Active, reactive and apparent power, the power triangle, and why cos φ matters in practice. A low power factor draws more current than the active power alone would require — and that costs you in cables and equipment.
Introduction
In an AC system with inductive or capacitive loads, current and voltage are phase-shifted by the angle φ. This gives three power concepts that are linked in the power triangle.
The three effects
- 01Active power P [W] — the power that does real work (heat, light, mechanical work).
- 02Apparent power S [VA] — the product of the RMS values of voltage and current.
- 03Reactive power Q [var] — the power that oscillates back and forth in inductance/capacitance without doing any net work.
Single-phase and three-phase
Example: A symmetrical three-phase load at 400 V draws 16 A at cos φ = 0.9. The active power is P = √3 · 400 · 16 · 0.9 ≈ 9977 W ≈ 10 kW, while the apparent power is S = √3 · 400 · 16 ≈ 11085 VA.
Why cos φ matters
A low power factor means that more current is drawn than the active power alone would call for. This causes greater losses in cables and transformers and may require larger cross-sections and equipment. That is why phase-shift compensation (capacitor banks) is often used on installations with many motors.
Phase shift compensation
A low power factor costs in current and in losses. The solution is to supply the reactive power locally with capacitors, so the grid does not have to deliver it back and forth. The required capacitor power Qc to raise the power factor from cos φ1 to cos φ2:
Eksempel: Et anlæg trækker P = 50 kW ved cos φ1 = 0,75 og skal hæves til cos φ2 = 0,95. tan(arccos 0,75) = 0,882 og tan(arccos 0,95) = 0,329. Qc = 50 · (0,882 − 0,329) = 50 · 0,553 = 27,7 kvar. Der skal altså installeres et kondensatorbatteri på ca. 28 kvar.
Capacitor bank and automatic control
With varying load, a fixed capacitor bank would over-compensate at low load and make the system capacitive (cos φ tilts to the other side). Therefore, stepped capacitor banks controlled by a power factor regulator are used, which switches stages in and out according to the measured cos φ. Fixed compensation is only used at constant load — typically a single motor, where the capacitor is connected directly across the motor terminals.
Network company requirements
Common Regulations 2026 requires that the power factor at the point of connection is maintained between 0.9 (inductive) and 1.0, measured as a 15-minute average. If the limit is exceeded, the grid operator can invoice the reactive power or demand compensation. (Source: Common Regulations 2026, elnet.dk — see also the article on Common Regulations.)
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