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GATE EE 2015 Official Paper: Shift 2

CT 1: Ratio and Proportion

2672

10 Questions
16 Marks
30 Mins

**Concept:**

Torque equation:

The torque equation of a three-phase induction motor is given by,

\(T = \frac{{180}}{{2\pi {N_s}}}\left( {\frac{{sV^2{R_2}}}{{\left( {R_2^2 + {s^2}X_2^2} \right)}}} \right)\)

Where Ns is the synchronous speed

V = supply voltage

R2 = rotor resistance

X2 = rotor reactance

s is the slip

By the above expression, we can say that the torque of an induction motor depends on rotor resistance and slip.

Condition for maximum torque:

The condition to get the maximum torque at starting is,

\({s_m} = \frac{{{R_m}}}{{{X_m}}} = 1\)

Xm = Rm

Where,

Rm = Motor resistance per phase

Xm = Motor reactance per phase

At the starting of the three-phase slip ring induction motor

Slip (s) = 1 (At the starting Nr = 0)

Therefore, \({s_m} = \frac{{{R_m}}}{{{X_m}}} = 1\)

⇒ Xm = Rm

the starting torque of an induction motor is maximum when rotor resistance equals rotor reactance.

__Explanation:__

**The condition for maximum torque in three phase induction motor,**

\(\begin{array}{l} {s_m} = \frac{{{r_2}}}{{x_2}} \\ 0.15 = \frac{{{r_2}}}{{x_2}} = \frac{{0.03}}{{{x_2}}} \Rightarrow {x_2} = 0.2{\rm{\Omega }} \end{array}\)

For \({T_{est}} = {T_{emax}}\)

\(\begin{array}{l} \frac{{{T_{est}}}}{{{T_{em}}}} = \frac{2}{{\frac{1}{{{S_{mT}}}} + {S_{mT}}}} = 1 \Rightarrow {s_{mT}} = 1\\ 1 = \frac{{r_2'}}{{{x_2}}} \Rightarrow r_2' = {x_2} = 0.2{\rm{\Omega }} \end{array}\)

Extra resistance =0.2 – 0.03 = 0.17 Ω /p