The synchronous reactance Xs in the equivalent circuit model of a synchronous machine is equal to: (X1 is leakage reactance and Xa is magnetizing reactance)

Explanation:

Synchronous Reactance in Synchronous Machines

Definition: In the equivalent circuit model of a synchronous machine, the synchronous reactance (denoted by X_s) is the combined reactance that accounts for the effects of both the leakage reactance (X₁) and the magnetizing reactance (X_a). It is a crucial parameter in the analysis and operation of synchronous machines, such as synchronous generators and motors.

Formula: The synchronous reactance is expressed as:

X_s = X₁ + X_a

Here:

• X₁: Leakage reactance, which represents the reactance due to leakage flux that does not contribute to the main magnetic field.
• X_a: Magnetizing reactance, which represents the reactance due to the main magnetic flux linking the stator and rotor.

Working Principle:

The equivalent circuit of a synchronous machine includes the synchronous reactance as a single reactance to simplify the analysis. When an alternating current flows through the stator winding of the synchronous machine, part of the flux produced by the current leaks out and does not link with the rotor. This is accounted for by the leakage reactance (X₁). The remaining flux links with the rotor, and its effect is represented by the magnetizing reactance (X_a). Together, these reactances form the synchronous reactance (X_s).

Importance of Synchronous Reactance:

• Synchronous reactance plays a significant role in determining the voltage regulation and power factor of synchronous machines.
• It impacts the steady-state performance of the machine, particularly under load conditions.
• Higher synchronous reactance leads to reduced fault current levels, which is advantageous for system protection.

Correct Option Analysis:

The correct option is:

Option 1: X_s = X₁ + X_a

This option is correct because it accurately represents the synchronous reactance as the sum of the leakage reactance (X₁) and the magnetizing reactance (X_a). The addition of these two reactances provides a complete representation of the reactance in the equivalent circuit of the synchronous machine. This formula is derived from the fact that both leakage and magnetizing reactances are connected in series within the machine's equivalent circuit.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: X_s = X₁ × X_a

This option is incorrect because it implies that the synchronous reactance is the product of the leakage reactance and the magnetizing reactance. In the equivalent circuit model, the reactances are in series, and their effects are additive, not multiplicative. Therefore, this representation does not hold true for synchronous machines.

Option 3: X_s = X₁ - X_a

This option is also incorrect because it suggests that the synchronous reactance is the difference between the leakage reactance and the magnetizing reactance. However, the synchronous reactance is a sum of these two reactances, as they both contribute positively to the total reactance in the equivalent circuit.

Option 4: X_s = X₁ ÷ X_a

This option is incorrect because it implies that the synchronous reactance is the division of the leakage reactance by the magnetizing reactance. This does not represent the physical or mathematical relationship between these reactances in the equivalent circuit of the synchronous machine.

Conclusion:

Understanding the concept of synchronous reactance is vital for analyzing the performance of synchronous machines. The synchronous reactance (X_s) is the sum of the leakage reactance (X₁) and the magnetizing reactance (X_a). This additive relationship is fundamental to the equivalent circuit representation and helps in understanding the operation of synchronous machines under various load conditions. Incorrect options, such as multiplication, subtraction, or division of reactances, do not accurately describe the concept of synchronous reactance and are not applicable in this context.