A three phase, four- pole, 30-hp, 220-V, 60-Hz, Y-connected induction motor draws a current of 77 A from the line source at a power factor of 0.88. At this operating condition, the motor losses are known to be the following:

Question

A three phase, four- pole, 30-hp, 220-V, 60-Hz, Y-connected induction motor draws a current of 77 A from the line source at a power factor of 0.88. At this operating condition, the motor losses are known to be the following:

Stator copper losses = 1033 W Stator core losses = 485 W
Rotor copper losses = 1299 W Rotational losses = 540 W

Determine the power transferred across the air gap, the internally developed torque, the slip, the horsepower output, the motor speed in rpm, the torque corresponding to the rotational losses, and the efficiency of operation at the stated condition.

Answer 1

To determine the various parameters of the motor at the stated condition, we will use the following formulas:

1. Power transferred across the air gap (PAG):
PAG = P-in - Stator copper losses - Stator core losses - Rotor copper losses - Rotational losses

2. Internally developed torque (TID):
TID = (PAG * 746) / (2 * pi * Motor speed in RPM)

3. Slip (S):
S = (Ns - N) / Ns

where Ns is the synchronous speed of the motor and can be calculated using:
Ns = (120 * Frequency) / Number of poles

4. Horsepower output (HP output):
HP output = TID * Motor speed in RPM / 5252

5. Motor speed in RPM (N):
N = (1 - S) * Ns

6. Torque corresponding to rotational losses (TR):
TR = (Rotational losses * 5252) / Motor speed in RPM

7. Efficiency of operation (Efficiency):
Efficiency = HP output / Total input power

Now, let's calculate these parameters step by step:

Step 1: Calculate synchronous speed (Ns):
Ns = (120 * 60) / 4 = 1800 RPM

Step 2: Calculate power transferred across the air gap (PAG):
PAG = 30 hp * 746 W/hp - 1033 W - 485 W - 1299 W - 540 W
= 22380 W

Step 3: Calculate slip (S):
N = (1 - S) * Ns
S = (Ns - N) / Ns
Let's assume the motor speed is at 90% of synchronous speed:
N = 0.9 * 1800 RPM = 1620 RPM
S = (1800 RPM - 1620 RPM) / 1800 RPM
= 0.1

Step 4: Calculate internally developed torque (TID):
TID = (PAG * 746) / (2 * pi * N)
= (22380 * 746) / (2 * 3.14 * 1620)
= 782 lb-ft

Step 5: Calculate horsepower output (HP output):
HP output = TID * N / 5252
= 782 lb-ft * 1620 RPM / 5252
= 240 hp

Step 6: Calculate the torque corresponding to rotational losses (TR):
TR = (Rotational losses * 5252) / N
= (540 W * 5252) / 1620 RPM
= 1752 lb-ft

Step 7: Calculate the efficiency of operation (Efficiency):
Total input power = P-in
Efficiency = (HP output / Total input power) * 100

Now, you can substitute the given values into the formulas to find the final answers.