Which of the following is correct

Question

Examoogle · Accepted Answer

Option B: $X''_d < X'_d < X_d$

Explanation & Concepts:
In a synchronous machine, the direct-axis reactances represent the stator flux linkage response to changes in excitation. Due to the demagnetizing effect of armature reaction and the time constants associated with damper windings and field windings, the sub-transient reactance ($X''_d$) is the smallest (immediate response), followed by the transient reactance ($X'_d$), with the steady-state synchronous reactance ($X_d$) being the largest.

Working Principle: Immediately following a short circuit, sub-transient currents are limited by the leakage reactances of both stator and damper windings, resulting in a low $X''_d$. As damper currents decay, the transient state is dominated by the field winding leakage, leading to the intermediate $X'_d$. Finally, when all transient effects die out, the steady-state reactance $X_d$ (dominated by armature reaction) limits the fault current.

Key Formulas: $X''_d

Key Points: Sub-transient reactance ($X''_d$) determines the initial high-magnitude fault current.; Transient reactance ($X'_d$) governs the current during the intermediate period after damper winding decay.; Synchronous reactance ($X_d$) limits the steady-state current after all transient effects vanish.; The values are defined for the direct-axis (d-axis) which is aligned with the rotor field poles.

Advantages: Provides a clear framework for calculating fault currents at different time intervals.; Allows for accurate sizing of circuit breakers based on initial sub-transient stress.

Disadvantages: Requires complex modeling to determine specific values for different machine designs.; Does not account for quadrature-axis reactances which differ significantly.

Applications: Short circuit studies in power systems.; Coordination and setting of protective relays and circuit breakers.

Additional Info: Typical values for machines are $X''_d \approx 0.1-0.2$ pu, $X'_d \approx 0.2-0.4$ pu, and $X_d \approx 1.0-2.0$ pu.. Option B correctly identifies that the reactance increases as the transient time increases.

Answer: B is correct — The sub-transient reactance is the lowest as it accounts for the low-impedance path of damper windings, while the steady-state synchronous reactance is the highest.

Exam Tip: Always remember that 'sub-transient' means 'very fast' and thus offers the least opposition to current, whereas 'steady-state' means 'slower/stable' and represents the highest impedance to fault current flow.

Examoogle · Answer

Option A: $X''_d = X'_d = X_d$

Examoogle · Answer

Option C: $X''_d = \frac{X_d}{2}$

Examoogle · Answer

Option D: $X'_d = \frac{X_d}{2}$

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