these induced voltages can be used to determine the rotor speed.

a. What is the rotor speed in r/min of a 50-Hz, six-pole induction motor if the frequency of the induced voltage is 0.89 Hz?

b. Calculate the frequency of the induced voltage corresponding to a four-pole, 60-Hz induction motor operating at a speed of 1740 r/min. What is the corresponding slip?

6.3A three-phase induction motor runs at almost 1198 r/min at no load and 1112 r/min at full load when supplied from a 60-Hz, three-phase source.

a. How many poles does this motor have?

b. What is the slip in percent at full load?

c. What is the corresponding frequency of the rotor currents?

d. What is the corresponding speed of the rotor field with respect to the rotor? With respect to the stator?

6.4 Linear induction motors have been proposed for a variety of applications including high-speed ground transportation. A linear motor based on the induction-motor principle consists of a car riding on a track. The track is a developed squirrel-cage winding, and the car, which is 4.5 m long and 1.25 m wide, has a developed three-phase, 12-pole-pair armature winding. Power at 75 Hz is fed to the car from arms extending through slots to rails below ground level.

a. What is the synchronous speed in km/hr?

c. What is the slip if the car is traveling 95 km/hr? What is the frequency of the track currents under this condition?

d. If the control system controls the magnitude and frequency of the car currents to maintain constant slip, what is the frequency of the armaturewinding currents when the car is traveling 75 km/hr? What is the frequency of the track currents under this condition?

6.5 A three-phase, variable-speed induction motor is operated from a variablefrequency, variable-voltage source which is controlled to maintain constant peak air-gap flux density as the frequency of the applied voltage is varied. The motor is to be operated at constant slip frequency while the motor speed is varied between one half rated speed and rated speed.

a. Describe the variation of magnitude and frequency of the applied voltage with speed.

b. Describe how the magnitude and frequency of the rotor currents will vary as the motor speed is varied.

c. How will the motor torque vary with speed?

6.6 Describe the effect on the torque-speed characteristic of an induction motor produced by (a) halving the applied voltage and (b) halving both the applied voltage and the frequency. Sketch the resultant torque-speed curves relative to that of rated-voltage and rated-frequency. Neglect the effects of stator resistance and leakage reactance.

6.7 Figure 6.1 shows a system consisting of a three-phase wound-rotor induction machine whose shaft is rigidly coupled to the shaft of a three-phase synchronous motor. The terminals of the three-phase rotor winding of the induction machine are brought out to slip rings as shown. With the system supplied from a three-phase, 60-Hz source, the induction machine is driven by the synchronous motor at the proper speed and in the proper direction of rotation so that three-phase, 120-Hz voltages appear at the slip rings. The induction motor has four-pole stator winding.

a. How many poles are on the rotor winding of the induction motor?

b. If the stator field in the induction machine rotates in a clockwise direction, what is the rotation direction of its rotor?

c. What is the rotor speed in r/min?

d. How many poles are there on the synchronous motor?

e. It is proposed that this system can produce dc voltage by reversing two of the phase leads to the induction motor stator. Is this proposal valid?

Figure 6.1 Interconnected induction and synchronous machines.

6.8 A system such at that shown in Fig. 6.1 is used to convert balanced 50-Hz voltages to other frequencies. The synchronous motor has four poles and drives the interconnected shaft in the clockwise direction. The induction machine has six poles and its stator windings are connected to the source in such a fashion as to produce a counterclockwise rotating field (in the direction opposite to the rotation of the synchronous motor). The machine has a wound rotor whose terminals are brought out through slip rings.

a. At what speed does the motor run?

b. What is the frequency of the voltages produced at the slip rings of the induction motor?

c. What will be the frequency of the voltages produced at the slip rings of the induction motor if two leads of the induction-motor stator are interchanged, reversing the direction of rotation of the resultant rotating field?

6.9 A three-phase, eight-pole, 60-Hz, 4160-V, 1000-kW squirrel-cage induction motor has the following equivalent-circuit parameters in ohms per phase Y referred to the stator:

R1R1 size 12{R rSub { size 8{1} } } {} = 0.220 R2R2 size 12{R rSub { size 8{2} } } {} = 0.207 X1X1 size 12{X rSub { size 8{1} } } {} = 1.95 X2X2 size 12{X rSub { size 8{2} } } {} = 2.42 XmXm size 12{X rSub { size 8{m} } } {} = 45.7

Determine the changes in these constants which will result from the following proposed design modifications. Consider each modification separately.

a. Replace the stator winding with an otherwise identical winding with a wire size whose cross-sectional area is increased by 4 percent.

b. Decrease the inner diameter of the stator laminations such that the air gap is decreased by 15 percent.

c. Replace the aluminum rotor bars (conductivity 3.5 × 107107 size 12{"10" rSup { size 8{7} } } {} mhos/m) with copper bars (conductivity 5.8 × 107107 size 12{"10" rSup { size 8{7} } } {} mhos/m).

d. Reconnect the stator winding, originally connected in Y for 4160-V operation, in ΔΔ size 12{Δ} {} for 2.4 kV operation.

6.10 A three-phase, Y-connected, 460-V (line-line), 25-kW, 60-Hz, four-pole induction motor has the following equivalent-circuit parameters in ohms per phase referred to the stator:

R1R1 size 12{R rSub { size 8{1} } } {}=0.103 R2R2 size 12{R rSub { size 8{2} } } {}=0.225 X1X1 size 12{X rSub { size 8{1} } } {}=1.10 X2X2 size 12{X rSub { size 8{2} } } {}=1.13 XmXm size 12{X rSub { size 8{m} } } {}=59.4

The total friction and windage losses may be assumed constant at 265 W, and the core loss may be assumed to be equal to 220 W. With the motor connected directly to a 460-V source, compute the speed, output shaft torque and power, input power and power factor and efficiency for slips of 1, 2 and 3 percent. You may choose either to represent the core loss by a resistance connected directly across the motor terminals or by resistance RCRC size 12{R rSub { size 8{C} } } {} connected in parallel with the magnetizing reactance XmXm size 12{X rSub { size 8{m} } } {}.

6.11 Consider the induction motor of Problem 6.10.

a. Find the motor speed in r/min corresponding to the rated shaft output power of 25 kW. (Hint: This can be easily done by writing a MATLAB script which searches over the motor slip.)

b. Similarly, find the speed in r/min at which the motor will operate with no external shaft load (assuming the motor load at that speed to consist only of the friction and windage losses).

c. Write a MATLAB script to plot motor efficiency versus output power as the motor output power varies from zero to full load.

d. Make a second plot of motor efficiency versus output power as the motor output power varies from roughly 5 kW to full load.

6.12 Write a MATLAB script to analyze the performance of a three-phase induction motor operating at its rated frequency and voltage. The inputs should be the rated motor voltage, power and frequency, the number of poles, the equivalent-circuit parameters, and the rotational loss. Given a specific speed, the program should calculate the motor output power, the input power and power factor and the motor efficiency. Exercise your program on a 500-kW, 4160 V, three-phase, 60-Hz, four-pole induction motor operating at 1725 r/min whose rated speed rotational loss is 3.5 kW and whose equivalent-circuit parameters are:

R1R1 size 12{R rSub { size 8{1} } } {} =0.521 R2R2 size 12{R rSub { size 8{2} } } {}=1.32 X1X1 size 12{X rSub { size 8{1} } } {} =4.98 X2X2 size 12{X rSub { size 8{2} } } {}=5.32 XmXm size 12{X rSub { size 8{m} } } {}=136

6.13 A 15-kW, 230-V, three-phase, Y-connected, 60-Hz, four-pole squirrel-cage induction motor develops full-load internal torque at a slip of 3.5 percent when operated at rated voltage and frequency. For the purposes of this problem, rotational and core losses can be neglected. The following motor parameters, in ohms per phase, have been obtained:

R1R1 size 12{R rSub { size 8{1} } } {}= 0.21 X1=X2X1=X2 size 12{X rSub { size 8{1} } =X rSub { size 8{2} } } {} = 0.26 XmXm size 12{X rSub { size 8{m} } } {} = 10.1

Determine the maximum internal torque at rated voltage and frequency, the slip at maximum torque, and the internal starting torque at rated voltage and frequency.

6.14 The induction motor of Problem 6.13 is supplied from a 230-V source through a feeder of impedance ZfZf size 12{Z rSub { size 8{f} } } {} = 0.05 + j0.14 ohms. Find the motor slip and terminal voltage when it is supplying rated load.

6.15 A three-phase induction motor, operating at rated voltage and frequency, has a starting torque of 135 percent and a maximum torque of 220 percent, both with respect to its rated-load torque. Neglecting the effects of stator resistance and rotational losses and assuming constant rotor resistance, determine:

a. the slip at maximum torque.

b. the slip at rated load.

c. the rotor current at starting (as a percentage of rotor current at rated load).

6.16 When operated at rated voltage and frequency, a three-phase squirrel-cage induction motor (of the design classification known as a high-slip motor) delivers full load at a slip of 8.7 percent and develops a maximum torque of 230 percent of full load at a slip of 55 percent. Neglect core and rotational losses and assume that the rotor resistance and inductance remain constant, independent of slip. Determine the torque at starting, with rated voltage and frequency, in per unit based upon its full-load value.

6.17 A 500-kW, 2400-V, four-pole, 60-Hz induction machine has the following equivalent-circuit parameters in ohms per phase Y referred to the stator:

R1R1 size 12{R rSub { size 8{1} } } {} = 0.122 R2R2 size 12{R rSub { size 8{2} } } {} = 0.317 X1X1 size 12{X rSub { size 8{1} } } {} = 1.364 X2X2 size 12{X rSub { size 8{2} } } {} = 1.32 XmXm size 12{X rSub { size 8{m} } } {} = 45.8

It achieves rated shaft output at a slip of 3.35 percent with an efficiency of 94.0 percent. The machine is to be used as a generator, driven by a wind turbine. It will be connected to a distribution system which can be represented by a 2400-V infinite bus.

a. From the given data calculate the total rotational and core losses at rated load.

b. With the wind turbine driving the induction machine at a slip of -3.2 percent, calculate (i) the electric power output in kW, (ii) the efficiency (electric power output per shaft input power) in percent and (iii) the power factor measured at the machine terminals.

c. The actual distribution system to which the generator is connected has an effective impedance of 0.18 + j0.41 ΩΩ size 12{ %OMEGA } {}/phase. For a slip of -3.2 percent, calculate the electric power as measured (i) at the infinite bus and (ii) at the machine terminals.

6.18 Write a MATLAB script to plot the efficiency as a function of electric power output for the induction generator of Problem 6.17 as the slip varies from -0.5 to -3.2 percent. Assume the generator to be operating into the system with the feeder impedance of part (c) of Problem 6.17.

6.19 For a 25-kW, 230-V, three-phase, 60-Hz squirrel-cage motor operating at rated voltage and frequency, the rotor I2RI2R size 12{I rSup { size 8{2} } R} {} loss at maximum torque is 9.0 times that at full-load torque, and the slip at full-load torque is 0.023. Stator resistance and rotational losses may be neglected and the rotor resistance and inductance assumed to be constant. Expressing torque in per unit of the full-load torque, find

a. the slip at maximum torque.

b. the maximum torque.

c. the starting torque.

6.20 A squirrel-cage induction motor runs at a full-load slip of 3.7 percent. The rotor current at starting is 6.0 times the rotor current at full load. The rotor resistance and inductance is independent of rotor frequency and rotational losses, stray-load losses and stator resistance may be neglected. Expressing torque in per unit of the full-load torque, compute

a. the starting torque.

b. the maximum torque and the slip at which the maximum torque occurs.

6.21 A A-connected, 25-kW, 230-V, three-phase, six-pole, 50-Hz squirrel-cage induction motor has the following equivalent-circuit parameters in ohms per phase Y:

R1R1 size 12{R rSub { size 8{1} } } {} = 0.045 R2R2 size 12{R rSub { size 8{2} } } {} = 0.054 X1X1 size 12{X rSub { size 8{1} } } {} = 0.29 X2X2 size 12{X rSub { size 8{2} } } {} = 0.28 XmXm size 12{X rSub { size 8{m} } } {} = 9.6

a. Calculate the starting current and torque for this motor connected directly to a 230-V source.

b. To limit the starting current, it is proposed to connect the stator winding in Y for starting and then to switch to the ΔΔ size 12{Δ} {} connection for normal operation. (i) What are the equivalent-circuit parameters in ohms per phase for the Y connection? (ii) With the motor Y-connected and running directly off of a 230-V source, calculate the starting current and torque.

6.22 The following data apply to a 125-kW, 2300-V, three-phase, four pole, 60-Hz squirrel-cage induction motor:

Stator-resistance between phase terminals = 2.23 ΩΩ size 12{ %OMEGA } {}

No-load test at rated frequency and voltage:

Line current = 7.7 A Three-phase power = 2870 W

Blocked-rotor test at 15 Hz:

Line voltage = 268 V Line current = 50.3 A

Three-phase power = 18.2 kW

a. Calculate the rotational losses.

b. Calculate the equivalent-circuit parameters in ohms. Assume that X1=X2X1=X2 size 12{X rSub { size 8{1} } =X rSub { size 8{2} } } {}.

c. Compute the stator current, input power and power factor, output power and efficiency when this motor is operating at rated voltage and frequency at a slip of 2.95 percent.

6.23 Two 50-kW, 440-V, three-phase, six-pole, 60-Hz squirrel-cage induction motors have identical stators. The dc resistance measured between any pair of stator terminals is 0.204 ΩΩ size 12{ %OMEGA } {}. Blocked-rotor tests at 60-Hz produce the following results:

Determine the ratio of the internal starting torque developed by motor 2 to that of motor 1 (a) for the same current and (b) for the same voltage. Make reasonable assumptions.

6.24 Write a MATLAB script to calculate the parameters of a three-phase induction motor from open-circuit and blocked-rotor tests.

Input:

Rated frequency

Open-circuit test: Voltage, current and power

Blocked-rotor test: Frequency, voltage, current and power

Stator-resistance measured phase to phase

Assumed ratio X1X2X1X2 size 12{ { {X rSub { size 8{1} } } over {X rSub { size 8{2} } } } } {}

Output:

Rotational loss

Equivalent circuit parameters R1,R2,X1,X2,XmR1,R2,X1,X2,Xm size 12{R rSub { size 8{1} } ,R rSub { size 8{2} } ,X rSub { size 8{1} } ,X rSub { size 8{2} } ,X rSub { size 8{m} } } {}

Exercise your program on a 2300-V, three-phase, 50-Hz, 250-kW

induction motor whose test results are:

Stator-resistance between phase terminals = 0.636 ΩΩ size 12{ %OMEGA } {}

No-load test at rated frequency and voltage:

Line current = 20.2 A Three-phase power = 3.51 kW

Blocked-rotor test at 12.5 Hz:

Line voltage = 142 V Line current = 62.8 A

Three-phase power = 6.55 kW

You may assume that X1=0.4(X1+X2)X1=0.4(X1+X2) size 12{X rSub { size 8{1} } =0 "." 4 $$X rSub { size 8{1} } +X rSub { size 8{2} }$$ } {}.

6.25 A 230-V, three-phase, six-pole, 60-Hz squirrel-cage induction motor develops a maximum internal torquelof 288 percent at a slip of 15 percent when operated at rated voltage and frequency. If the effect of stator resistance is neglected, determine the maximum internal torque that this motor would develop if it were operated at 190 V and 50 Hz. Under these conditions, at what speed would the maximum torque be developed?

6.26 A 75-kW, 50-Hz, four-pole, 460-V three-phase, wound-rotor induction motor develops full-load torque at 1438 r/min with the rotor short-circuited. An external non-inductive resistance of 1.1 ΩΩ size 12{ %OMEGA } {} is placed in series with each phase of the rotor, and the motor is observed to develop its rated torque at a speed of 1405 r/min. Calculate the rotor resistance per phase of the motor itself.

6.27 A 75-kW, 460-V, three-phase, four-pole, 60-Hz, wound-rotor induction motor develops a maximum internal torque of 225 percent at a slip of 16 percent when operated at rated voltage and frequency with its rotor short-circuited directly at the slip rings. Stator resistance and rotational losses may be neglected, and the rotor resistance and inductance may be assumed to be constant, independent of rotor frequency. Determine

a. the slip at full load in percent.

b. the rotor I2RI2R size 12{I rSup { size 8{2} } R} {} loss at full load in watts.

c. the starting torque at rated voltage and frequency in per unit and in N. m. If the rotor resistance is doubled (by inserting external series resistance at the slip rings) and the motor load is adjusted for such that the line current is equal to the value corresponding to rated load with no external resistance, determine

d. the corresponding slip in percent and

e. the torque in N. m.

6.28 Neglecting any effects of rotational and core losses, use MATLAB to plot the internal torque versus speed curve for the induction motor of Problem 6.10 for rated-voltage, rated-frequency operation. On the same plot, plot curves of internal torque versus speed for this motor assuming the rotor resistance increases by a factor of 2, 5 and 10.

6.29 A 100-kW, three-phase, 60-Hz, 460-V, six-pole wound-rotor induction motor develops its rated full-load output at a speed of 1158 r/min when operated at rated voltage and frequency with its slip rings short-circuited. The maximum torque it can develop at rated voltage and frequency is 310 percent of full-load torque. The resistance of the rotor winding is 0.17 ΩΩ size 12{ %OMEGA } {}/phase Y. Neglect any effects of rotational and stray-load loss and stator resistance.

a. Compute the rotor I2RI2R size 12{I rSup { size 8{2} } R} {} loss at full load.

b. Compute the speed at maximum torque in r/min.

c. How much resistance must be inserted in series with the rotor windings to produce maximum starting torque?

With the rotor windings short-circuited, the motor is now run from a 50-Hz supply with the applied voltage adjusted so that the air-gap flux wave is essentially equal to that at rated 60-Hz operation.

d. Compute the 50-Hz applied voltage.

e. Compute the speed at which the motor will develop a torque equal to its rated 60Hz value with its slip-tings shorted.

6.30 A 460-V, three-phase, six-pole, 60-Hz, 150-kW, wound-rotor induction motor develops an internal torque of 190 percent with a line current of 200 percent (torque and current expressed as a percentage of their full-load values) at a slip of 5.6 percent when running at rated voltage and frequency with its rotor terminals short-circuited. The rotor resistance is measured to be 90 m ΩΩ size 12{ %OMEGA } {} between each slip ring and may be assumed to remain constant. A balanced set of Y-connected resistors is to be connected to the slip rings in order to limit the rated-voltage starting current to 200 percent of its rated value. What resistance must be chosen for each leg of the Y connection? What will be the starting torque under these conditions?

6.31 The resistance measured between each pair of slip rings of a three-phase, 60-Hz, 250-kW, 16-pole, wound-rotor induction motor is 49 m ΩΩ size 12{ %OMEGA } {}. With the slip tings short-circuited, the full-load slip is 0.041. For the purposes of this problem, it may be assumed that the slip-torque curve is a straight line from no load to full load. The motor drives a fan which requires 250 kW at the full-load speed of the motor. Assuming the torque to drive the fan varies as the square of the fan speed, what resistance should be connected in series with the rotor resistance to reduce the fan speed to 400 r/min?

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