Power-angle equation and stability
P = (E_f·V_t / X_s)·sin δ — the heartbeat of synchronous-machine analysis, with pull-out at δ = 90°.
Step 1 — From the equivalent circuit to power transfer P(δ)
Reference notes
Use Next → on the narrator above to step through six configurations: from the equivalent-circuit derivation, to the full P-δ curve, to the stability limit at δ = 90°.
The power-angle equation (cylindrical rotor)
Take the per-phase equivalent circuit, neglect Ra (it's usually tiny next to Xs), and compute the real power flowing from the source Ef to the terminals Vt. The result is the famous power-angle equation:
For a 3-phase machine, multiply by 3. δ is the load angle from the previous lessons — the angle by which Ef leads Vt.
The curve and its operating regions
- Maximum power: Pmax = Ef·Vt / Xs, reached at δ = 90°. This is the pull-out power.
- Stable region: 0° < δ < 90°. dP/dδ > 0 — if the rotor swings forward, it generates more power and slows down, returning toward the operating point (a restoring "synchronising torque"). The grid is rigid; the machine adjusts to it.
- δ = 90°: the operating limit. Any further shaft input pushes the operating point past the peak — and on a sinusoidal curve, past the peak means LESS power generated for MORE input torque. The rotor decelerates relative to the field, δ grows further, power falls further... the machine slips a pole.
- Beyond 90°: unstable. The machine has lost synchronism. Protection trips it out.
How to increase the pull-out capability
Two knobs: Pmax = Ef·Vt / Xs.
- Raise Ef by increasing the rotor field current. More excitation → bigger pull-out.
- Lower Xs — design choice, sets the machine's short-circuit ratio (SCR). Hydro alternators have high SCR (~1.0–1.5) and large pull-out margin; turbo-alternators have low SCR (~0.5–0.8) and operate closer to their limit.
Salient-pole machines: add reluctance power
For salient-pole rotors (typical of low-speed hydro generators), the air gap is non-uniform — the d-axis (direct, through the pole) and q-axis (quadrature, between poles) have different reluctances. The two-reaction theory of Blondel splits Ia into d- and q-axis components and gives:
The second term is reluctance power — present even with zero excitation. It peaks at δ = 45° and shifts the maximum power angle slightly below 90°.
Synchronising power coefficient
The slope of the P-δ curve at the operating point:
This is the "stiffness" of the machine's connection to the grid. A bigger Psyn means small disturbances are corrected faster — the machine returns to its operating angle more strongly. Psyn goes to zero at δ = 90° (no stiffness, no restoring force), then becomes negative (unstable).
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