Power in AC Circuits and Power Factor: The Complete Guide

In a DC circuit, power is simply voltage times current. In an AC circuit this is true only when voltage and current rise and fall together. When there is a phase difference φ between them, only part of the current actually contributes to useful work, and the average power becomes P = Vrms Irms cos φ. The factor cos φ is called the power factor, and it decides how efficiently an AC load uses the electricity supplied to it.

Power factor matters in money and in engineering. Electricity boards supply current, but homes and factories do useful work only with the in-phase part. Motors, fans and tube lights draw extra lagging current that heats cables without doing work. A poor power factor means thicker wires, bigger transformers and penalty charges for industries, which is why this single number gets so much attention.

Key Concepts

• Average power (P): the mean rate of doing work over a cycle, P = Vrms Irms cos φ, measured in watts.
• Power factor (cos φ): the ratio of average power to the product Vrms Irms; it lies between 0 and 1.
• Apparent power (S): the product Vrms Irms, measured in volt-amperes (VA); it is what generators and transformers must be sized for.
• Reactive power (Q): the power that merely oscillates between source and load, measured in VAR; it does no net work.
• Lagging and leading: inductive loads make current lag and give a lagging power factor; capacitive loads make it lead.
• Unity power factor: φ = 0, achieved by a pure resistor or a circuit at resonance; all supplied current does work.

Formulas

The average power is P = Vrms Irms cos φ. The three powers form a right triangle: S² = P² + Q², where S = Vrms Irms is the apparent power in VA and Q = Vrms Irms sin φ is the reactive power in VAR. The power factor is also given by cos φ = R / Z for a series circuit, where R is the resistance and Z the impedance in ohms.

For a pure resistor φ = 0 and P = Vrms Irms. For a pure inductor or capacitor φ = 90 degrees, cos φ = 0 and the average power is zero, even though current flows; such a current is sometimes called wattless current.

Worked Example

A motor is connected to a 230 V RMS supply and draws 5 A RMS at a power factor of 0.6 lagging. Find the useful power consumed, the apparent power, and the current the same power would need at unity power factor.

1. Apparent power S = Vrms × Irms = 230 × 5 = 1150 VA.
2. Useful power P = S × cos φ = 1150 × 0.6 = 690 W.
3. For the same 690 W at unity power factor, I = P / V = 690 / 230 = 3 A.

The motor needs only 3 A of useful current but actually draws 5 A. The extra 2 A of effectively wattless current still heats the supply cables and loads the transformer, which is the practical cost of a power factor of 0.6.

Common Misconceptions

• Low power factor means the appliance wastes the extra energy as heat inside itself. Wrong. The extra current is mostly reactive: energy is borrowed and returned each cycle. The real losses appear as heating in the transmission lines and transformers, not extra work lost in the appliance.
• Power factor can be greater than 1. Wrong. Cos φ cannot exceed 1. A value stated above 1 always indicates an error in measurement or calculation.
• Domestic meters charge us for apparent power. Wrong. Home meters record real energy in kilowatt-hours, that is, average power over time. Industries, however, may be penalised for poor power factor because of the extra current burden they impose.

Real-World Applications

Factories install banks of capacitors to cancel the lagging effect of motors and bring the power factor close to unity, reducing bills and penalties. Ceiling fans, refrigerators and washing machines all have inductive motors, so their combined power factor shapes the load on the neighbourhood transformer. Electricity utilities specify power factor standards for large consumers, and modern electronic devices use power factor correction circuits so that chargers and LED drivers draw current in phase with the voltage, easing the load on the grid.

Average power in AC is Vrms Irms cos φ, and the power factor cos φ measures what fraction of the supplied current does real work. Improving it with capacitors where inductance dominates cuts line losses and frees capacity, which is why power factor is watched from the home fan to the national grid.