Question #203134

A 3 phase, 400 V, 50 Hz, 4 pole, star connected SCIM has the following per phase constants

in ohms referred to stator:

r1= 0.15, x1=0.45, x2=0.45, Xm= 28.5

Rotational losses= 400W.

Compute stator current, rotor speed, output torque and efficiency when the motor is operated at

rated voltage and frequency at 4% slip.

Expert's answer

"I'_2=\\frac{v_{pn}}{(R_s+\\frac{R_r}{S})+ j(X_{is}+X_{ir})}=\\frac{230.94 \\angle0}{(0.15+\\frac{0.12}{0.04})+ j(0.45+0.45)} =70.494 \\angle -15.95^0"

"I_m=\\frac{V_{pm}}{jX_m}=\\frac{230.94 \\angle0}{j28.5} A = 8.103 \\angle -90^0"

a. State current, "I_s=I_m+I'_s=73.137 \\angle -22.06^0"

b. Synchronuos speed "N_s=\\frac{120f}{P}=\\frac{120*50}{4}=1500 rpm"

"N_r=N_s(1-s)=1500(1-0.04)=1440 rpm"

c. Air gap power, "=3 I_2'^2* \\frac{R_r}{s} \\implies P_g=3(70.414)^2*(\\frac{0.12}{0.04})=44.7246 kW"

Rotor power developed =(1-s)P_{g} =(1-0.04)(44.7246)kW=42.936 kW

Rotational loss =400 W = 0.4 kW

Shaft power = 42.936 -0.4 =42.536 kW

Output torque, "T_{sh}=\\frac{60}{2 \\pi N_s}* Shaft power output=\\frac{60}{2 \\pi *1440}*42.536*1000=282.07 Nm"

d. Total power output = 42.536 kW

Total power input = "3v_{pm}I_s cos\\phi_s=3(230.94)(73.137) cos 22.06 W= 46.961 kW"

"Efficiency = \\frac{42.536}{46.961}*100=90.577 \\%"

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