Power Triangle Formula (AC Circuits)
The power triangle relates real power (W), reactive power (VAR), and apparent power (VA) in AC circuits.
Essential for power factor correction and electrical billing.
The Formulas
|S| = √(P² + Q²)
Power factor: PF = P/S = cos(φ)
In AC circuits, power is not simply voltage times current (as in DC circuits) because voltage and current may be out of phase. The power triangle captures all three aspects of AC power: real power P (what actually does work), reactive power Q (energy stored and returned by inductors/capacitors), and apparent power S (what the source must supply).
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| S | Apparent power (complex power) | volt-amperes (VA) or kVA |
| P | Real power (active power — does useful work) | watts (W) or kW |
| Q | Reactive power (stored in inductors/capacitors, not consumed) | volt-ampere reactive (VAR) |
| PF | Power factor = cos(φ) | dimensionless (0 to 1) |
| φ | Phase angle between voltage and current | degrees |
| j | Imaginary unit (√−1) — Q is the imaginary component | — |
Inductive loads (motors, transformers) have lagging power factor and positive Q. Capacitive loads have leading power factor and negative Q. Power utilities bill industrial customers on apparent power (kVA) or charge penalties for low power factor (< 0.9).
Example 1 — Industrial Motor
A motor draws P = 1000 W at a power factor of 0.8 lagging. Find Q, S, and the phase angle.
φ = arccos(PF) = arccos(0.8) = 36.87°
S = P/PF = 1000/0.8 = 1250 VA
Q = √(S² − P²) = √(1250² − 1000²) = √(562500) = 750 VAR
S = 1250 VA, Q = 750 VAR (inductive). The utility must supply 1250 VA even though only 1000 W does useful work.
Example 2 — Power Factor Correction
The motor above (Q = +750 VAR inductive) needs correction to PF = 0.95. What capacitor VAR rating is needed?
Target Q_new = P × tan(arccos(0.95)) = 1000 × tan(18.19°) = 1000 × 0.329 = 329 VAR
Required capacitor Q_C = Q_old − Q_new = 750 − 329 = 421 VAR
Install a 421 VAR capacitor bank. New S = √(1000² + 329²) = 1053 VA — reduced from 1250 VA, saving utility current.
When to Use It
Use the power triangle when:
- Sizing electrical generators, transformers, and switchgear (rated in kVA)
- Calculating capacitor banks for power factor correction
- Understanding utility billing for industrial facilities
- Analyzing motor efficiency and sizing motor starters
- Designing UPS (uninterruptible power supply) systems