A 150 kW, 3000 V, 50 Hz, 6 – pole star connected induction motor has a star-connected slip ring rotor with a transformation ratio of 3.6 (stator/rotor). The rotor resistance is 0.1 Ω/phase and its per phase leakage inductance is 3.61 mH. The stator impedance may be neglected. The starting torque on rated voltage with short-circuited slip rings is ______ (in N-m)



Let's calculate the starting torque of this induction motor step by step, using the given parameters.

### Given Data:
- Rated power \( P = 150 \, \text{kW} \)
- Stator line voltage \( V_s = 3000 \, \text{V} \)
- Supply frequency \( f = 50 \, \text{Hz} \)
- Number of poles \( p = 6 \)
- Stator/rotor transformation ratio \( k = 3.6 \)
- Rotor resistance \( R_r = 0.1 \, \Omega \)
- Rotor leakage inductance \( L_r = 3.61 \, \text{mH} = 3.61 \times 10^{-3} \, \text{H} \)

### Step 1: Calculate synchronous speed

The synchronous speed \( N_s \) in RPM is given by:

\[
N_s = \frac{120 \times f}{p}
\]

For \( f = 50 \, \text{Hz} \) and \( p = 6 \):

\[
N_s = \frac{120 \times 50}{6} = 1000 \, \text{RPM}
\]

Now, convert this to radians per second (\( \omega_s \)):

\[
\omega_s = \frac{2 \pi N_s}{60} = \frac{2 \pi \times 1000}{60} = \frac{2000 \pi}{60} \approx 104.72 \, \text{rad/s}
\]

### Step 2: Rotor voltage at standstill

The rotor voltage at standstill \( E_r \) is related to the stator voltage by the transformation ratio \( k \). The stator line voltage \( V_s = 3000 \, \text{V} \), and for a star connection, the phase voltage \( V_{s\_ph} \) is:

\[
V_{s\_ph} = \frac{V_s}{\sqrt{3}} = \frac{3000}{\sqrt{3}} \approx 1732.05 \, \text{V}
\]

The rotor phase voltage at standstill \( E_r \) is:

\[
E_r = \frac{V_{s\_ph}}{k} = \frac{1732.05}{3.6} \approx 481.13 \, \text{V}
\]

### Step 3: Rotor reactance at standstill

The rotor reactance at standstill \( X_r \) is calculated as:

\[
X_r = 2 \pi f \cdot L_r = 2 \pi \cdot 50 \cdot 3.61 \times 10^{-3} \approx 1.135 \, \Omega
\]

### Step 4: Starting torque formula

The starting torque \( T_{start} \) is given by:

\[
T_{start} = \frac{3 \cdot E_r^2 \cdot R_r}{\omega_s \cdot (R_r^2 + X_r^2)}
\]

Let's calculate each term:

- \( E_r^2 = (481.13)^2 \approx 231486.08 \, \text{V}^2 \)
- \( R_r^2 + X_r^2 = (0.1)^2 + (1.135)^2 = 0.01 + 1.288225 = 1.298225 \, \Omega^2 \)
- \( \omega_s = 104.72 \, \text{rad/s} \)

Now, substitute these values into the starting torque formula:

\[
T_{start} = \frac{3 \cdot 231486.08 \cdot 0.1}{104.72 \cdot 1.298225}
\]

First, calculate the numerator:

\[
3 \cdot 231486.08 \cdot 0.1 = 69445.82
\]

Now, calculate the denominator:

\[
104.72 \cdot 1.298225 = 135.94
\]

Thus, the starting torque is:

\[
T_{start} = \frac{69445.82}{135.94} \approx 510.9 \, \text{N-m}
\]

### Final Answer:
The starting torque is approximately **510.9 N-m**.

Would you like more details on any part of the calculation, or is there anything specific you'd like to explore?

Here are 5 related questions for further understanding:

1. How does increasing the rotor resistance affect the starting torque of an induction motor?
2. What role does the transformation ratio play in determining rotor voltage?
3. How can we reduce the starting torque without changing the motor's specifications?
4. Why is the synchronous speed calculated using the number of poles and frequency?
5. What is the impact of rotor leakage inductance on the motor's performance?

**Tip**: Increasing rotor resistance increases the starting torque but reduces efficiency at higher speeds. Consider external resistance starters to manage this trade-off in slip ring induction motors.

A 150 kW, 3000 V, 50 Hz, 6 – pole star connected induction motor has a star-connected slip ring rotor with a transformation ratio of 3.6 (stator/rotor). The rotor resistance is 0.1 Ω/phase and its per phase leakage inductance is 3.61 mH. The stator impedance may be neglected. The starting torque on rated voltage with short-circuited slip rings is ______ (in N-m).

This page covers the detailed solution to calculate the starting torque of a 150 kW, 3000 V, 50 Hz, 6-pole induction motor with short-circuited slip rings. Key concepts include motor theory, synchronous speed, rotor reactance, and torque calculation using step-by-step formulas.

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