Power Transfer and Power Angle Formulas
Power transfer formulas: real power from the angle between two voltages, the maximum power at 90 degrees and the reactive power at the receiving end.
The formulas
| To find | Formula (three-phase, line voltages) |
|---|---|
| Real power | P = V1 × V2 × sin δ / X |
| Maximum power | Pmax = V1 × V2 / X, at δ = 90° |
| Reactive power at the receiving end | Q = V2 × (V1 × cos δ − V2) / X |
V1 and V2 are the line-to-line voltages at the sending and receiving ends in volts, X the reactance between them in ohms per phase and δ the angle between the voltages. The result is in watts or VAR for a three-phase circuit. The formulas assume a reactance with negligible resistance, which is a fair estimate for a transmission line or a transformer. Power flows from the end with the leading voltage. A line is never run near 90° because the system would lose stability, so the practical limit is much lower.
Worked example: 11 kV at both ends, X = 10 Ω, δ = 15°
- P = 11,000 × 11,000 × sin 15° / 10 = 3.132 MW
- Pmax = 11,000² / 10 = 12.1 MW
- Q = 11,000 × (11,000 × cos 15° − 11,000) / 10 = -0.4123 MVAR, so the receiving end sends reactive power back to the line
Power for different angles (same circuit)
| Angle | Power |
|---|---|
| 5° | 1.055 MW |
| 15° | 3.132 MW |
| 30° | 6.05 MW |
| 60° | 10.48 MW |
| 90° | 12.1 MW |
Frequently asked questions
What controls the real power flow?
The angle between the two voltages. A larger angle sends more power, up to a maximum at 90°.
What controls the reactive power flow?
The difference in voltage magnitude. Reactive power flows from the higher voltage to the lower one.
Why is the maximum power not used in practice?
At and beyond 90° the machines lose synchronism. A margin is kept to stay stable after a disturbance.
Does the resistance matter?
Yes, for a distribution cable, where the resistance is not small compared with the reactance. This page is for the reactive case.
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