Electric power (P) is the rate of energy transfer in a circuit, measured in watts (W). In DC circuits, power is the product of voltage and current: P = V × I. In AC circuits, especially with sinusoidal waveforms, power has three components: active (real) power P in kW, reactive power Q in kVAR, and apparent power S in kVA. These relate to each other by the power factor (cosφ). The key formulas for AC systems (balanced conditions) are:
Here, U is the line-to-line voltage (in volts), V is the line-to-neutral voltage, and φ is the phase angle between current and voltage. These equations indicate that active power is the portion doing useful work (heat, motion, lighting), whereas reactive power oscillates between source and load (e.g., energy stored in inductors/capacitors). Apparent power S is the product of RMS voltage and current without considering phase angle, and the power factor (cosφ = P/S) indicates the efficiency of power usage.
Consider a 3-phase 400 V (line) industrial motor drawing 50 A at 0.8 power factor lagging. The active power is calculated using the three-phase formula. The reactive power (due to motor inductance) would be determined next.
Since sin(arccos 0.8) = 0.6, Q = 20.8 kVAR. The apparent power is then computed as S = √(P² + Q²) = 34.6 kVA, which also equals √3 × 400 × 50 = 34.6 kVA as expected. This example shows how to compute the kW, kVAR, and kVA of a load. In a simple DC case (or AC with unity power factor), these distinctions vanish; for example, a 12 V battery supplying 2 A gives P = 12 × 2 = 24 W of DC power.
Power calculations extend to many scenarios. In AC traction or aerospace, complex power calculations in polyphase systems use vector sums of P and Q. In the audio industry, “power” might refer to amplifier output in dBW or dBm. For short pulses or non-sinusoidal waveforms, engineers calculate instantaneous power p(t) and then average it. Another niche case is skin effect and I²R loss: the formula P = I²R (derived from combining P = V × I and V = I × R) is used to compute heat dissipation in conductors. Data centers also track apparent power (kVA) to size UPS systems and generators, since they must handle the total current including the reactive portion.
Power calculations are performed in every electrical project – from sizing a household solar inverter to designing national grid systems. Utilities specify billing in kW (demand) and may impose penalties for low power factor (excess kVAR). Equipment ratings (transformers, generators) are given in kVA because they must accommodate the total current regardless of power factor. Engineers must ensure that motor drives, UPS, and cables are rated for the apparent power and that excess reactive power is managed. Standards for energy efficiency (like IEC 60364 or building codes) require calculating energy consumption, which builds on these power formulas.
Fundamental power equations are reinforced in various standards and guides. The IEC’s Electrical Installation Guide explicitly lists formulas for single- and three-phase power (active, reactive, apparent). IEEE Std 1459 provides definitions for power in systems with non-sinusoidal waveforms. BS 7671 (UK Wiring Regulations) implicitly uses P = V × I cosφ when recommending cable sizes (since current depends on load power and power factor). In North America, the NEC includes notes explaining that proper conductor sizing can improve efficiency by reducing I²R losses. While the formulas are universal, these standards ensure consistent usage across applications.
Many software tools assist with power calculations: