Question

A 3-phase star connected slip ring induction motor has the following parameters referred to the stator: \[ R_s = 3 \, \Omega, \, X_s = 2 \, \Omega, \, X_r' = 2 \, \Omega, \, R_r' = 2.5 \, \Omega \] The per phase stator to rotor effective turns ratio is 3:1. The rotor winding is also star connected. The magnetizing reactance and core loss of the motor can be neglected. To have maximum torque at starting, the value of the extra resistance in ohms (referred to the rotor side) to be connected in series with each phase of the rotor winding is ___________ (rounded off to 2 decimal places).

Hint:

For maximum torque at starting, ensure that \( R_r' + R_{ext} = X_r' \), where \( R_{ext} \) is the external resistance added.

Answer 1

Correct Answer: 0.277

Step 1: Condition for maximum starting torque

For maximum torque at starting, the total rotor resistance must equal the rotor reactance:

\[ R_2' + R_{\text{ext}} = X_2' \]

---

Step 2: Compute equivalent rotor impedance term

Given:

\[ \frac{R_2'}{s} = \sqrt{R_1^2 + (x_1 + x_2)^2} \]

\[ = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Thus,

\[ \frac{R_2'}{s} = 5 \]

---

Step 3: Apply starting condition

For starting,

\[ s = 1 \]

Hence,

\[ R_2' + R_{\text{ext}} = 5 \]

Given rotor resistance:

\[ R_2' = 2.5\ \Omega \]

Therefore,

\[ R_{\text{ext}} = 5 - 2.5 = 2.5\ \Omega \]

---

Step 4: Refer external resistance to rotor side

External resistance referred to rotor side is:

\[ R_{\text{ext (referred)}} = \left(\frac{1}{3}\right)^2 R_{\text{ext}} \]

\[ = \frac{1}{9} \times 2.5 = 0.277\ \Omega \]

---

Final Answer:

External resistance referred to rotor side = 0.277 Ω