If Xa is the armature reactance of a synchronous machine and Xl is the leakage reactance of the same machine,

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If Xa is the armature reactance of a synchronous machine and Xl is the leakage reactance of the same machine, then synchronous reactance Xs is

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Option C: $X_s = X_a + X_l$

Explanation & Concepts:
The synchronous reactance ($X_s$) of a synchronous machine is defined as the sum of the armature leakage reactance ($X_l$) and the armature reaction reactance ($X_a$). It represents the total reactance offered to the stator current under steady-state operating conditions.

Working Principle: In a synchronous machine, the flux produced by the stator current (armature reaction) interacts with the main air-gap flux, which is represented by the reactance $X_a$. Additionally, the leakage flux that does not link the rotor, but only the stator windings, produces a leakage reactance $X_l$. Because both effects cause a voltage drop in the stator phase that is in quadrature with the stator current, they are additive.

Key Formulas: $X_s = X_a + X_l$ — Fundamental definition of synchronous reactance; $Z_s = R_a + jX_s$ — Synchronous impedance equation

Key Points: Synchronous reactance accounts for the total internal voltage drop due to magnetic effects in the stator.; It is a fictitious reactance used to model the machine behavior in steady-state analysis.; It depends on the operating saturation level of the magnetic circuit.

Advantages: Simplifies steady-state performance analysis of synchronous machines.; Allows for easy calculation of voltage regulation using the EMF method.

Disadvantages: Does not account for non-linear magnetic saturation accurately at all loads.; Only applicable for steady-state analysis, not for transient behavior.

Applications: Voltage regulation calculations for synchronous generators.; Determination of power angle characteristics.

Additional Info: Note: $X_a$ is often significantly larger than $X_l$ in most industrial synchronous machines.. Option B is conceptually incorrect as it suggests a scaling relationship between reactance components that does not exist in standard machine models.

Answer: C is correct — The synchronous reactance $X_s$ is the total steady-state reactance consisting of the armature reaction reactance and the leakage reactance.

Exam Tip: Always remember that synchronous impedance $Z_s$ is a complex quantity $\sqrt{R_a^2 + X_s^2}$, where $R_a$ is usually negligible for large synchronous machines.

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Option A: $X_s = 0.5 X_a$

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Option B: $X_a = 0.5 (X_a - X_l)$

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Option D: $X_s

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