Compute apparent power (kVA), active power (kW), reactive power (kVAR), and power factor for single‑phase and three‑phase AC systems. Visualize the power triangle and get instant insights — ideal for electrical engineers, technicians, students, and facility managers.
In electrical engineering, apparent power (S) — measured in kVA (kilovolt‑amperes) — represents the total power flowing in an AC circuit. It is the vector sum of active power (P) in kilowatts (kW) and reactive power (Q) in kilovolt‑amperes reactive (kVAR). The relationship between these three quantities forms the power triangle, a fundamental concept for understanding AC power systems, transformer sizing, generator loading, and power factor correction.
S² = P² + Q²
P = S · PF | Q = S · sin(θ) | PF = cos(θ)
where θ is the phase angle between voltage and current.
Apparent power (kVA) is the rating used for transformers, generators, UPS systems, and switchgear because it reflects the total thermal and magnetic stress on equipment, regardless of the load's power factor. A 100 kVA transformer can deliver 100 kW at unity power factor, but only 80 kW at 0.8 PF — the difference is wasted as reactive power that circulates without doing useful work. This is why utilities penalize industrial customers with low power factors: the distribution network must carry the full kVA, but only the kW portion is billable.
Understanding the power triangle helps engineers size cables, select breakers, and design compensation banks. For example, adding capacitors to improve PF from 0.7 to 0.95 can reduce line current by over 25%, freeing capacity and reducing losses.
Three‑phase systems are more efficient for high‑power loads because they deliver constant power and use less conductor material for the same power level.
Power factor is a critical indicator of electrical system efficiency. The table below shows typical PF ranges and their implications:
| PF Range | Rating | Typical Applications | Impact |
|---|---|---|---|
| 0.95 – 1.00 | Resistive heaters, LED lighting, modern VFDs | Maximum efficiency, low losses | |
| 0.85 – 0.94 | Induction motors, transformers, HVAC | Acceptable, may need minor correction | |
| 0.70 – 0.84 | Older motors, welders, some industrial loads | Significant reactive power, likely utility penalties | |
| < 0.70 | Arc furnaces, large inductors, highly reactive loads | High losses, voltage drop, urgent correction needed |
A manufacturing plant operates a 500 kVA transformer feeding multiple induction motors. The average PF is 0.78, resulting in an active power draw of 390 kW and reactive power of 313 kVAR. By installing a 200 kVAR capacitor bank, the PF improves to 0.95. The transformer loading drops from 500 kVA to 410 kVA (a 18% reduction), freeing capacity for future expansion and reducing monthly demand charges. The plant saves approximately $4,200 annually in utility penalties and loses less energy in distribution cables. This example demonstrates why KVA analysis is not just academic — it has direct financial and operational benefits.