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The science
Swing equation
A synchronous generator on a strong grid keeps its rotor locked to the grid frequency at a load angle δ that depends on the power it exports. In per unit the rotor obeys (2H/ω0) d²δ/dt² = Pm - Pe(δ) - D Δω, with H the inertia constant (stored kinetic energy per rating, 2-9 s), Pm the turbine power and ω0 = 2π50 rad/s. This is the classical model: constant EMF E' behind the transient reactance X'd, an infinite bus of 1.00 pu voltage, no governor or exciter response.
Power-angle curve
Electrical power through a reactance X is Pe = (E' V / X) sin δ = Pmax sin δ. Pre-fault X = X'd + XL/2 (two lines in parallel). A three-phase fault on line 1 at fraction f of its length grounds a node; eliminating it by a star-delta transformation gives the transfer reactance Xf = X'd + XL + X'd XL / (f XL), which grows without bound as the fault nears the generator (f → 0, so Pe → 0). After the breakers open line 1 the transfer reactance is X'd + XL, so Pmax is lower than before the fault.
Equal-area criterion
Multiplying the swing equation by dδ/dt and integrating shows that the rotor kinetic energy gained while Pm > Pe (area A1, accelerating) must be given back while Pe > Pm (area A2, decelerating): ∫ (Pm - Pe) dδ = 0 over the first swing. The machine is stable if A2 available up to the unstable angle δu = π - asin(Pm/Pmax,post) is at least A1. For a bolted fault (Pe = 0) the critical angle is cos δcr = (Pm/Pmax,post)(δu - δ0) + cos δu and the critical clearing time tcr = √(4H(δcr - δ0)/(ω0 Pm)). The page finds the same time by bisection on a time-domain simulation.
Small oscillations
Linearising about δ0 gives an undamped swing frequency f = (1/2π) √(ω0 Pmax cos δ0 / 2H), about 1.3 Hz here, the familiar local electromechanical mode of a power plant. Damping D removes the swing energy.
Reading the display
The rotor dial is a stroboscopic view in the frame rotating at synchronous speed: the white arrow is the grid field, the red/blue rotor poles lead it by δ. A fault pushes the poles forward, the clearing re-establishes braking torque, and the poles swing back. Past the purple unstable angle, or past 180 degrees, torque reverses and the unit slips a pole; after three slips protection would trip it, and the model resynchronises. The sweep graph runs the whole simulation for many clearing times, so its cliff is the critical clearing time for the current settings.
Honest simplifications
Classical model only: no saliency, flux decay, AVR, governor, load voltage dependence or network resistance; D is a lumped constant; the fault is a bolted three-phase fault and breakers open instantly at the set time. Real studies use full multi-machine models and tabulated clearing times of 4-8 cycles. Values are typical teaching numbers, not a model of any real plant.