3.	a) Derive the expression for the maximum torque of a 3 phase induction motor.

### Deriving the Expression for Maximum Torque of a 3-Phase Induction Motor

To derive the expression for the maximum torque (\( T_{\text{max}} \)) of a 3-phase induction motor, we start by understanding the relationship between the torque and slip (\( s \)) in an induction motor.

### 1. Torque-Slip Relationship

The torque (\( T \)) of a 3-phase induction motor is given by:

\[
T = \frac{k \cdot s \cdot E_2^2}{R_2 + s \cdot X_2^2}
\]

where:
- \( k \) is a constant that depends on the motor parameters.
- \( s \) is the slip.
- \( E_2 \) is the rotor-induced EMF per phase.
- \( R_2 \) is the rotor resistance per phase.
- \( X_2 \) is the rotor reactance per phase.

### 2. Finding the Condition for Maximum Torque

The maximum torque occurs when the denominator of the torque equation is minimized concerning slip. To find the slip at which the torque is maximum (\( s_{\text{max}} \)), differentiate the torque expression with respect to slip and set it to zero:

\[
\frac{dT}{ds} = 0.
\]

However, for simplicity, using calculus:

\[
\text{Condition for maximum torque: } s_{\text{max}} = \frac{R_2}{X_2}.
\]

### 3. Substituting \( s_{\text{max}} \) into the Torque Equation

Substituting \( s_{\text{max}} = \frac{R_2}{X_2} \) into the torque equation:

\[
T_{\text{max}} = \frac{k \cdot \left(\frac{R_2}{X_2}\right) \cdot E_2^2}{R_2 + \left(\frac{R_2}{X_2}\right) \cdot X_2^2}.
\]

Simplify the above equation:

\[
T_{\text{max}} = \frac{k \cdot R_2 \cdot E_2^2}{2 \cdot X_2}.
\]

### 4. Final Expression for Maximum Torque

\[
\boxed{T_{\text{max}} = \frac{k \cdot E_2^2}{2 \cdot X_2}.}
\]

This expression shows that the maximum torque depends on the square of the induced EMF, the rotor reactance, and a proportionality constant.

### Would you like to explore more details or have any questions?

### Related Questions:
1. What is the physical significance of slip in an induction motor?
2. How does rotor resistance affect the maximum torque?
3. What are the effects of supply voltage on the maximum torque?
4. How can maximum torque be achieved without altering motor design?
5. What role does rotor reactance play in torque generation?

### Tip:
Always ensure that motor parameters are balanced and maintained to achieve optimal performance and efficiency in real-world applications.

Derive the expression for the maximum torque of a 3-phase induction motor.

Learn how to derive the expression for the maximum torque of a 3-phase induction motor. This tutorial covers key concepts such as the torque-slip relationship, rotor resistance, and the condition for maximum torque. Suitable for undergraduate electrical engineering students.

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