Chapter 6 Additional Problems
X6.1 The three-phase, wound-rotor induction motor of Fig. X6.1 is operating in the steady-state with a balanced set of three-phase voltage applied to the stator winding when the two-pole switch is quickly moved from position 1 to position 2. Does the motor speed increase or decrease in value?
Prior to movement of the switch, the rotor coils were shorted. After movement of the switch, a balanced set of three-phase resistances have been added to the rotor winding so that the rotor resistance per phase is now . Based on Example 6.14 and the resulting Fig. 6.23, it is concluded that the developed torque in the region of operation is reduced; thus, the motor speed must decrease in value.
X6.2 The three-phase, wound-rotor induction motor of Fig. X6.2 has balanced three-phase voltages applied. It is operating in the steady-state when the three-pole switch is suddenly moved from position 1 to position 2. Does the motor reverse direction of rotation?
Study of the switch arrangement shows that after the switch moves, the impressed voltages are still connected in phase sequence in a counter-clockwise direction of the stator winding. The stator-produced mmf wave does not change direction of rotation; thus, the rotor does not change rotation direction.
X6.3 Replace the three resistors for the wound-rotor induction motor of Fig. X6.1 by three identical capacitors sized so that (the rotor leakage reactance). Make a qualitative sketch of the developed torque-speed curve for the motor with the switch in position 1 and in position 2.
Based on [6.52], [6.56], and [6.58], with the switch in position 2,
It is concluded that for the switch in position 2, the developed torque is larger at all speeds, and the slip at which maximum developed torque occurs has a larger value when compared to the case for the switch in position 1. Fig. X6.3 displays a qualitative sketch of the torque-speed curves.
X6.4 Replace the three resistors for the wound-rotor induction motor of Fig. X6.1 by three identical inductors. Assume the values of inductance are such that is about the same order of magnitude as . Make a qualitative sketch of the developed torque-speed curve for the motor with the switch in position 1 and in position 2.
Based on [6.52], [6.56], and [6.58], with the switch in position 2,
It is concluded that for the switch in position 2, the developed torque is smaller at all speeds, and the slip at which maximum developed torque occurs has a smaller value when compared to the case for the switch in position 1. Fig. X6.4 shows a qualitative sketch of the torque-speed curves.
X6.5 A three-phase, 6-pole, 10 HP, 400 Hz induction motor has a slip of 3% at rated output power. Friction and windage losses are 300 W at rated speed. The rated condition total core losses are 350 W. . . If the motor is operating at rated output power, speed, and frequency, find (a) rotor speed, (b) frequency of rotor currents, (c) total power across the air gap, (d) efficiency, and (e) applied line voltage. Use the approximate equivalent circuit for analysis.
(a)
(b)
(c)
(d) The reflected secondary current is found by
(e)
X6.6 The induction motor of Problem X6.5 operates from a 120 V, 400 Hz source. It is known that due to rotor current distribution and saturation, the starting values of and are, respectively, 1.3 and 0.8 times the values at full-load condition given in Problem X6.5. Find the ratio of developed starting torque to developed full-load torque for this motor if full-voltage is applied for both cases. Use the approximate equivalent circuit for analysis.
From Problem X6.5, the rated-load reflected secondary current is . For start conditions,
X6.7 A 4-pole, 230 V, three-phase induction motor has a value of secondary resistance such that the motor produces maximum developed torque at stall. Neglect core losses and use the Thevenin equivalent circuit for analysis. (See Fig. 6.56). Known equivalent circuit values are:
Find (a) the reflected value of and (b) the total developed torque at stall.
(a) Based on [6.50] with ,
Since , [6.56] leads to
Whence,
Substitute known values and apply the quadratic formula.
where the negative value was discarded as extraneous.
(b) By [6.49] for ,
By [6.58],
X6.8 A three-phase,. 4-pole, 600 V, 60 Hz induction motor is modeled by , , and . Find the shaft speed at which maximum torque occurs if the motor is absorbing power from the three-phase lines at rated frequency.
By [6.56],
This motor is operating in the braking or plugging mode. See Fig. 6.22.
X6.9 A three-phase, 230 V induction motor is operating at no-load condition with rated voltage applied. Equivalent circuit parameters are
It is known that the rated voltage core losses are equal to the rotational losses. Assume that for this no-load condition the coil resistive voltage drops and the leakage reactance voltage drops can be neglected. Determine (a) no-load slip, (b) no-load input power factor, and (c) no-load line current.
(a) Under the stated assumption, the per phase equivalent circuit can be drawn as shown by Fig. X6.5. Since , and reasoning that the converted power must be the rotational losses,
Whence,
(b) From (1), it is seen that . Thus,
(c)
X6.10 A three-phase, 4-pole, NEMA design B, 3 HP, 440 V, 60 Hz induction motor is known to have a per phase stator resistance of . The following test data have been recorded for the motor:
No-load / Blocked rotorDetermine the per phase equivalent circuit parameters for this motor.
Based on [6.30]-[6.32],
By [6.34] and [6.35],
By use of Table 6.2,
Using [6.36]-[6.43],
X6.11 A three-phase, 50 HP, 460 V, 60 Hz, 865 rpm induction motor is operating at rated conditions and has and . It is known that and rotational losses at rated speed are 1050 W. Determine (a) full-load power factor, (b)full-load efficiency, (c) total rotor coil ohmic losses, and (d) total core losses.
(a)
(b)
(c)
(d)
X6.12 A three-phase induction motor is supplying 100 HP to a coupled mechanical load. At this point of operation, the rotational losses are 2046 W, the total core losses are 3320W, the total stator coil ohmic losses are 2690 W, and the slip is 3%. Determine (a) the efficiency at this point of operation and (b) the total rotor coil ohmic losses.
(a)
(b)
X6.13 A three-phase, 230 V, 60 Hz, 6-pole induction motor has the following equivalent circuit parameters:
The rotational losses are known to be 332 W at synchronous speed and vary with speed raised to the 2.8 power. Edit to enter the equivalent circuit parameter values. Edit to enter value. Execute to determine the rated speed and efficiency.
Fig. X6.6 shoes the resulting MATLAB plot of vs. where it is seen that the output power is 15 HP for a speed of 1172 rpm. Fig. X6.7 displays the resulting efficiency vs. speed. For the rated speed of 1172 rpm, .
X6.14 The stator phase coils of a three-phase squirrel-cage induction motor are reconnected as shown in Fig. X6.8. The capacitor is added for current phase shift in coil b with intent to build up a self-starting single-phase induction motor. Will the motor self-start when the sinusoidal source is applied?
Yes. Coils a and b combined form a pulsating magnetic field along a vertical axis. Coil c establishes a pulsating magnetic field along a horizontal axis. These two fields are separated by 90º in space. Owing to the capacitor C, the two fields peak at approximately 90º difference in time-phase. If C is properly sized, then the air gap traveling mmf wave of [6.89] results.
X6.15 The stator phase coils of a three-phase squirrel-cage induction motor are reconnected as shown in Fig. X6.9 with intent to build up a self-starting single-phase induction motor. Will the motor self-start when source is applied?
No. Coils a and b combine to form a pulsating magnetic field along a horizontal axis. Coil c also establishes a pulsating magnetic field along the same horizontal axis. With no separation in space between these two magnetic fields, a revolving mmf wave cannot be established.
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