Correct Answer - Option 2 : 150

\(\sqrt{2}\) A

**Concept:**

**Torque developed in Induction Motor:**

- In the Induction motor, the torque is proportional to the product of flux per stator pole and rotor current.
- In addition to current and flux, the power factor also comes into existence.

And we know that, T ∝ ϕ_{2}I_{2}cosϕ_{2}

Or, T = kϕ_{2}I_{2}cosϕ_{2}

Where I_{2} is rotor current at standstill

Φ_{2} is the angle between rotor EMF and rotor current

k is a constant

We have, E_{2} ∝ ϕ_{2}

⇒ T = k_{1}E_{2}I_{2}cosϕ_{2} …. (1)

Where E_{2} is rotor EMF at a standstill and k_{1} is another constant

If R_{2} is rotor resistance per phase and X_{2} is rotor reactance per phase at standstill,

Therefore rotor impedance at standstill per phase is given as,

\({Z_2} = \sqrt {R_2^2 + X_2^2} \)

\({I_2} = {\frac{{{E_2}}}{Z_2}} = \frac{{{E_2}}}{{\sqrt {R_2^2 + X_2^2} }}\)

\(cos{\phi _2} = \frac{{{R_2}}}{{{Z_2}}} = \frac{{{R_2}}}{{\sqrt {R_2^2 + X_2^2} }}\)

From equation (1)

\(T = {T_{st}} = {k_1}\frac{{E_2^2{R_2}}}{{R_2^2 + X_2^2}}\)

If the Induction motor is under the running condition and run with slip s,

E_{r} = sE_{2}

And, X_{r} = sX_{2}

Under this condition torque given as,

\(T = {T_r} = {k_1}\frac{{sE_2^2{R_2}}}{{{{\left( {{R_2}} \right)}^2} + {{\left( {s{X_2}} \right)}^2}}}\)

**For developing maximum running torque,**

\(\frac{{dT}}{{ds}} = 0\)

s = s_{m}

The** torque at any load in an induction motor is the function of slip.**

After solving it,

R_{2} = s_{m }X

And, value of current at maximum starting torque is given as,

\({I_m} = \frac{{{E_r}}}{{\sqrt {R_2^2 + {{\left( {s_m{X_2}} \right)}^2}} }} = \frac{{E_r}}{{\sqrt { {{2R_2}^2}} }} \)

**Calculation:**

Given,

R_{2} = 0.1 Ω

X_{2} = 0.5 Ω

\(s_m = \frac{{{R_2}}}{{{X_2}}} = \frac{{0.1}}{{0.5}} = 0.2\)

E_{2} = 150 volt

E_{r} = 0.2 × 150 = 30 volt

For maximum torque,

R_{2} = s_{m}X_{2} = 0.1 Ω

\({I_m} = \frac{{{E_r}}}{{\sqrt {R_2^2 + {{\left( {s_m{X_2}} \right)}^2}} }} = \frac{{E_r}}{{\sqrt { {{2R_2}^2}} }} = \frac{{30}}{{\sqrt { {{2(0.1)}^2}} }} =150\sqrt 2 \;A\)