Generator Power Angle Calculator (P = EV sin δ / Xs)
By Santaji GadepatilinUpdated November 2, 2026
The power a synchronous generator delivers to the grid depends on the angle δ between its internal EMF and the terminal voltage, while its excitation decides the reactive power. This generator power angle calculator gives the active power, the steady-state stability limit and the reactive power for a round-rotor machine — see how the excitation system and AVR control the EMF.
How to use the generator power angle calculator
Enter the generator’s internal EMF and the terminal (bus) voltage, both line-to-line, the synchronous reactance per phase and the power angle δ. The calculator returns the three-phase active power, the maximum (pull-out) power and the reactive power delivered — negative Q means the machine is absorbing reactive power. To find the load angle for a given output, enter P and clear δ.
Power angle equations
P = (E × V ÷ Xs) × sin δ · Pmax = E × V ÷ Xs
Q = (E × V × cos δ − V²) ÷ Xs
Example
An 11 kV generator with E = 12 kV, Xs = 10 Ω per phase, running at δ = 20°:
- Pmax = 12,000 × 11,000 ÷ 10 = 13.2 MW
- P = 13.2 × sin 20° = 4.515 MW
- Q = (12,000 × 11,000 × cos 20° − 11,000²) ÷ 10 = 0.304 MVAr (exporting)
Power and reactive power vs load angle
Same machine (E = 12 kV, V = 11 kV, Xs = 10 Ω):
| δ | sin δ | Active power | Reactive power |
|---|---|---|---|
| 0° | 0.000 | 0.00 MW | +1.10 MVAr |
| 10° | 0.174 | 2.29 MW | +0.90 MVAr |
| 20° | 0.342 | 4.51 MW | +0.30 MVAr |
| 30° | 0.500 | 6.60 MW | -0.67 MVAr |
| 45° | 0.707 | 9.33 MW | -2.77 MVAr |
| 60° | 0.866 | 11.43 MW | -5.50 MVAr |
| 75° | 0.966 | 12.75 MW | -8.68 MVAr |
| 90° | 1.000 | 13.20 MW | -12.10 MVAr |
Notes
- Armature resistance is neglected — typically less than 1% of Xs for large machines.
- In per-unit, P = E V sin δ ÷ Xs with E, V and Xs in pu; see the per-unit calculator.
- During a fault the transient reactance X′d replaces Xs in transient-stability studies.
Frequently asked questions
What is the power angle equation?
P = (E × V ÷ X_s) × sin δ for a round-rotor machine with armature resistance neglected. With line-to-line voltages it gives the total three-phase power.
What is the steady-state stability limit?
The maximum power E V ÷ X_s, reached at δ = 90°. Beyond that, more mechanical input makes the machine slip poles, so generators normally run at δ = 20–40°.
How does excitation affect reactive power?
Q = (E V cos δ − V²) ÷ X_s. Raising the field current raises E and the generator exports more reactive power (over-excited, lagging PF); weakening it makes the machine absorb reactive power.
Does this apply to salient-pole machines?
Only approximately. Salient-pole generators add a reluctance term, V² (1/X_q − 1/X_d) sin 2δ ÷ 2, which this calculator does not include.
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