A three-phase 12-pole, 220-V, 60-Hz, 50-hp squirrel-cage motor, operating at rat

Question

A three-phase 12-pole, 220-V, 60-Hz, 50-hp squirrel-cage motor, operating at rated load, has a speed of 595 r/min, an efficiency of 89 percent, and a power factor of 81 percent lagging. the locked-rotor current and locked-rotor torque at rated voltage and rated frequency arc 725 A and 120 percent rated torque, respectively. Determine (a) rated line current; (b) rated torque; (c) minimum voltage required to obtain a blocked-rotor torque equal to 70 percent rated torque; (d) autotransformer bank ratio required to provide the minimum voltage in (c); (e) stator current at blocked rotor with autotransformer in circuit; (f) corresponding input current to the transformer when starting the motor at the voltage in (c).

Explanation / Answer

a)

Real power = 40 HP = 40*745.7 = 29828 W (this is mechanical power)

Real in power = Real Power/ efficiency =29828/0.89 =33514.6 W

(mechanical power corrected with efficiency)

Real in power/Apparent power = Power factor    P/Pa = f =0.81

Pa = P/f =29828/0.81 = 41376 VA

Pa = i*V    it means i = Pa/V =41376/220= 188 A

b)

Torque is T = I*omega       (here I is momentum of inertia)

omega is the angular frequency of rotor =2*pi*F1,   F1 =595/60 =9.917 Hz

P = E/T = E*F2    E is energy, F2 is frequency of the current, F2 = 60 Hz

E = P/F2 = I*omega^2/2

P/F2 = T*omega/2

T = 2*P/(omega*F2)) = 2P/(2*pi*F1*F2)) = P/(pi*F1*F2) = 29828/(pi*9.917*60) = 15.957 N*m

c) locked rotor torque is

Tb = 0.7*T = 11.17 N*m

maximum locked rotor torque Tbmax = 1.20*T = 19.148 N*m

The torque is proportional to the magnetic field, which in turn is proportional to the Voltage

It means

Tbmax/Tb = Vmax/V

V = Vmax*Tb/Tmax =220*11.17/19.148 =128.33 Volts

d) transfomer bank ratio = Vmax/V = 128.33/220 =0.583

e)

Pa = 41376 VA (see point a) above)

Pa =V*i2

i2 = 41376/128.33 =322.4 A

f) i1 =i2*transformer bank ratio = 322.4*0.583 =188 A

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