The expression for critical power angle is

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Option A: $\left(\frac{\pi}{4} - \frac{	heta}{2}ight)$

Explanation & Concepts:
The critical power angle, often denoted as $\delta_{cr}$, represents the maximum rotor angle displacement at which a synchronous machine remains stable following a transient disturbance. In the context of the Equal Area Criterion for a power system, this angle is derived by equating the accelerating and decelerating energies.

Working Principle: During a fault, the mechanical input power exceeds the electrical output power, causing the rotor to accelerate. Stability is maintained if the area under the power-angle curve (the decelerating area) is sufficient to absorb the kinetic energy gained during the fault. The critical angle is defined such that the system reaches the limit of its decelerating capacity at $\delta_{cr}$.

Key Formulas: $\delta_{cr} = \cos^{-1} [\frac{P_m(\pi - 2\delta_0) + P_{max2} \cos \delta_0 - P_{max3} \cos \delta_{max}}{P_{max3}}]$ — General stability criterion; $\delta_{cr} = \frac{\pi}{4} - \frac{	heta}{2}$ — Simplification for specific lossless power systems

Key Points: The power angle $\delta$ is the angle between the excitation voltage ($E$) and terminal voltage ($V$).; A transient stability fault results in an increase in the rotor angle due to mismatch between $P_m$ and $P_e$.; If $\delta$ exceeds $\delta_{cr}$, the system loses synchronism and becomes unstable.

Advantages: Predicts transient stability limits.; Provides a mathematical basis for setting protection relay timings.

Disadvantages: Simplistic models ignore machine damping.; Assumes constant mechanical power during fault transients.

Applications: Transient stability analysis of synchronous generators.; Designing clearing times for circuit breakers.

Additional Info: The expression provided in option A is a common derivation for a system where the post-fault power curve is specific to the generator's geometry.. Options B and C do not satisfy the energy balance required by the equal area criterion.

Answer: A is correct — The expression represents the critical rotor angle threshold for maintaining transient stability in a synchronous generator.

Exam Tip: Always ensure the power angle $\delta$ is in radians when performing energy integration calculations in transient stability problems.

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Option B: $\left(\frac{\pi}{4} + \frac{	heta}{2}ight)$

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Option C: $\left(\frac{\pi}{4} + \frac{	heta}{4}ight)$

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Option D: None of above

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