Q2Power System - I
Question
Q.2. Explain the switching operation in an R-L circuit. Also explain sub-transient period, transient period and steady-state period.
Answer
Switching an R-L circuit onto an AC supply produces a total current with a steady-state (forced) sinusoidal component and a decaying DC offset (transient) component whose magnitude depends on the switching instant; in a synchronous machine short-circuit, the fault current similarly has sub-transient, transient and steady-state periods corresponding to progressively decreasing effective reactance (and hence decreasing current) as damper/rotor flux decays.
Switching operation in an R-L circuit: consider a series R-L circuit suddenly connected to an AC source v(t) = Vm sin(ωt+θ) at t=0. The circuit differential equation is L(di/dt) + Ri = Vm sin(ωt+θ). Solving this linear first-order ODE gives the total current as the sum of a steady-state (particular) solution and a transient (complementary) solution:
where Z = √(R²+(ωL)²) and φ = tan⁻¹(ωL/R). The first term is the steady-state sinusoidal current lagging the voltage by φ; the second term is a unidirectional (DC) transient that decays exponentially with time constant τ = L/R. The magnitude of this DC offset depends critically on the instant of switching (θ): if switching occurs when θ-φ = 0, there is no transient at all (current starts directly in steady state); if switching occurs when θ-φ = ±90°, the DC offset is maximum, equal to the full peak of the steady-state current, giving the worst-case asymmetrical fault current.
Sub-transient, transient and steady-state periods (in the context of a synchronous machine fault): when a 3-phase short circuit suddenly occurs at the terminals of an unloaded synchronous generator, the resulting symmetrical short-circuit current envelope can be divided into three distinct periods based on the effective reactance opposing the fault current, because different parts of the machine (damper windings, field winding, main armature reaction) respond with different time constants.
Sub-transient period
Immediately after the fault (first few cycles, typically 0-2 cycles), the current is limited only by the sub-transient reactance Xd″, the smallest of the three reactances, because the damper winding and field winding both initially oppose the sudden change in armature flux by inducing currents that keep flux linkage constant, resulting in the highest fault current of the three periods, decaying with the shortest time constant Td″ (a few cycles, typically 0.03-0.05 s).
Transient period
As the damper-winding currents decay away (they have a much shorter time constant than the field winding), the fault current is now limited by the larger transient reactance Xd′, since only the field winding continues to oppose the change in flux linkage; this period lasts roughly from a few cycles up to a few tenths of a second, with time constant Td′ typically 0.5-2 s, and the current magnitude during this period is intermediate between the sub-transient and steady-state values.
Steady-state period
Once the field-winding transient currents have also decayed away, the fault current settles to its final steady-state value, limited by the full synchronous reactance Xd (the largest of the three reactances), which now includes the full demagnetizing effect of armature reaction acting directly on the constant-excitation field, giving the lowest and longest-lasting fault current magnitude of the three periods.