Single-Phase Motors--Polyphase Induction Motors (part 4)

SUMMARY

In this section we shed some light on a three-phase induction motor, which essentially consists of a stator and a rotor. The stator is wound using double-layer winding just like the stator of a synchronous machine. There are two types of rotors: a squirrel-cage rotor and a wound rotor. A wound rotor, although expensive, is wound for the same number of poles as the stator. It provides means to add external resistance in series with the rotor circuit. A squirrel-cage rotor uses bars in the slots that are shorted at either end by the end-rings. For a low-horsepower motor, the bars and the end-rings are formed in a die-casting process.

When the three-phase stator winding is connected to a balanced three-phase source, it sets up a revolving field that rotates around the periphery of the rotor at the synchronous speed given by the following equation:

...where f is the frequency of the applied voltage source and P is the number of poles in the stator.

The uniform revolving field induces emf in the rotor conductors. Since the rotor winding forms a closed circuit, the induced emf gives rise to a current in the rotor conductors. The interaction of the current in the rotor conductors with the magnetic field in the motor creates a torque in accordance with the Lorentz force equation. Therefore, the rotor starts rotating and attains a speed slightly less than the synchronous speed. For this reason, an induction motor is also called an asynchronous motor. The difference between the synchronous speed and the rotor speed is called the slip speed. The per-unit slip is then defined as By using the transformer analogy we developed an equivalent circuit of an induction motor as referred to the stator side. The rotor circuit parameters were transformed to the stator side by using the a-ratio given as where m, and m2 are the number of phases, k,, and kw2 are the winding factors, and N1 and N2 are the number of turns in each phase of the stator and the rotor windings. For squirrel-cage rotor, k, = 1, m^2 = Q/P, and N2 = P/2, where Q is the number of bars in the rotor. For a wound-rotor induction motor rn, = m^2.

We defined R,(1 s)/s as the dynamic (or effective) resistance because the power developed is proportional to it. Note that R, is the rotor resistance. From the per-phase equivalent circuit, we can compute the power input as where V, and I, are the per-phase applied voltage and the input current. 0 is the power factor angle between the two.

The stator copper loss can be computed as ...

Using the approximate equivalent circuit, we found out that the efficiency of an induction motor is maximum when ...

The motor develops maximum torque at a slip known as the breakdown slip such that ...

....where X, = X, + X,. The expression for the maximum torque developed is When the stator winding impedance is negligible, the approximate expressions for the breakdown slip and the breakdown torque are ...

If Td is the torque developed at a slip s, then ...

The power developed by an induction motor is maximum when ...

...where Z, = IR, + jXe( and Re = R, + R,). The maximum power developed by the motor is ...

The motor circuit parameters can be determined by performing the blocked-rotor test, the no-load test, and the stator-resistance test. From the blocked-rotor test where Pbr and I, are the per-phase power input and the current. The test is conducted when the rotor is held stationary and the motor draws the rated current from a carefully applied low voltage, Vbra The magnitude of the stator and the rotor winding impedance is ...

The individual values of the leakage reactances are ...

and the rotor resistance is R, = Re -R,. The no-load test is conducted at the rated voltage when the rotor is free to rotate without load. If W, Ioc, and V, are the power input, the current, and the applied voltage on a per-phase basis, then ...

… where Pfib is the per-phase friction and windage loss.

The core-loss resistance is ...

The magnetization reactance is ...

where ...

We have also examined the effect of changes in the rotor resistance on the speed-torque characteristic of an induction motor. An increase in the rotor resistance increases the starting torque, reduces the starting current, and enables the operation of the motor at a somewhat lower speed. In a wound motor, the rotor resistance is increased by adding external resistance to the rotor circuit via slip-rings.

In a squirrel-cage induction motor, the change in the rotor resistance is realized by using a multicage rotor.

We also examined various schemes that enable us to control the speed of an induction motor. Some of the methods we have discussed are frequency control, changing stator poles, rotor resistance control, stator voltage control, and injecting an emf in the rotor circuit.

QUIZ

Explain the principle of operation of an induction motor.

Describe the construction of a squirrel-cage induction motor.

Explain the construction of a wound-rotor induction motor.

What are the advantages and drawbacks of a wound-rotor induction motor? What are the advantages and drawbacks of a squirrel-cage induction motor? At what speed does the revolving field rotate in an induction motor? How can it be determined? Explain why an induction motor cannot operate at its synchronous speed.

Explain slip speed, per-unit speed, per-unit slip, and percent slip.

What is the rotor frequency when the rotor is (a) locked and (b) rotates at 5% slip? Define starting torque, breakdown torque, breakdown slip, rated torque, torque developed, and shaft torque.

How can you minimize the rotor reactance? Describe the no-load test, blocked-rotor test, and stator-resistance test.

Define stator copper loss, rotor copper loss, air-gap power, dynamic resistance, and effective rotor resistance.

What losses are measured by (a) the no-load test and (b) the blocked-rotor test? What are the various techniques used to control the speed of induction motors? What are the various classes of squirrel-cage induction motors? List several applications for each class of squirrel-cage induction motor.

How can the starting current be controlled in a wound motor? Cite possible reasons why a three-phase induction motor fails to start.

Why is an induction motor called an asynchronous motor? How can the direction of a three-phase induction motor be reversed? What happens to the speed of an induction motor if the load is increased? What is a consequent-pole winding? What happens when a 6-pole induction motor is reconnected as a consequent-pole motor? What is the effect of increase in rotor reactance on the starting current? the maximum torque? Explain the nature of the speed-torque characteristic of an induction motor.

What happens to the speed-torque characteristic of an induction motor if the rotor resistance is increased? How does the increase in the rotor resistance affect the breakdown slip? the starting torque? the breakdown torque?

Is it always possible to start an induction motor by applying the rated voltage? Is it possible for an induction motor to operate as an induction generator? If yes, how can it be done?

Problems

The frequency of the induced emf in the secondary winding of an %pole, three-phase induction motor is 10 Hz. At what speed does the magnetomotive force (mmf) of the secondary revolve with respect to the secondary winding? A 2-pole, 230-V, 50-Hz, three-phase induction motor operates at a speed of 2800 rpm. Determine (a) the per-unit slip and (b) the frequency of the induced emf in the rotor.

A 12-pole, 440-V, 400-Hz, three-phase induction motor is designed to operate at a slip of 5% on full load. Determine (a) the rated speed, (b) the rotor frequency, and (c) the speed of the rotor revolving field relative to the rotor.

A three-phase induction motor operates at a slip of 3% and has a rotor copper loss of 300 W. The rotational loss is 1500 W. Determine (a) the air-gap power and (b) the power output. If the rotor impedance is 0.2 + j0.8 Rl_phase, what is the magnitude of the induced emf per phase in the rotor? A 10-hp, 6-pole, 440-V, 60-Hz, A-connected, three-phase induction motor is designed to operate at 3% slip on full load. The rotational loss is 4% of the power output. When the motor operates at full load, determine (a) the rotor copper loss, (b) the air-gap power, (c) the power developed, and (d) the shaft torque.

A 4-hp, 230-V, 60-Hz, 6-pole, three-phase, Y-connected, induction motor operates at 1050 rpm on full load. The rotational loss is 300 W. Determine the per-phase rotor resistance if the rotor current is not to exceed 100 A. The per-phase equivalent circuit parameters of a 208-V, 4-pole, 60-Hz, three-phase, Y-connected, induction motor are R, = 0.4 SZ, XI = 0.8 R, R, = 0.3 a, X, = 0.9 0, and X,, = 40 Q. The core loss is 45 W, and the friction and windage loss is 160 W. When the motor operates at a slip of 5%, determine (a) the input current, (b) the power input, (c) the air-gap power, (d) the power developed, (e) the power output, (f) the shaft torque, and (9) the efficiency of the motor. Draw its power-flow diagram.

Calculate the starting torque developed by the motor of Problem 7.

A 4-pole, 230-V, 60-Hz, Y-connected, three-phase induction motor has the following parameters on a per-phase basis: R, = 10.12 R, X, = 38.61 R,.

10. R, = 21.97 SZ, X, = 11.56 Q, and X, = 432.48 Q. The core loss is 10.72 W, and the friction and windage loss is 5.9 W. When the motor operates at its rated speed of 1550 rpm, determine (a) the stator current, (b) the magnetization current, (c) the rotor current, (d) the power input, (e) the stator copper loss, (f) the rotor copper loss, (g) the power output, (h) the shaft torque, and (i) the efficiency.

Plot the speed versus torque, power input versus speed, and power developed versus speed characteristics of the induction motor of Problem 9.

What is the starting torque? Also determine the power input and the power developed at the time of starting.

A 230-V, 60-Hz, 6-pole, Y-connected, three-phase induction motor has the following parameters in ohms/phase as referred to the stator: R, = 12.5, XI = 21.3, R, = 28.6, X, = 13.6, R, = 4200, and X, = 1800. The friction and windage loss is 3% of the power developed. If the motor speed is 1125 rpm, determine (a) the power input, (b) the stator copper loss, (c) the rotor copper loss, (d) the air-gap power, (e) the power developed, (f) the shaft torque, and (g) its efficiency.

A 440-V, 60-Hz, $-pole, A-connected, three-phase induction motor has the following parameters in ohms/phase: R, = 0.3, XI = 0.9, R, = 0.6, X, = 0.9, R, = 150, X,, = 60. If the rotational loss is 4% of the power developed, determine the efficiency of the motor when it runs at 4% slip.

Using the data of Problem 7, calculate (a) the slip at which the motor develops maximum power, (b) the maximum power developed by the motor, and (c) the corresponding torque developed.

Calculate the slip at which the motor of Problem 9 develops maximum power. What is the maximum power developed by the motor? What is the torque developed by the motor at that slip? Using the data of Problem 11, calculate (a) the slip at which the motor develops maximum power, (b) the maximum power developed, and (c) the associated torque developed by the motor.

Calculate the maximum power developed by the motor of Problem 12.

What is the torque developed by it at that slip? Using the data of Problem 7, calculate (a) the slip at which the motor develops maximum torque, (b) the maximum torque developed by the motor, and (c) the corresponding power developed.

Calculate the slip at which the motor of Problem 9 develops maximum torque. What is the maximum torque developed by the motor? What is the power developed by the motor at that slip? Using the data of Problem 11, calculate (a) the slip at which the motor develops maximum torque, (b) the maximum torque developed, and (c) the associated power developed by the motor.

Calculate the maximum torque developed by the motor of Problem 12.

What is the power developed by it at that slip? A 6-pole, 60-Hz, Y-connected, three-phase induction motor develops a maximum torque of 250 N-m at a speed of 720 rpm. The rotor resistance is 0.4 fl/phase, and the stator winding impedance is negligible. Determine the torque developed by the motor when it operates at a speed of 1125 rpm.

A 6-pole, 50-Hz, A-connected, three-phase induction motor has a rotor impedance of 0.05 + j0.5 R/phase. The stator winding impedance is negligible. Determine the additional resistance required in the rotor circuit so that it develops maximum torque at starting.

An 8-pole, 230-V, 60-Hz, A-connected, three-phase induction motor has a rotor impedance of 0.025 + j0.l K?/phase. The stator winding impedance is negligible. Determine (a) the speed at which the motor develops the maximum torque, (b) the maximum torque of the motor, and (c) the starting torque as a percentage of maximum torque. What additional resistance must be inserted in the rotor circuit to make the starting torque equal to 75% of the maximum torque? A 120-V, 60-Hz, 6-pole, three-phase induction motor operates at a speed of 1050 rpm on full load and develops 5 hp. Under reduced load the speed increases to 1125 rpm. Determine the torque and the power developed by the motor at reduced load.

An 8- pole, 50-Hz, 208-V, A-connected, three-phase induction motor develops 20 hp at full-load slip of 5%. Determine the torque and the power developed at the same slip when a reduced voltage of 120 V is applied.

What must be the new slip for the motor to develop the same torque? The starting current of a 208-V, three-phase, A-connected, induction motor is 120 A when the rated voltage is applied to the stator windings. Determine the starting current when the applied voltage is reduced to 120 V. If the starting current is not to exceed 50 A, what must be the applied voltage? The following test data were obtained on a 460-V, 60-Hz, A-connected, three-phase induction motor: No-load test: power input = 380 W, line current = 1.15 A at rated voltage.

Blocked-rotor test: power input = 14.7 W, line current = 2.1 A at the line voltage of 21 V. The friction and windage loss is 21 W, and the winding resistance between any two lines is 1.2 SZ. Determine (a) the equivalent circuit parameters of the motor and (b) its efficiency at a slip of WO. The following test data apply to a 208-V, $-pole, Y-connected, three-phase induction motor: Running without load at its rated voltage, the line current and the power input are 2 A and 360 W. With blocked rotor the current is 20 A and power input is 600 W when the applied voltage is 30 V. The friction and windage loss is 36 W. The resistance between any two lines is 0.2 W. Obtain the equivalent circuit parameters of the motor.

A 440-V, 4-pole, Y-connected, three-phase induction motor gave the following readings when tested: No load: 440 V, 6.2 A, power factor = 0.1 lagging Blocked-rotor test: 100 V, 12.5 A, 750 W The winding resistance between any two lines = 1.2 a. The friction and windage loss is 30 W. Determine the equivalent circuit parameters of the motor.

The following are the test results on a 440-V, 50-Hz, slip-ring, three-phase, Y-connected, induction motor: No load: 440 V, 7.5 A, 1350 W (including 650-W friction and windage loss)

Blocked-rotor test: 100 V, 32 A, 1800 W. The stator and rotor copper losses are equal under blocked-rotor condition.

Determine the equivalent circuit parameters of the motor.

The equivalent impedances of the inner and outer cages of a 4-pole, 60-Hz, Y-connected, three-phase induction motor are 0.5 + j2 fl/phase and 2 + j0.5 Cl/phase at standstill. Calculate the ratio of torques developed by the two cages (a) at starting and (b) at 4% slip.

If the stator winding impedance of the motor in Problem 31 is 1 + j3 fl/phase and the applied voltage is 208 V, determine the torque developed (a) at starting and (b) at 4% slip.

The inner and outer cages of a rotor have standstill impedances of 0.5 + j4 Sll_phase and 2.6 + j1.2 fl/phase. Find the ratio of the torques developed by the two cages (a) at starting and (b) at a slip of 0.05.

The outer and inner cages of a double-cage induction motor have standstill impedances of 2.6 + j2 O/phase and 1.8 + j3 fl/phase. If the full-load slip is 5%, determine the ratio of the starting torque to the full-load torque.